Tetrahedral Frameworks of Zeolites, Clathrates and Related Materials

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ISSN 1615-2018 (Physical Chemistry) ISBN 3-540-61743-4 Springer-Verlag Berlin Heidelberg New York

Library of Congress Cataloging in Publication Data: Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology, New Series. Editor in Chief: W. Martienssen. Vol. IV/14A: Edited by W.H. Baur and R.X. Fischer. Springer-Verlag Berlin, Heidelberg, New York 2000. Includes bibliographies. 1. Physics - Tables. 2. Chemistry - Tables. 3. Engineering - Tables. I. Börnstein, Richard (1852-1913). II. Landolt, Hans (1831-1910). III. Physikalisch-chemische Tabellen. IV. Numerical Data and Functional Relationships in Science and Technology. QC 61.23 502'.12 62-53136 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law. © Springer-Verlag Berlin, Heidelberg, New York 2000, a member of BertelsmannSpringer Science+Business Media GmbH. Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Product Liability: The data and other information in this handbook have been carefully extracted and evaluated by experts from the original literature. Furthermore, they have been checked for correctness by authors and the editorial staff before printing. Nevertheless, the publisher can give no guarantee for the correctness of the data and information provided. In any individual case of application, the respective user must check the correctness by consulting other relevant sources of information. Typesetting: Author, Editors and Redaktion Landolt-Börnstein, Darmstadt Printing: Computer to plate, Mercedes-Druck, Berlin Binding: Lüderitz & Bauer, Berlin SPIN: 10478522

63/3020- 5 4 3 2 1 0 – Printed on acid-free paper

Editors Fischer, Reinhard X. FB Geowissenschaften, Universität Bremen, Germany Baur, Werner H. Department of Geophysical Sciences University of Chicago, USA

Author Smith, Joseph V. Department of Geophysical Sciences and Center for Advanced Radiation Sources, University of Chicago, USA

Landolt-Börnstein Editorial Office Gagernstraße 8 D-64283 Darmstadt, Germany fax: +49-(0)6151-171760 email: [email protected] Internet science.springer.de/newmedia/laboe/lbhome.htm Helpdesk email: [email protected]

Foreword

Zeolites and zeolite-like materials became important because of their ion exchange capacities and their outstanding catalytic properties. Millions of tons of zeolites have been produced in the past years for the oil refining industry alone and, in even greater quantities, as ion-exchanging softening agents for detergents. Numerous other applications, e.g., in environmental protection, farming, gas separation, medicine, and pharmacy, are known, making zeolites almost a necessity for daily life. Consequently, there are many research activities dealing with zeolite properties and characterization. However, a strictly systematic description of zeolite-type crystal structures was not available but is now presented in this series of volumes. It is designed as a reference work for zeolite chemists and materials scientists, but it also serves as a tool to interpret structural similarities and to derive new structures from known topologies. It presents a complete overview of the different framework topologies, the crystal structures, and the chemistry of zeolites and other microporous materials. In the first volume J.V. Smith analyses the topology of zeolitic and related frameworks including many which do not conform to the definition used by the Structure Commission of the International Zeolite Association (SC-IZA), which deals only with open tetrahedral frameworks. The related frameworks are found in various minerals denser than the zeolites, and in clathrates and other synthetic materials. The following volumes by W.H. Baur and R.X. Fischer systematically present the crystal structures and chemistry of all zeolites and microporous materials for which the topology was accepted by the SC-IZA and given a zeolite structure code (130 of them as of this writing). The study of the topology of the bonding arrangements in crystal structures was pioneered in numerous papers, beginning in 1954 by A.F. Wells (Acta Crystallogr. 7, 1954, 535) and summarized in two monographs (Three-dimensional Nets and Polyhedra, Wiley, New York, 1977; Am. Crystallogr. Assoc. Monograph No. 8, 1979). Among the many three-dimensional nets invented by Wells was his net 8 (Acta Crystallogr. 7, 1954, 545), subsequently, in the 1979 monograph, called by him 42.63.8-a. It was later identified as the threedimensional net of primary bonds in the topology of the ABW structure type, see volume B of this series. This is just one of many examples where a predicted net was subsequently identified in an experimentally determined crystal structure type. J.V. Smith is prominent among the researchers who took up the challenge posed by Wells and continued the study of three-dimensional nets, especially as they apply to zeolites, other silicates and to various microporous materials. The present volume is a culmination of the first stage of his efforts in this direction. Very valuable to pure mathematicians is the literature review of the mathematical principles, and of the multifarious applications in chemistry. Smith’s unpublished Catalog of Theoretical Nets comprises over 1300 examples, of which over two dozen were invented prior to their discovery in actual materials. A particularly illuminating example of the prediction of a structural topology concerns the AFI-type. In 1978 J.V. Smith predicted the topology of the AFI net as net #81 (Am. Mineral. 63, 1978, 960) see also volume B of this series. E. M. Flanigen recognized the relationship between net #81 and the topology of the AFI-type structure. In deriving net #81 a three-dimensional linking of the plane 4.6.12 net was used. Net 4.6.12 itself was described already in 1619 by Johannes Kepler in his Harmonice mundi (see W.H. Baur: Nova Acta Leopoldina 79, 1999, 47). Thus, we have here a continuation of efforts which have gone on for almost four hundred years.

Smith has even linked some of his nets to the Platonic and Archimedean solids thereby coupling modern structural chemistry to the earliest formal mathematics of Ancient Greece. In contrast to the subsequent volumes describing the framework structures, this volume does not aim for completeness in the compilation of structural units. Completeness would require a strict definition of such units which is not available yet. Thus, this is an open system which can be added to. However, this volume is the most comprehensive published compilation of building units, and as such is a great asset in the structural characterization and description of zeolite-type structures. We sincerely hope that our efforts will prove useful to zeolite scientists and zeolite science.

Chicago/ Bremen, April 2000

Werner H. Baur Reinhard X. Fischer

Ref. p. 251]

1

Introduction

1.1

General

1 Introduction

1

Zeolite molecular sieves, clathrates and various related materials are important for human welfare. The first zeolite was identified by Axel Fredrick Cronstedt as a mineral (stilbite) that boiled when heated with borax [1756Cro1]. Research in the first half of the twentieth century demonstrated that each zeolite mineral is based on a coherent three-dimensional framework of thermally stable chemical units, mainly SiO4 and AlO4 tetrahedra, sharing oxygen atoms. Each 3D-framework contains a network of holes, which are occupied by water molecules and electrically charged species (cations) or molecules. The cations balance the negative electrostatic charge of the tetrahedral framework. Upon heating, the extra thermal energy causes the water molecules to escape from the holes via channels and windows. Upon cooling, water molecules can refill the holes. Alternatively, other molecules or cations can be deliberately introduced and removed. During dehydration, the framework retains its connectivity, but the positions of the occluded atoms and molecules, and of the cations, change to minimize the thermochemical energy. The cations can be exchanged with others that can enter through the windows, either when the zeolite is hydrated or dehydrated: e.g. one divalent Ca can be exchanged with a divalent Sr or two monovalent Na or K. Some adjustment of framework geometry, water content, and cation position generally occurs, but the framework retains its tetrahedral connectivity. Hence, the physicochemistry of zeolites is fundamentally based on mathematical principles of 3D geometry. Dehydration/rehydration, ion-exchange and molecular adsorption/desorption are controlled in general by the size of windows, pores and channels, and the three-dimensional connectivity. Each type of framework has specific features ranging from the three-dimensional channel system of faujasite with access through 12-rings for molecules up to 0.7 nanometer wide to the simple pores of analcime that contain Na and water that can be exchanged or removed only with great difficulty. The range of variation among the geometrical features of the various zeolites has been exploited for human well-being. Gasoline is made very efficiently from petroleum by a combination of industrial processes utilizing molecular adsorption and shape-selective catalysis. Toxic elements in nuclear and various other industrial wastes are concentrated by zeolites using selective adsorption. Various gases are removed from air by zeolites. Oxygen can be separated from nitrogen using zeolites. This volume is concerned with the topology of the zeolite frameworks, and not with the chemical theory and engineering practice. It is essentially an exercise in pure mathematics. The mineralogical term zeolite has been extended by chemists to cover any material that is structurally analogous to, but different in chemical composition, from a mineral in the zeolite family. Historically, Donald W. Breck and I looked at the chemical substitutions in the feldspar minerals to obtain possible substituents for Si and Al in mineralogical zeolites. We found that in Nature the following elements definitely occurred: Be, Mg, P, Ti, Mn, Fe, Zn, Ga, Ge; others, including Li, Co, Ni and Cu seemed to be likely prospects. Systematic exploration over the next several decades has generated a massive range of chemical analogs of natural minerals. Particularly important are the aluminophosphate molecular sieves, with an electrostatically neutral tetrahedral framework, some of which are structurally analogous to a natural mineral except for replacement of 2 Si by Al plus P, while the others are new topologic types. Nature has generated a wide range of framework structures, of which only some are zeolites. Thus melanophlogite has a tetrahedral framework containing small pores that enclose various volcanic gases: it is a clathrasil with all tetrahedra containing Si. Various other clathrasils have been synthesized from silica gels with structure-directing organic molecules, and mineral equivalents are not known. From the chemical viewpoint, it is fascinating that water can enclose a molecule when cold to make a variety of icemolecule clathrates.

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Further, there is a hierarchy of structures that begin with diamond (a variety of carbon), advance in complexity to cristobalite (a polymorph of silica, with Si positions matching C in diamond, and O lying near to the mid-point of C-C bonds), and on to the zeolite faujasite in which each C is replaced by a truncated octahedron of (Si,Al)-O linked tetrahedra. In addition, the mineral blende, ZnS, uses the atomic positions of diamond, as do other sulfide minerals, including chalcopyrite. For a recent review of this topic see Schindler et al. [99Sch1]. Because of these structural linkages, this compilation covers all materials based on tetrahedral frameworks whatever the chemical composition. It covers feldspathoid minerals that lack water. It covers incomplete tetrahedral frameworks in which all but one of the oxygen atoms are 2-connected to tetrahedral atoms; the break in the connectivity is typically the result of replacement of one O by hydroxyl OH bonded to only one tetrahedral atom instead of two. However, it does not cover framework structures containing sub-units other than tetrahedra, namely triangles, octahedra, and other polyhedra. There is a caveat: tetrahedral Al in as-synthesized aluminophosphates may be linked to one or more additional ligands (hydroxyl and water) thus becoming pentagonal or hexagonal in coordination. These materials are considered here because mathematical removal of the extra ligands generates a tetrahedral net. Furthermore, careful heating may remove the extra ligands, though some materials do recrystallize into a new material with a different type of topology. With this introduction to the mineralogical and chemical background, it is appropriate to consider the mathematical description of tetrahedral nets. Specifically, each zeolite structure is based on a threedimensional 4,2-connected net with tetrahedral geometry, whose connectivity is characterized using simple concepts taken from the mathematical discipline of topology. Each Si and Al atom occupies a 4connected vertex (denoted T) of the net from which four edges go out to the adjacent oxygen atoms (Fig. 1a). Each O atom lies at a 2-connected vertex from which two edges go out to adjacent T vertices occupied by Si/Al atoms (Fig. 1b). In zeolites, the four chemical bonds to adjacent O atoms have tetrahedral or near-tetrahedral geometry, and the four O atoms lie at the vertices of a tetrahedron (Fig. 1c). The line segments between the four O atoms comprise the edges of this geometrical solid (Fig. 1d) that is one of the five Platonic regular solids (= all edges and all vertices have the same connectivity). To a first approximation, the O atoms can be considered as non-perfect spheres that are in contact (Fig. 1e), though smaller spheres are commonly used in drawings to reduce overlap (Fig. 1a, b, c). The chemical bonding at the O atom is tolerant of changes in the bending angle T-O-T, where T is the mathematical symbol for a tetrahedrally-linked vertex occupied by Si, Al and other chemical elements. This chemical tolerance permits the adjacent tetrahedra to hook up in different ways to generate an infinity of 3D nets whose subunits have different connectivities and geometries. Very important conceptually is the typical simplification of the 4,2-connected 3D net into a 4-connected net in which only the T-T edges are shown (Fig. 1f). Each 2-connected vertex for an O atom must lie near the mid-point of a T-T edge (Fig. 1b, f). Its real position in an actual zeolite can be determined by a diffraction experiment, and a theoretical position can be estimated by a geometrical calculation.

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1 Introduction

a Silicon (T) atom at tetrahedral center of four Oxygen (O) atoms.

3

b 4,2-connected T-O tetrahedral net showing 4T ring.

c Regular tetrahedron.

d Regular tetrahedron.

e Four Oxygen spheres touching each other.

f

4-connected net abstracted from 4,2connected T-O tetrahedral net.

Fig. 1 Geometry and topology of a fragment of a 4-connected net abstracted from a framework structure containing tetrahedrally-connected Silicon (T) atoms and Oxygen (O) atoms.

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[Ref. p. 251

To conclude this general introduction, it is important to emphasize here how the microporous crystal structure of zeolites controls various physicochemical processes important for human well-being: reversible loss and gain of water, ion-exchange of non-framework cations, adsorption of atoms and molecules on internal surfaces of dehydrated zeolites, and heterogeneous catalysis of adsorbed molecules. These processes have led to major advances in human welfare [99Smi1; 99Mum1; 99She1]. Gasoline/petrol is obtained from petroleum by a sequence of processes involving zeolite catalysts and selective adsorbents. Natural gas is cleaned by zeolite adsorbents. Consumer products include zeolite adsorbents for removal of toxic materials from drinking water and air, controlled delivery of drugs, and clean-up of air inside double-glazed windows. Various toxic elements and molecules in lakes and rivers resulting from industrial and agricultural processes are removable by zeolite adsorbents. Preliminary applications to agriculture and food production, including controlled delivery of fertilizer and pesticides, are indicative of potential large-scale modification of farm land and forests in the future. Many geological processes at the Earth’s surface involve sedimentary beds of zeolites grown from volcanic ash. Biochemical evolution on Earth may have begun by catalytic assembly of simple molecules into polymers inside silica-rich organophilic molecular sieves, followed by generation of proto-cells with inter-tangled polymers and energy-storage molecules inside honeycomb channels of weathered feldspars [99Smi2]. Since this introduction was first written about two years ago, the number of distinct synthetic zeolitic materials has doubled as more and more chemical variants are tested. In the past half-year, a dozen new 3D tetrahedral nets have been found, along with a host of complex nets. The chemical literature continues to grow as new uses are being developed for synthetic materials. Just beginning is the exploitation of zeolitized beds of volcanic ash.

The 3D net of LTA (the approved three-letter code of the Structure Commission of the International Zeolite Association – briefly IZA-SC) has its T-T edges at the vertices of three polyhedra: regular hexahedron = cube (University of Chicago Consortium for Theoretical Frameworks three-letter code cub, Fig. 2 d), truncated octahedron (toc, Fig. 2 e), and great rhombicuboctahedron (grc, Fig. 2 f). The cube, of course, is one of the five fully regular Platonic solids. The toc and grc polyhedra are two of the 14 Archimedean semi-regular solids. Table 3.1 on p. 78 of Chapter 3 “Regular Polyhedra” in [82Smi1] lists the following properties: Face Symbol 6

Vertex Symbol

Schläfli Symbol

Face Angle(s)

8

4.4.4

90°

Cube = hexahedron

4

truncated octahedron

4668

324

4.6.6

90°,120°

great rhombicuboctahedron = truncated cuboctahedron

4126886

348

4.6.8

90°,120°, 135°

3

For convenience, formal definitions are given here: Face symbol = number (superscript) of polygonal faces comprising the polyhedron (specified by the number of vertices around the circumference of each type of polygon). Thus the cube has 6 (hexa) square faces (hedron; plural is hedra), each with a circuit of 4 vertices around its circumference. The truncated octahedron has 6 square faces and 8 hexagonal ones.

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1 Introduction

5

Vertex Symbol = number (superscript) of vertices at which a particular number of faces meet. All three polygons above have only one type of vertex, but other polygons have more than one type. Thus the cube has eight vertices each of which has three faces meeting there. Schläfli Symbol = sequence of faces meeting at a vertex, each specified by the number of edges of each face. Thus it is 4.4.4 for the cube, corresponding to three square faces. More complex polyhedra have more than one type of vertex, and it is necessary to specify the relative multiplicity of each type. Face Angles are the angles in the polygon at each vertex. For the cube, the angle is 90° for each vertex of each square face. The truncated octahedron has 90° for the square faces, and 120° for the hexagonal ones. To reiterate, the cube has: 6 (hexa) square faces (hedra), each with a circuit of 4 vertices; 8 vertices, each at the intersection of three edges and three faces, and with a circuit of 4.4.4-faces. The (vertex)truncated octahedron is obtained by slicing away the 6 vertices of a regular octahedron until each of the eight (octa) triangular faces becomes a regular hexagon [82Smi1, Fig. 3.8]. Each of the six vertices of an octahedron becomes a square (circuit 4; face angle 90°), and each of the eight triangular faces becomes a regular hexagon (circuit 6; face angle 120°). There are 24 congruent vertices each with one 4- and two 6faces. The great rhombicuboctahedron has three types of faces but again has only one type of vertex as befitting an Archimedean semi-regular solid: details in [82Smi1], p. 67-69. The last column lists the face angles. All three polyhedra are convex, which means that all edges are more distant from the centroid (volume center) than adjacent points on the faces. These three polyhedra are very simple from the topologic viewpoint. Many polyhedra have more than one type of face and vertex. Thus the polyhedron listed on the first page of Table 16.3.1 (mrr) has six different types of square faces, five types of pentagons, four types of hexagons, two hexagons & two nonagons. This topological distinction must be retained. In the chemical literature, many faces and vertices are lumped together, thus making it difficult to characterize a polyhedron simply from its listed face symbol. Thus mtn and doh have different face symbols (51264 and 5126262) when the two types of hexagons are distinguished. The 3D net of LTA type is 4-connected because four edges radiate from each vertex. The net is obtained uniquely by sharing each of two opposing 4-faces of each cube with a 4-face of a truncated octahedron. From the two sets of three edges, two pairs merge to yield the 4-connectivity. Of course, a simple rule like this generates an infinite net if all vertices are treated equally. In practice, crystals of LTA grow as cubes, and one can guess that the outer surface is terminated ideally by 4-faces lying in cubic planes. At this point it is important to emphasize the degree of mathematical abstraction in the last paragraph, and return to ‘real’ chemistry. For example, the ideal chemical formula of as-synthesized LTA is Na12Al12Si12O48.27H2O. To minimize the electrostatic energy, the Al and Si cations lie on alternate vertices of the 4-connected LTA net, and a mathematician would describe this as a 2-color decoration. There are twice as many 2-connected vertices in the (4,2)-connected net as 4-connected ones: hence there are 48 oxygen anions to match the 12 + 12 = 24 T atoms. Using the formal electric charges of +3, +4, and -2 for Al, Si and O ions, each Si balances two O, but Al lacks one charge. This is provided by a sodium Na ion. Hence there are 12 Na per 12 Al. Six divalent calcium Ca would also provide a balance, and can indeed be obtained by ion-exchange of the as-synthesized Na type. A crystallographer simplifies the description of a crystal structure by selecting a unit cell that generates the infinite edifice by simple translation in 3 directions. In LTA, a cube with vertices at the centers of adjacent truncated octahedra is the obvious unit cell. The 12 Na are electrostatically attracted to the O atoms. In dehydrated Linde Type A zeolite, eight Na become bonded to six O, three to eight O, and the remaining one to only four O. In hydrated Na-A zeolite, there is enough space to accommodate 27 water molecules that move rapidly because the thermal energy is higher than the bonding energy. Returning to mathematical abstraction, the FAU type of net, which occurs in the mineral faujasite and various synthetic molecular sieves (Table 16.2.1), can be assembled by sharing half of the 6-rings of a truncated octahedron (toc) with a hexagonal prism (hpr, Fig. 2g). The hexagonal prism consists of two Landolt-Börnstein New Series IV/14

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1 Introduction

[Ref. p. 251

hexagons at top and bottom joined by six squares at the sides. This is actually a semi-regular solid that is not listed as one of the 14 Archimedean solids. It is a member of an infinite series of regular prisms in which the top and bottom faces can have any number of vertices greater than 2: 3, trigonal prism; [4, cube = Platonic solid]; 5, pentagonal prism, etc. Note that this is a series of regular prisms with regular polygons as faces including squares as side faces rather than non-regular prisms with rectangular side faces. The sharing of 6-ring hexagonal faces by tco and hpr generates a channel system with 3D access by 12-ring windows.

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1 Introduction

a Sodalite SOD.

b Linde Type A LTA.

c Faujasite FAU. Fig. 2 Stereoview of fragment of SOD, LTA and FAU 3D nets together with polyhedral subunits.

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8

1 Introduction

d Cube.

e Truncated octahedron.

f Truncated cuboctahedron.

g Hexagonal prism.

[Ref. p. 251

h fau polyhedron. Fig. 2 (continued). Stereoview of fragment of SOD, LTA and FAU 3D nets together with polyhedral subunits.

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Ref. p. 251]

1 Introduction

9

This channel system can be assembled from a polyhedron coded as fau (Fig. 2h) with face symbol 4124664124. The 4 hexagons replace the vertices of a regular tetrahedron. Joining them are six triplets of edge-sharing 4-rings. Each of the four 12-rings uses edges from three 6-rings and nine 4-rings. The 4connected 3D net of FAU type can be built simply by strictly alternating 4-rings of adjacent toc across each hpr (trans). Remarkably this net can also be obtained by replacing the Zn and S atoms of the zinc blende structure with toc and hpr respectively, and the ZnS blende structure is itself a derivative of the diamond structure. Just as diamond has a polytypic relation with lonsdaleite, the FAU 3D net has a polytypic relation with the EMT net (see later). In the early chemical literature on synthetic zeolites, the cube and hexagonal prism are called double4-rings (D4R) and double-6-rings (D6R), but a cube actually is constructed from six congruent 4-faces, while the two 6-faces of the hpr are suspended by six congruent 4-faces. The SOD 4-connected 3D net is obtained simply by sharing the 3-connected vertices of toc. This linkage of toc is related to an equipartition of space [1894Kel1]. Strictly speaking, sodalite is regarded as a feldspathoid rather than a zeolite by mineralogists because a typical chemical variety Na4Al3Si3O12Cl contains no water, and has the toc cage occupied by a chlorine anion. Nevertheless, it is important to consider the topochemistry of the SOD class of materials because of the communality of subunits occurring in nets of zeolites and other materials, exemplified here by toc, and the importance of hydroxysodalite in certain zeolite synthesis procedures. This examination of three nets based on the linkage of regular and semi-regular polyhedra demonstrates the value to chemists of the formal concepts of topology. We shall extend this introductory treatment to less-regular polyhedra, and consider other subunits including 3-connected 2D nets, chains, columns, tubes and helices. In addition to the 4,2-3D nets found in zeolites (sensu stricto), we shall also consider the 4,2-3D nets in clathrates and dense materials. An infinity of novel frameworks can be invented by cross-linking known subunits in new ways, and by inventing new subunits. These new frameworks become targets for synthesis of new materials with chosen properties. In addition to the practical value of such theoretical development, the sheer intellectual thrill of advancement of the theory of pure mathematics justifies the research on the geometry and topology of open tetrahedral nets. Indeed, the advances in mathematical theory in the twentieth century link up with the ancient Greek concepts of Platonic and Archimedean polyhedra, the first drawings of two-dimensional nets by the astronomer Kepler, and the various pioneering attempts to understand the space packing of solid phases through the eighteenth and nineteenth centuries.

1.3

Specific aims of this volume

The creative application of X-ray diffraction in the second decade of this century led to specific atomic positions for simple close-packed crystal structures. The pioneering crystal-structure studies of zeolite minerals in the nineteen-thirties generated the concept of a microporous framework together with the first glimmerings of mathematical inter-relations between sub-units (e. g. the cross-linking of chains in the fibrous zeolites). The systematic mathematic theory of the connectivity and topology of tetrahedral and other types of frameworks has developed over the past five decades. Like all branches of pure mathematics, it has an endless frontier, and the present treatment will grow as new topologies of nets are discovered and analyzed.

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The present aim is to: •

describe the mathematical principles of the topology and connectivity of subunits in known nets as simply as possible while retaining full mathematical rigor,

•

characterize each tetrahedral net in a known crystal structure by its sub-units (rings; polyhedra and cages; chains, helices and columns; two-dimensional nets),

•

assign each three-dimensional net and its sub-units to a uniquely retrievable place in databases that can be checked rigorously for isostructural phases,

•

list all materials that contain the same net to permit informed speculation on the chemical relationships among isostructural materials,

•

for each subunit, list all nets and materials that contain it.

To achieve these goals, the Consortium for Theoretical Frameworks (abbreviated CTF) was established at the University of Chicago in 1980 with the following aims that continue past 2000: •

Examine known crystal structures, both topologically and topochemically using mathematical principles [88 Smi1], to obtain 3D nets and subunits. Place the nets and derived subunits systematically and uniquely into logically-arranged tables that are accessible as a computer database. Nets invented by other scientists are incorporated into the databases when sufficient information is available. Duplicates are combined after recognition from the unique placement in the computer database.

•

Determine the linkage/connectivity of subunits in each type of parent net, establish general algorithms, and invent new nets by systematic enumeration.

•

Test the crystallographic properties predicted for the theoretical nets against observed ones for materials with unknown structure. The space group and cell dimensions provide the first possible match. A calculated powder diffraction pattern provides a second possible match. Because the actual geometry for a material with a particular net depends on the details of chemical occupancy and the response to thermal energy, the matching procedure may not be simple.

•

Compare subunits in phases that crystallize under similar physicochemical conditions to allow speculation on crystallization mechanisms. Test the speculations by observation of stacking faults, twin mechanisms and other growth indicators [93Smi1].

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2 Mathematical description, classification and invention of nets

11

2

Mathematical description, classification and invention of nets

2.1

Mathematical description

The 4:2 connectivity of a tetrahedrally-bonded structure is a common feature of various types of chemical materials, ranging from the compact crystal structures in the minerals coesite and feldspar to the open crystal structures of the the molecular-sieve zeolites which contain pores connected by windows large enough for transport of molecules. The nature of the chemical linkages is absolutely unimportant to a mathematician specializing in topology, and specific values of bond distances, bond angles and chemical occupancy do not matter so long as the specified connectivity is maintained. Each atom of a net is regarded mathematically as occupying a position in space, namely a vertex (plural: vertices), and each pair of adjacent vertices is joined by an edge. A circuit of edges from a chosen vertex through a continuous sequence of edges back to the starting vertex is a ring. Because all (4,2)-nets have an oxygen atom between each pair of adjacent T atoms, the O positions can be ignored from the viewpoint of topology. Specifying the connectivity between adjacent T vertices is sufficient because the connectivity of the O vertices can be recovered simply by taking the mid-point of each T-T edge. To understand the chemical crystallography, the specific type of geometry of the 4-connectivity around a T vertex and of the 2-connectivity around the O atoms must be determined. In zeolites, clathrates and other framework materials considered here, the geometry is tetrahedral, but not necessarily with ideal isometric symmetry. In many 4,2-connected 3D nets, some perturbation from ideal tetrahedral geometry is required merely from the necessity of closure of all the rings. Moreover, some perfectly valid 4-connected nets, including that for NbO, have square geometry. Thus each Nb in NbO is surrounded by 4 O, and there are 3 different orientations of the squares to permit closure of the rings in the (4,2)-3D net. I list various types of (4,2)-nets in Table 16.2.1, but concentrate on those with tetrahedral or near-tetrahedral geometry. Useful is an idealized geometry in which T positions are obtained by minimizing the deviations from equality of the bond distances and angles. Such an idealized net, whose vertex positions are commonly determined by distance-least-squares modeling with an electronic computer program, is also useful to crystallographers trying to solve the crystal structures of new materials. Specific values of the bond distances and bond angles are needed to begin an exploration of the chemical forces in each particular material with a chosen (4,2)-3D net. Such data are obtained from diffraction and spectroscopic studies, and there can be a considerable range for each parameter as the chemical occupancy of the T vertices and the non-framework atomic positions change. The specific details for particular materials are not listed here, but are given in primary references locatable from crystallographic databases and in the following volumes of this series [2000Bau1]. There are complications. • Certain elements that occupy T sites have extra ligands. Thus Al, which ideally has 4-connected neartetrahedral geometry in aluminosilicate zeolite molecular sieves, has one or two extra ligands in several aluminophosphate molecular sieves. Each extra ligand occupies a solid angle that forces the edges of the (4,2)-net to be flattened away from near-tetrahedral geometry. Hence a distance-leastsquares model for an idealized near-tetrahedral net has rather different positions of the vertices than that for an actual material with extra-ligand framework atoms. •

Mixed occupancy of tetrahedral sites without extra ligands (e.g. 4-connected Al and Si) must cause geometrical perturbations in vertex positions from idealized geometry. Particularly interesting from the viewpoint of a mathematician is regular occupancy of a specific T site by two or more chemical

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species. The strict regularity of Al and P is important to chemists, and is specified mathematically as a 2-color feature. Such strict regularity is possible only for a net with even-numbered rings: 4, 6, 8, etc., and is strictly forbidden for odd numbers. The strict alternation of Al and P forces a 2-fold reduction in the crystallographic symmetry. •

2.2

The actual geometry of a net in a real material may not have the highest symmetry simply because chemical and physical forces are causing a structural perturbation. The forces may involve nonframework species including cations, water, and various sorbed molecules and atoms. In general, the closest approach to ideal tetrahedral geometry is for materials containing no extra-framework cations, or ones with low ionic potential. Furthermore, raising the temperature tends to move a framework towards the idealized geometry and symmetry simply because of the additional vibrational energy. Movement of cations and adsorbed molecules during heating causes complex changes in framework geometry that generally tend to approach the idealized geometry, but not necessarily so.

Invention of (4,2)-3D nets

Such nets can be invented systematically by several techniques, including (i) condensation of polyhedra by sharing of vertices, edges and faces, and (ii) conversion of all or some edges of a stack of parallel 3connected 2D nets into chains and columns. They can also be generated by computer searches, in which starting positions for a chosen number of vertices in a unit cell move under various controls including the space-group symmetry, near-geometrical uniformity and crystal-chemical energy. Such searches can be random (Monte-Carlo) or constrained, or both. The potential of a new topology for occurrence in the crystal structure of a particular material is typically assessed by the deviation of its distance-least-squares model from perfect regularity of its distances and angles, or by its lattice energy. Although such criteria are a good start for choosing whether to add a new theoretical net to a selected database, all nets should be listed in a computer database simply because extra ligands and unusual chemical bonding have generated bizarre geometry in certain framework materials. Can the multiple-infinity of 4-connected 3D nets be enumerated by a single algorithm that radiates outwards from one single net, or from just a few simple nets? This mathematical poser has not been answered rigorously, but the answer may be no. However, it would be nice to see a firm decision backed by a strict mathematical proof. In the meantime, a practical approach is: •

identify 3D nets and subunits in known structures,

•

classify the nets and subunits of each type (0D, 1D, 2D and 3D) into topological groups,

•

enumerate systematically members of each topological group,

•

place all subunits of each type into a single database with an accessible unique position for each member,

•

examine how the subunits are connected in 3D nets of known structures,

•

select the “most interesting” connectivities from both the topological and topochemical viewpoints obviously, a subjective choice,

•

use the connectivities to invent new 4,2-connected 3D nets systematically from all relevant subunits, and

•

select the “most interesting” for incorporation into an active database of 3D nets.

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Each net and subunit must be given a unique topologic/geometric signature to permit (i) rigorous search for nets with specific topological and geometric properties, and (ii) identification of duplicates obtained from enumeration of different types of linkages. A “bootstrap” feature is important. As subunits and connectivities yield new 3D nets in systematic enumerations, further subunits and types of linkages appear automatically. From a practical viewpoint this builds up the databases of useful subunits and nets. But equally important is the mathematical significance. Is this a “fractal” method for extending our understanding of the topologic geometry of 4connected nets? Or are there nets that cannot be discovered by bootstrapping of known nets? Are some nets discoverable only by random methods, or by crystal structure determination of new materials, or by sheer invention? Enumeration of the combinations of 3-connected 2D nets and chains [99Han1, 2, 3] is a first step in systematizing the CTF Catalog of Nets. Some of these enumerations are upgrades of the opportunistic pioneering studies begun three decades ago. Enumerations based on polyhedral and other units are just beginning. The discovery of new subunits and connectivity-algorithms enlivens the tedium of the enumerations and DLS calculations. The enumerations are open-ended, and it appears that synthesis chemists will keep on whetting our appetite with new units and connectivities, and with new geometrical perturbations of known units and connectivities.

2.3

Number and connectivity of known nets

Over 200 topologically-distinct 4-connected nets with near-tetrahedral geometry have been recognized in chemical species, another thousand have been invented by systematic application of geometrical algorithms, and some thousands sit in computer databases after generation by various search procedures. Over 150 non-tetrahedral nets have been recognized in chemical species with n- & n,m-connected vertices, of which the octahedral-tetrahedral (114) & triangle-tetrahedral (11) subgroups are prominent in minerals and synthetic chemicals. Some nets have unusual geometry because of Jahn-Teller distortion or dangling ligands. Three nets with square-tetrahedral geometry have been recognized in chemical species, and two each with square & trigonal pyramids. The current CTF Catalog of 3D Nets is a chronologic compilation of entries obtained from diverse sources. About one-half are based on rigorous enumeration of connectivity-algorithms. I systematically check the literature on known framework structures (including zeolites, clathrates, anhydrous silicates, etc.) to identify new types of nets (243 entries in Table 16.2.1 on December 19, 1999). Each net is dissected for useful subunits. All “interesting” 3D nets enumerated by crystallographers at the University of Chicago are entered routinely into the CTF Catalog of 3D Nets. I examine all nets described in the public literature by other scientists, and add new ones to the CTF Catalog (1323 entries on December 19, 1999). Ideally, the atomic coordinates for an idealized geometry of each net should be incorporated into the CTF Catalog of Nets following distance-least-squares (DLS) modeling [77Bae1] and lattice-energy calculations for chosen chemical species. As time permits, such idealized DLS atomic coordinates are calculated for the earlier nets. Because the cell dimensions of ‘isostructural’ chemical species depend on bond distances and angles for chemical elements, I selected a standard reference near the middle of the

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3 Chemical background

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chemical range [note: connectivity is the common criterion for isostructural species, but can be hard to define for certain structures with complex bonding]. Based on the structures loosely classified as zeolites and related materials, I chose T-O = 1.64 Å and O-T-O = 147°. The major problem in matching with actual materials is the angular deformation. Thus the O-T-O angles in framework structures vary from ~115 to 180° [95Bau1], with median values for individual structures ranging from ~140 to 155°, depending on several chemical features. Some structures with “hinges” are more flexible to chemical and physical changes than others [92Bau1]. Hence it is wise to check for possibly significant changes in cell geometry and atomic coordinates by redoing the DLS calculations for any structure type between extreme values of ~137 and 157°, and varying the edges of the reference cell by several per cent.

3

Chemical background

The main drive behind development of net theory was the need to understand the crystal chemistry of zeolite molecular sieves. The late Donald W. Breck of Union Carbide Corporation reviewed the early history of research on zeolite molecular sieves [74Bre1]; further references [88Smi1]. Mineralogist/chemists set the scene: A. F. Cronstedt invented the name zeolite (boiling stone) [1756Cro1)]; A. Damour reported reversible dehydration [1840Dam1]; G. Friedel in 1896 and F. C. Grandjean in 1909 adsorbed liquids and gases in dehydrated zeolites [1896Fri1; 09Gra1]. These macroscopic evidences for microporosity were explained at the atomic level by the X-ray studies of W. H. Taylor & L. Pauling on analcime [30Pau1; 30Tay1], the fibrous zeolites, sodalite and cancrinite [33Tay1]; also important was the determination of Si,Al order-disorder in the framework structure of anhydrous potassium feldspar [33Tay2]. McBain invented “molecular sieve” after discovering that only small molecules entered dehydrated chabazite [32McB1]. Following various sporadic attempts over the past century, R. M. Barrer [review: 78Bar1] began systematic hydrothermal synthesis of zeolites, generally at elevated temperature, and characterized many zeolites and chemical analogs. In 1948, R. M. Milton began industrial synthesis of molecular absorbents at the Linde Division of Union Carbide Corporation [68Mil1]. His low-temperature gel technique yielded many microporous materials, including Linde Type A zeolite that is still not recorded as a mineral. The Linde Types X and Y zeolites turned out to have the same net as the very rare mineral faujasite [58Ber1]. Cracking catalysts based on zeolites replaced amorphous ones in the 1960’s, and a host of shape-selective absorbents and catalysts have revolutionized the fine-chemicals business. The linked-polyhedral topology of the above Linde materials triggered off the new wave of studies of mathematical topology of framework structures, including a structural classification of zeolites [63Smi1]. All recent reviews of the physicochemistry of zeolites and related materials, including clathrasils, are firmly based on an expanding knowledge of the crystallographic relationships between the many hundreds of chemical subspecies: [92Cat1; 91Der1; 89Ker1; 83Lie1; 85Lie1; 86Lie1; 89Smi1; 92Szo1; 91van1; 96Tre1; 98Akp1; 98Fra1; 98Zon1; 98Gie1]. In retrospect, the discovery of a common sub-unit (the fib chain) in the nets of the fibrous zeolites by W. H. Taylor [33Tay1], and the systematic enumeration of the different ways of cross-linking it into the nets of edingtonite, natrolite and thomsonite [33Tay2; updates: 75Alb1; 83Smi1], foreshadowed the later comprehensive development of net theory. A. F. Wells wrote a series of papers from 1954-1984 [summaries: 77Wel1; 79Wel1; 86Wel1] that interpreted the stereogeometry of various framework and other structures in terms of nets and other building units. The concept of a secondary building unit, “SBU”, [68Mei1] was used in the early editions of the Atlas of Zeolite Structure Types [authorized by the Structure Commission of the International Zeolite Association: 4th edition, 96Mei1], and was extended to include growth units which repel each other during crystal nucleation of an ordered framework [92Bru1]. However, the value of the SBU’s in physicochemical speculation about the geometrical stages in zeolite crystallization [review: 73Fla1] has been questioned [90Kni1]. To explain growth of 5-ring zeolites, the assembly of tertiary building units

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was proposed [91van1]. Whatever the uncertainties in the mechanisms for zeolite nucleation and growth, some type of subunits must be involved, and mathematical analysis must be useful as a basis for selection of particular subunits for chemical speculation. The classical concept of an aluminosilicate zeolite based on a (4,2)-connected 3D net has mushroomed out into a host of related concepts that do not necessarily involve simple 3D nets. Important general ones are: supramolecular architecture [92Bei1]; inclusion compounds [including hydrates & carbohydrates, 84Jef1, 92Jef1; 91Atw1]; and nanoscale inclusion chemistry [92Stu1]. The classification of chemical species containing some kind of structural unit related to a 3D net encompasses many concepts: clathrate hydrates [67Jef1; 72Kin2; 95Kos1; 96Kos1]; aluminophosphate molecular sieves [82Wil1; 86Ben1], clathrasils [83Lie1]; various materials with open and interpenetrating nets, including cyanides, selenides, porphyrins [82Gar1; 90Abr1; 90Hos1; 92Rob1; 92Kim1; 94Abr1; 95Bat1; 95Hos1]; silica minerals & glasses [88Mar1, 88Mar2, 90Mar1, 90Mar2; 90Mar3]. Also of interest are: mesoporous molecular sieves prepared with liquid crystal templates [92Bec1]; carbon polymorphs, including polyhedra (fullerenes) & shells ([87Kro1; 88Kro1; 93Kro1; 95Ame1]; non-carbon fullerene analogs [94Nes1]; carbon wires inside mesoporous phases [94Wu1]; octahedral-tetrahedral linkages in oxide hydrates [81Bar1]; molecular container compounds [92Cra1]; self-assembling organic nanotubes based on a cyclic peptide architecture [93Gha1; 95Gha1]; molecular confinement (93Har1); quasicrystalline alloys and decagonal tilings [92Ing1; 88Jar1]; plane nets in crystal chemistry [80O’Ke1]; symmetry from a chemist’s viewpoint [86Har1; 93Hei1]; curved surfaces in biological materials [huge literature, e. g., 87von1]; coordination polymers based on a (10,3)-a net [98Abr1]; molecular scaffolds [98War1]; self-organized nanostructures [97Stu1; tailored porous materials (99Bar2)].

4

Mathematical characterization of a crystal structure

The characterization of crystal structures based on 3D nets must be integrated into the mainstream of crystallography. A standardized description is needed for each structure type, including a specific choice of reference axes and space group of symmetry elements [63Sch1; 84Par1, 85Par1]. Each standardized structure type must be fitted into an overall nomenclature for crystal families, lattices and classes [76Lim1; 85deW1]. For zeolite structures, the first ever standardization is being prepared by W. H. Baur and R. X. Fischer [2000Bau1]. Classification of crystal structures is greatly facilitated by recognition of various “types”, especially those that can be organized into hierarchies associated with chemical and mineralogical families. The classic monographs and compilations are well-known, and not listed here. Papers since 1980 include: attempted exact, systematic, geometrical description of crystal structures based on Hilbert’s geometry and various transformation techniques [82And1]; tetrahedral AX2 structures [83Wel1]; description of complex inorganic structures [83And1]; nomenclature of inorganic structure types, including the concepts of isopointal, isoconfigurational, homeotypic, polytypic, recombination structures by unit-cell twinning, crystal shear planes, intergrowth of blocks or slabs, & periodic out-of-phase boundaries [90Lim1]; hierarchy of simple-to-complex structures [91Veg1]; towards a grammar of crystal packing [94Bro1]; searching for representatives of a specific structure type [99Ber1]. A method for searching for similarity between crystal structures uses affine normalization and reduced symmetry [94Dzy1] .

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5 Concise history of relevant mathematics / 6 Mathematical reference books

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Concise history of relevant mathematics

Early Man used geometry, but written records are recent: e.g. polyhedra [88Sen1]. Egyptians used icosahedral dice. Fragments of Greek manuscripts describe the Platonic and Archimedean convex polyhedra. Arabic geometrical patterns are valuable for teaching symmetry [77Mak1]. The atomic nature of matter [95Rou1] was discussed by early philosophers: concept of atom, Leucippus, c. 470-420 BC, Democritus, c. 460-370 BC; matching of air, earth, fire & water with the octahedron, cube, tetrahedron and icosahedron in Timaeus by Plato, 427-347 BC; linkage of atoms by hooks in the poem De Rerum Natura by Lucretius, 95-55BC, based on Epicurus (342-270 BC), and by Réne Descartes, 1596-1650, and Pierre Gassendi, 1592-1655; aggregation of ‘solid, massy, hard, impenetrable movable particles’ in Opticks by Isaac Newton, 1643-1727. Ultimately, John Dalton, 17661844, invented ball-and-stick models of chemically-bonded atoms. The genius Kepler, 1571-1630, found time from laborious service work (casting horoscopes for petty princes engaged in religious wars) to make creative drawings that foreshadow modern work on convex polyhedra, 2D nets, closest sphere packing, structural harmony, six-cornered snowflake, etc. [40Kep1, 66Kep1; various commentaries, 75Bee1; 75Cox1; 75Sha1; 81Mac1; 82Mac1; 88Sch1; 92Mül1]. Crystallography began in the Renaissance [83Sch1]. Descartes, 1596-1650, worked on polyhedra, and Euler, 1707-1783, on topology ~1750 [82Fed1]. The Gauss concept of a lattice, and modern attempts to find a sphere packing denser than the familiar hcp/ccp polytypic series are covered in 92Mül1. Another burst of creativity at the end of the nineteenth century is represented by the following selected studies: ideas on atomic packing [1879Soh1, 1888Soh1, 1894Kel1]; various ideas on sphere packing and crystal symmetry that provoked a polemic with Sohncke [1883Bar1; 01Bar1; 06Bar1; 07Bar1]; various ‘Strukturarten’, ‘Kristallisationsgesetz’ and parallelohedra [1894von1; 1897von1]; homogeneous division of space into truncated octahedra [1894Kel1]. Much of the later twentieth-century work owes a debt to pioneering topologic structure analyses [27Nig1] and to Niggli’s successors, especially Fritz Laves [29Haa1; 31Lav1; 39Sin1; 43Sin1]. The systematic studies of circle packings established a new tone of rigor and completeness lacking in the wonderful but spotty inventions of earlier centuries. Finally, the following books offer a bouillabaisse of riches: growth and form, revised edition [92Tho1]; symmetry [52Wey1]; symmetry in science & art [74Shu1]; space structures, their harmony & counterpoint [76Loe1]; shaping space, a polyhedral approach, with contributions from many distinguished geometers [88Sen1]; connections, the geometric bridge between art & science [91Kap1]; patterns in Nature [74Ste1]; patterns and symmetry in crystal structures [96O’Ke2].

6

Mathematical reference books

The following books on geometry and topology are useful: sphere packing, lattices and groups [88Con1]; introduction to geometry [69Cox1]; mathematical models, especially of polyhedra [61Cun1]; regular figures [64Fej1]; convex polytopes [67Grü1]; tilings & patterns [87Grü1]; graph theory [69Har1]; optimal form [85Hil1]; inorganic crystal structure [89Hyd1]; graphs [63Ore1]; convex figures and polyhedra [63Lyu1]; N-dimension geometry [58Som1]; convex polyhedra with regular faces [69Zal1].

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7 Various mathematical conetps and tools

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Various mathematical concepts and tools

The following concepts and tools are useful in the description and classification of nets and subunits, and in the invention of new ones. Graph. A graph is a net if it has unlimited connected cycles, and a radiating tree if it lacks cycles. The cycle of a graph is a closed path (circuit). Each net has an abstract graph whose embedding comprises each center (vertex) and connecting straight edges [84Chu1; 87Goe1; 91Goe1; 92Beu1, 94Beu1]. A graph is n-connected at the vertices. Tetrahedral vertices are denoted T. A graph is n-dimensional and mperiodic. A regular (crystallographic) graph contains point lattices, whose finite graph becomes the quotient graph of the 3-periodic net. A graph is simple if there are no loops or multiple edges. A ring is a cycle without a shortcut. A strong ring is not the sum of smaller cycles. A cycle is a very strong ring if it contains some edge that is not a member of some shorter cycle. A minimal covering set does not permit removal of a ring. A fundamental circuit is defined in [90Mar1; 91O’Ke1; 91O’Ke2]. Counting rings is non-trivial in certain 3D nets: a prescription is given in 90Gut1. The connectivity of a net is specified by an adjacency matrix [87Kle1] and the eigenvalue spectrum. Graph theory can describe Al/Si distribution over the vertices of nets in framework silicates (74Kle1). Concentric clusters for all combinations of 4- & 5-rings were enumerated by graph theory [96Sat1]. Coordination sequence of a vertex in a net. The Wachstumreihe is defined as Nk = number of vertices at a topological distance of k edges [71Bru2]. The Coordination Sequence (CS) for each vertex consists of an infinite cascade of increasing numbers [Kaskadefolgen, 94Beu1]. It is useful for classification of 3D nets in zeolites [79Mei1; 96Mei1]. Its asymptotic behavior for increasing k is almost parabolic [94Her1], and the coefficient of the quadratic term correlates well but not exactly with the framework density [96O’Ke1]. The CS is a periodic function that can be described by p quadratic equations, where p ranges from 1 for the Kelvin/sodalite net to very large for certain nets containing several sizes of rings, including 5 [96Gro1]. Tectosilicates were classified in terms of the CS [87Sat1], and zeolites in terms of consecutive concentric clusters [94Sat1]. The cycle class sequence [94Beu1; 97Thi1] distinguishes LTA from RHO whose coordination sequences are identical [96Mei1]. Framework, topological and metrical densities. The framework density (FD) is defined for zeolites as the number of tetrahedral (T) atoms per 1000 Å3 of cell volume [96Mei1]. For general application to 3D nets, it might prove convenient to idealize the cell geometry of a net with a distance-least-squares calculation. The FD ranges from ~12 for open zeolites to 26 for dense tectosilicates [96Mei1, Fig. 1], and correlates inversely with the size of the smallest ring [56Hol1; 79Bru1; 89Bru1; 90Sti1: 14-18 for 3, 1226 for 4, 17-20 for 5, and 23-26 for 6]. Applications of graph theory to zeolites were reviewed [89Mei1]. Like the FD, the topological density (TD) is a measure of the density of vertices of a net, and it can be expressed quantitatively as the metrical density (MD) [91O’Ke2]. Zeolites are separated from the denser frameworks with a band running from ~20-21 at 4 to ~22 at 6 [96Mei1]. Assuming that 3D nets tile triplyperiodic hyperbolic surfaces that are free of self-intersections, a plot of FD vs. ring size in silica frameworks is constrained by constant area per T vertex in hyperbolic 2D planes [94Hyd1]. Loop configuration of T-atom. Defined as a graph showing the number of 3- and 4-rings at the T atom [96Mei1, p. 6]. Sigma-transformation. A 4-connected net with a set of vertices lying appropriately in a plane may be expanded by splitting each vertex in two, and linking the two new vertices by a new edge [73Sho1]. An edge joining two vertices in the sigma-plane turns into a 4-ring. This transformation was applied to tridymite relatives [90And1]; cristobalite relatives [81Han1], and ferrierite [84Gra1]. Some twin boundaries correspond to a sigma-transformation.

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Other tools and concepts for interrelation of nets and invention of new ones. These include the following concepts in alphabetical order of author, some of which are described with different names: topologic distortion, crystallographic slip, and swinging crystallographic slip plane [74And1]; cyclic intergrowths [82And1]; slipping schemes, stellation, truncation and addition [82Sat1; 90Bos1]; swinging crystallographic shear and chemical twinning on the unit-cell level [77Bov1]; hierarchy of classification methods for patterns, with examples of Euclidean 2D patterns [81Grü1]; connectivity, triangulated coordination shells, lattice complex of Niggli, and Bauverbände (framework) plus application to cubic crystal structures [65Hel1; 81Koc1; 79Hel1; 81Hel1; 82Hel1; 85Hel1; 86Hel1; 74Koc1; 78Koc1; 84Koc1; 95Fis1]; modular & binary algebra applied to hexagonal and cubic nets and structures [62Loe1; 64Loe1; 60Mor1]; six regular polygons at a 4-connected vertex [82Mac1]; 0-,1-,2-,3-celloids, and topological curvature applied to zeolites [91Mül1]; examples of structural relations between dense and open structures [81Nym1; 91O’Ke1]; decoration of vertices and edges [91O’Ke1; 94Rei1]; space filling applied to simple structures [61Par1]; various operators for predicting zeolite structures, comprising translation (T), mirror (M, produces new edge), mirror (Mo, fuses two sheets of vertices), mould (Mz, combined inversion and translation of upper sheet to mould onto lower sheet) (92Pri1); combinatorics for generating new zeolite nets [93Tre1]; interruption and expansion of the SOD net [95Fen1]; regular polyhedral helices [96Lid1]. Polytypism, OD-structures, polysynthetic twinning and faults. In polytypism, a layer can be rotated at a surface plane to generate a new structure. The simplest example is the closest packing of spheres, whose A, B and C positions of a 2D layer may be selected at random, with the simplest being the AB sequence of hcp and ABC of ccp [enumeration: 81Igl1]. Mathematical tools for enumeration of graphs and polytypes are given in [37Pól1] and [81McL1]. The principle carries over to the ABC-6 family of 3D nets, in which the AB, ABC, AABB & AABBCC stackings of the 4.6.12 2D net are the simplest relatives of sphere packing [81Smi1]. In general, for a 4-connected net in a polytypic series, the vertices in the surface plane permit rotation of the adjacent layer to a new angular position suitable for reconnection of edges. The extensive bibliography includes: order-disorder (OD-) structures & twinning [56Dor1; 59Dor1; 61Dor1]; layer stacking [67Bec1]; polysynthetic unit-cell twinning [74And1]; symmetry at twin boundaries & 80 two-sided plane groups [58Hol1, 58Hol2]; twin composition plane as extended defect and structure-building entity [79Hyd1]; coincidence site lattice = unit-cell twinning [87Yan1]. Simulated annealing. This computer-intensive calculation procedure aims to produce the most likely nets that fit an assumed unit cell geometry, space group symmetry and number of T positions per unit cell [89Dee1; 92Dee2]. A random (Monte Carlo) search generates possible T positions optimized with a wiggle routine and a cost function. A figure-of-merit tests the closeness of distances and angles to the histograms for known zeolite structures. The procedure led to the correct net for most of the known zeolites, but not for AEI, AFT, BEA*, BOG, BPH, HEU & THO (three-letter codes in Table 1). Six nets were indicated for VFI. Some 5,000 hypothetical nets were generated. Using a cost function based on a hypothetical H6Si2O7 molecule, 50 nets were obtained for silica [94Boi1]. Color. The vertices of a net are uncolored for a simple topological analysis. Assignment of chemical species such as Al and P to the vertices can be described mathematically using color. Strict alternation of (say) Al and P in synthetic aluminophosphate molecular sieves forces the rings to be even-numbered. Three-color ordering of Al, Si and P in the synthetic MAPO variant of sodalite was evaluated in [90Han1].

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8 Polyhedra & related conepts

19

Polyhedra & related concepts

Some or all of the tetrahedral and other vertices of many 3D nets correspond to the vertices of polyhedra. Most of the recent mathematical developments involve 3-connected vertices of polyhedra abstracted from 4-connected 3D nets. Corresponding advances can be expected from study of 3D nets with various combinations of 4-, 5- & 6-connected vertices, and of ones containing triangles. The classical convex regular and semiregular polyhedra were explored originally for presence as subunits in 4-connected 3D nets. Further study demonstrated the theoretical and practical value of polyhedra lacking certain edges, or with single edges replaced by double or triple edges. General concepts. Proceeding on from Kepler’s description of the regular prism and regular antiprism, the following lists are useful: 92 convex polyhedra containing regular faces, but not necessarily regular vertices [65Joh1; 69Zal1]; complete catalog of polyhedra with 8 or fewer vertices [73Bri1]; trivalent polyhedra with 10 vertices [78Ran1]; 1028 enantiomorphic pairs and 221 single symmetric trivalent polyhedra up to 11 faces [85For1]; generalized space-filling polyhedra invented by Fedorov [59Bas1]; new family of non-parallel trivalent space-filling polyhedra [85Fer1]; space-filling heptahedra and various new space-filling polyhedra [76Gol1; 78Gol1]; space-filling polyhedra with 14 faces obtained from staggered packing of identical hexagonal prisms [86Ros1]. The mathematical properties of coordination polyhedra, metal clusters and boron hydrides are given in [69Kin1; 69Kin2; 70Kin1; 70Kin2; 72Kin2]. Cellular structures are common in biological organisms, which could be called polypolyhedra [85For2]. Various 14-face polyhedra occur in soap-bubble aggregates, plant cells, and polycrystalline metals [68Wil1]. Aggregates are described in terms of a Dirichlet region, a Voronoi polyhedron, a lattice domain and a Wigner-Seitz cell [64Smi1]. The overall mean geometrical coordination number for all aggregates is near 14, ranging from ~13-14 for compressed deformable spheres to 15.5 for random aggregates. A class of 20-faced space-filling polyhedra occurs in aggregates [65Smi1]. The experimental discovery of polyhedral varieties of carbon (fullerenes) triggered a flurry of theoretical papers [93Fow1; 93Rut1]. Polyhedra in clathrasils were described in [72Kin2] and [86Lie1]. Colored polyhedra were analyzed in terms of color symmetry [83Sen1]. Plane groups on polyhedra were covered in [62Paw1]. Polyhedra and Nets. Various 2D polygonal nets and 14 space-filling combinations of Platonic regular polyhedra and Archimedean semi-regular polyhedra (including the infinite series of prisms and antiprisms) were invented [07And1]: Figs. 12-25 therein are stereodrawings of the filling of space with combinations of these polyhedra, including the Kelvin/sodalite equipartition of space (Fig. 14), and the polyhedral equivalent of RHO (Fig. 24). Enumeration of 4-connected 3D nets whose vertices are based on Archimedean polyhedra yielded the Kelvin/sodalite, Linde Type A (LTA), faujasite (FAU), ZK-5 (KFI) and RHO nets [64Moo1]. Parameters for describing space-filling polyhedra, planar nets and frameworks are given in [81Bru1]: Fig. 1 therein includes the 6-handle tetrahedron and 6-edge-stellated tetrahedron (hes in the compilation of [94And1]). Removal of the control of regularity or semi-regularity on the polyhedron opens up an infinity of possible polyhedra and 3D nets based thereon. Alberti [79Alb1] invented many nets related to the 1,3stellated cube, denoted bru [93And1; 94And1], that are listed in [79Alb1] and [94Han1]. The edges of the cube, tetrahedron and hexagonal prism can be converted into derivative polyhedral units [94And1] by opening and stellation of certain edges, or by conversion into a handle (triple-edge). All recognized polyhedral units occurring in 3D nets of crystalline materials have been placed in a unique position in a computer data-base. Some of these derivative polyhedral units have been examined casually for assembly into new 3D nets, but systematic exploration remains a wonderful challenge. Miscellaneous. The following items are potentially useful in thinking about polyhedra and nets: crystal-chemical model of atomic interactions, including closest packing, Platonic & Archimedean

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polyhedra, coordination polyhedra and structural ensemble [88Asl1; 88Asl2; 89Asl1; 89Asl2]; generalization of Federov space-filling stereohedra to permit curved faces and re-entrant angles [59Bas1]; faceting the dodecahedron [74Bri1]; nested polyhedral units in alloys [81Cha1]: truncated octahedron in crystal structures [79Chi1; 80Chi1; 82Chi1]; closest packing of equal spheres on a sphere [86Cla1]; symbolism for coordination polyhedra [64Don1]; polyhedra related to Dirichlet domains with cubic symmetry [81Eng1; 81Eng2]; algorithm for enumeration of all combinatorial types of 3D polyhedra, and application to polyhedra with 4 to 11 faces [82Eng1]; polyhedra of three quasilattices associated with the icosahedral group, including Kepler’s great rhombicosihedron [87Haa1]; Edshammer polyhedron in betaK2SO4 [91Lid1]; 4-capped tetrahedron = stellae quadrangulae in scheelite and other structures [79Nym1; 83Nym1]; elongated rhombic dodecahedron in alloys [79Nym1]; bisdisphenoid in several structures [84Nym1]; snub cube and bipyramidal nets [63Smi1]; 3D polyhedra with tunnels [63Wel1]; growth polyhedra [08Wul1]; computer modeling of silica fragments with reference to nucleation, growth and templating during hydrothermal synthesis ([98Cat1].

9

Circle and sphere packing

Although most of the literature deals with closest and close packing of circles and spheres [67Pat1; 68Fis1; 71Bru1; 88Con1], open packings are important here as exemplars for the packing of oxygen atoms in zeolites, particularly for nets with edges of constant length [review: 83Sla1]. Sphere packings were described in terms of crystallographic point groups, rod groups and layer groups [78Koc2]. Open packing. The pursuit of the most open sphere packing [history: 54Wel1; 54Wel2; latest paper, 95Koc1] began with Heesch and Laves [33Hee1]. Three-packing is not mechanically stable [91O’Ke1], though it can be regarded mathematically as the simplest packing. Various 4-packings include spheres at the vertices of a 4-connected net containing truncated cubes joined by 4-rings. Truncation of 3-connected vertices causes triplication of a sphere packing. A (2,3)-connected sphere packing containing a tetragonal spiral has topological density 0.042 [42Mel1; 49Mel1]. Another sphere packing with density of 0.045 is a truncated version of a 3-connected sphere packing listed in [33Hee1]. Replacement of a 4-connected vertex by a tetrahedron in the underlying net of the sphere packing was described as decoration [91O’Ke1; 91O’Ke2]. A Kelvin/sodalite parent yields a decorated net of even lower density than for a diamond parent. Miscellaneous. Other relevant papers include: homogeneous sphere packing with tetragonal symmetry and associated 3D nets, including the (h,z)*fee 3D net [71Fis1]; periodic close packings [73Smi1]; 78 types of symmetrically-equivalent sphere packings for all cubic lattice complexes with less than 3 degrees of freedom [73Fis1]; 182 types of homogeneous sphere packing for all cubic lattice complexes with 3 degrees of freedom, including 4/3/c27 with topological density of 0.0789, lower than the Heesch-Laves open sphere packing [74Fis1]; 39 sphere packings with cubic symmetry, demonstrating 12 different types of interpenetration [76Fis1]; limiting forms and comprehensive complexes for regular and irregular packings of spheres, and curved voids relevant to the containment of fluids [35Gra1]; packing of ellipses [95Mat1]; relations between dense sphere packings with 6 to 11 neighbors [87Wel1]; densest packing of circles on sphere [91O’Ke3]; sphere packings and space fillings by congruent simple polyhedra [98O’Ke1].

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10

10/11 One-/Two-dimensional units / 12 Three-dimensional nets

21

One-dimensional units

One-dimensional building units extracted from selected vertices of 3D 4-connected nets are illustrated and classified according to the number of vertices in the rod repeat, the shortest circuit down the rod, the repeat distance, and the highest rod-group symmetry [93And2]. Relevant papers include: two-color cylindrical columns [70Rom1; 71Rom1]; enumeration of 4connected 3D nets from linkage of the fib chain in natrolite [75Alb1]; rod packings in crystal chemistry [77O’Ke1]; chiral wrapping of hex 2D net around cylinder [87Maj1]; towards a general description of hexagonal 3D framework structures using lateral connection of trigonal columns [90And1]; cubic cylinder packings [92O’Ke2]; rod packings and braids [95Lid1]; carbon tubes ([94Wea1].

11

Two-dimensional units

Two-dimensional building units extracted from selected vertices of 3D 4-connected nets are illustrated and classified according to the number of independent vertices and circuit symbols [93Plu1]. Most have 3connected vertices, but some have various selections of 2-, 3- and 4-connected ones. Relevant papers include: structure theory of plane symmetry groups [29Hee1]; connecting a fraction of the vertices in a closest-packed hexagonal net [73Fig1]; systematic generation of plane nets with 1, 2 and 3 vertices per unit cell [76Gui1]; plane nets in crystal chemistry [80O’Ke1]; ancient oriental patterns related to the fer 3-connected 2D net in ferrierite [86Gra1]; systematic enumeration and classification of 3D nets by linking 2D nets with translation, mirror, mirror + fusion, and mold operators [89Akp2; 92Woo1; 94Akp2].

12

Three-dimensional nets: topology

Topology is a key to prediction of crystal structures of minerals [99Bau3]. In this context, the following concepts are presented. The number of 3D nets is unlimited because vertices can be added to any net [92Beu1]. A summary of search procedures for discovery of new frameworks is given in [99Tre1]. The combination of a structural envelope with chemical information to solve complex zeolite structures from powder diffraction data is reviewed in [99Bre1]. The following papers cover both principles and net descriptions. Near-chronologic list of papers: 3-connected 3D nets [54Wel1; 72Wel1; 83Wel2]; 4-connected 3D nets generated from 2D nets by several procedures, open packing of tetrahedra, face-sharing of large polyhedra, and face-sharing of polyhedra with 5-rings in clathrate inert-gas hydrates [54Wel2; 72Kin1]; single, stuffed and interpenetrating nets in various structures, and cylindrical and helical structures obtained from the 63 2D net [54Wel3]; various 3-connected 3D nets, some interpenetrating [54Wel4; 55Wel1; 56Wel1; 76Wel1; 83Wel2]; 3-, 4-, 5-, and 6-connected 3D nets, Schläfli symbol, reciprocal nets, 3D polyhedra with tunnels, LTA net [63Wel1]; theoretical nets related to mordenite [63Ker1]; nets related to Archimedean polyhedra [64Moo1]; 3D nets containing vertices of hexagonal prisms, including CHA, ERI and GME [64Kok1]; plane and radiating nets, and various 3-, 4- and 3,4-connected 3D nets with

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[Ref. p. 251

application to borates [65Wel1; 86Wel2]; various 4- and p,q-connected nets [68Wel1]; further nets, including KFI [69Wel1]; 3D nets related to LTL containing the gml 2D net and a chain repeat n [69Bar1]; nets related to MOR [73She1]; nets in fibrous zeolites [75Alb1; 83Smi1]; enumeration of 3D nets with up and down connections between a stack of hex nets [77Smi1]; nets containing the bru polyhedral unit [79Alb1; 94Han1]; enumeration of 3D nets with perpendicular and near-perpendicular links between a stack of 4.82 fee, 3.122 ttw and 4.6.12 gml nets [78Smi1; 79Sat1]; enumeration of 3D nets obtained by combining helical, zigzag z, crankshaft c, and saw s chains with simple 2D nets [79Smi1]; the infinite set of ABC-6 nets, and the Archimedean and sigma-related nets [81Smi1]; enumeration of the zeolites of the chabazite group [89Set1]; 3D nets obtained from (528) 2(582)1 nets bik and biz (actually the 3D nets have four crystallographic types of vertices, (528)2 (528)1(528)1(582)2 [84Smi1]; two infinite series of 4connected 3D nets with channels of unlimited diameter based on the expansion of 4-rings in the hex and fee 2D nets whose edges were converted to crankshaft c chains [84Smi2; Fig. 2b therein predicted the VFI net with 18-ring channel]; 3D nets based on the gml net and its sigma-minus derivative (4.6.10)2(4.6.10)2(6210)1 ael net [85Ben1]; 3D nets based on double-bifurcated chains and the fee and gml 2D nets [86Smi1]; 3D nets based on insertion of 2-connected vertices into 3-connected plane nets [86Haw1]; 3D nets based on fer 2D net [86Gra1]; AB-5 and ABC-6 nets [87McC1]; 3D nets combining zigzag & saw chains with five 2D nets [88Haw1]; various hypothetical 3D nets with 2D and 3D channels [89Kok1: some of the data are hard to disentangle]; 3D nets in zeolites with 5-rings [89Akp1]; towards a general description of hexagonal 3D nets containing a trigonal unit [90And1; 90And2; 90Bos1]; expansion of zeolite-type nets based on a ‘supertetrahedron’ T8O17 that contains twelve 3-rings enclosing a 4-connected vertex, and on related units with 18, 19 and 20 O [90Han2; 90Han3]; 3D net in boggsite with both 10- and 12-rings [90Plu1]; units in ABC-6, AFI, LTL & MAZ 3D nets [92Akp1]; uninodal 4connected 3D nets with 3-rings [92O’Ke3]; uninodal 4-connected 3D nets lacking 3- and 4-rings [92O’Ke3]; extension of Price-Akporiaye formalism to three high-silica zeolites ZSM-12 (MTW), NU-87 (NES) and NU-86 [93Sha1]; stacking of triangular, hexagonal and kagomé nets [93Pea1]; enumeration of chiral nets, which might be synthesized, but in highly disordered polytypic materials [94Akp1]; sheets in nets of known structures [94Akp2]; 4-connected nets in mixed-coordination aluminophosphate framework structures [94And1]; 23 silica framework structures invented by simulated annealing using an energy function for a H6Si2O7 molecule [94Boi1]; comparison of linkage of hexagonal prism hpr in the 3D nets CHA, AEI and KFI [94Lil1]; faults, intergrowths and random phases in the ABC-6.hpr sub-family of zeolites, including babelite, Phi and Linde D [94Lil2; 94Szo1]; 4-connected nets in theoretical silica structures [95O’Ke1]; nets with three or four 4-rings at a vertex [95O’Ke2]; eight interpenetrating (10,3)-a nets [96Abr1]; 3-regular nets with 4 & 6 vertices per unit cell [97Bad1]; theoretical large-pore net [99Li1]; vertex symbols [97O’Ke1; 98 O’Ke2]; insertion of functional groups into square-planar units [97Sch1]; enumeration of 6,400 uninodal 4-connected graphs in 230 space groups [97Tre1]; faulting in CHA-GME [99Plé1]; invention of tectosilicate structures using molecular energy function and simulated annealing [99Boi1]; theoretical series of nets with 5-rings [99Li3]; systematic enumeration of the combination of a 3-connected 2D net from the CTF catalog with a chain, including the crankshaft c, zigzag z, and saw s types [99Han1; 99Han2; 99Han3]; faulted zeolite framework structures [99Gie1]; 4connected nets from packings of non-convex parallelohedra, etc. [99O’Ke1].

13

Three-dimensional nets: geometry & chemical stability of related materials

From the strictly mathematical viewpoint of topology and connectivity, the geometry of a net is irrelevant. From the crystal-chemical viewpoint, the geometry of an assemblage of atoms based on that net is extremely important [73Meg1; review of structural problems from a computational perspective: 92New1].

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The stable and metastable thermochemical parameters determine whether a crystalline nucleus can grow and survive. This section covers the various research studies that relate to the geometry of a net and the crystal chemistry of materials based on it. They range from simple optimization of bridge-building with near-rigid rods [82Gra1] to sophisticated force-field models which take into account chosen algorithms for chemical bonding, thermal motions, and thermodynamic disorder. The applicability of a mathematical model to a particular material is better for a ‘loose’ 4-connected framework with strong tetrahedral cations that is perturbed only lightly by non-framework species. It may prove to be unsatisfactory for a 4-connected framework in which the 4-connected cations have dangling ligands, including OH bridges and water molecules, or in which the species in the micropores cause strong geometrical perturbations, or both. Frameworks at high temperature tend to be more regular in geometry than at low temperature, and indeed a phase change may occur upon cooling. Aluminophosphate framework structures typically have strict alternation of Al3+ and P5+ over the vertices [86Ben1], whereas many aluminosilicate frameworks have disorder of the Al3+ and Si4+: the smaller difference of ionic charge in the latter is an obvious factor. A large literature on the details of Al/Si disorder in zeolites and related materials is not listed here. The essential mathematical assumption that each Al atom has only Si neighbors is consistent with all nuclear magnetic resonance and diffraction observations, but allows a wide range of positional arrangements for the number of Al atoms less than for Si. Further mathematical development is considering how best to position the Al and Si among the various rings of a net when not strictly fixed by atomic ratio or space group symmetry. The most recent paper evaluates the consequences of the assumption that a 5-ring contains only one Al in aluminosilicate zeolites [95Tak1]. Bridge building. The standard technique uses a distance-least-squares iteration [77Bae1], in which separate weights are assigned for permitted variation in T-O distance and T-O-T and O-T-O angles. This DLS technique works well for ‘loose’ high-silica frameworks, but not so well for ‘tight’ frameworks, such as LTA with triple 4-rings at the vertices of a cube. DLS works better when the O atoms are deliberately permitted to move away from the high-symmetry averaged positions in the cubic or hexagonal space groups to off-center ones in lower-symmetry groups. Aluminophosphate molecular sieves containing extra ligands on the Al, such as VPI-5 (VFI) and AlPO-17 (ERI) fit poorly with a DLS prediction based on a loose, strictly-tetrahedral net [86Ben1]. The simulated atomic positions can be fed into the POWD computer program [86Smi1] to obtain a calculated X-ray diffraction pattern for potential matching with patterns of materials with undetermined structure. Considerable tolerance is needed in checking for a match. Topologic factors related to stability of framework materials. At a simple level, certain topological properties of a net can be related to the thermochemical stability of corresponding materials: e. g. materials with large pores and wide channels tend to be metastable with respect to more compact ones, unless the framework is ‘inherently’ stable from a particularly ‘nice’ connectivity (e. g. FAU and MFI), or is stabilized by the chemical forces from encapsulated species, or both. At a more advanced level, it is desirable to find ways to prioritize the thousands of new nets coming from computer searches. Ideally, algorithms should be devised that can be programmed for automatic scanning of each new net. The following papers contain suggestions for relating stability to topologic and other features: evaluation of hypothetical zeolite frameworks [90Bru1; 93Bru1; 93Bru2] using the point group (the larger pores are centered on points of highest symmetry), ring size, framework density and loop configuration (a prediction that the VFI net was unfavorable because of its one-sided linkage at the ‘fused’ 4-rings was later negated by the discovery in synthetic aluminophosphate VPI-5 that the Al atom had two extra non-framework ligands on the open side; microporous silica and aluminosilicate materials have strictly tetrahedral bonding); templating cations such as tetra-methyl-ammonium should have optimal positions that match the periodic minimal surface of the pore system [90Blu1]; structural relationships and building units in 5-ring zeolites [92Van1]; transformation of linear-chain aluminophosphate to chain, layer and framework structures [98Oli1]. Force fields and various chemical parameters. The implied force fields in the simple DLS bridgebuilding are empirical, but can be adjusted by varying the statistical weights. Semi-empirical force fields can be derived from models for the electron distribution around atoms, and checked against actual distances and angles in appropriate crystal structures. Such force fields are being incorporated at Landolt-Börnstein New Series IV/14

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[Ref. p. 251

increasing levels of sophistication in commercial computer programs. The following chronologic list of papers provides an introduction to the complex literature: ab initio molecular orbital calculations on phosphates, and comparison with silicates (85O’Ke1); electronegativity equalization [87Mor1]; latticeenergy and extended Hückel calculations of lattice energy for dehydrated and hydrated zeolite frameworks indicating that Si-rich types differ little in energy, and that Al substitution for Si increases the energy range and tends to stabilize wider pores [88Oom1]; predicted lattice energies of theoretical silica frameworks of zeolite type decreasing as the number of vertices in the fourth shell of the coordination sequence increases [89Akp3]; molecular models [89Sau1]; force fields for zeolites [91Nic1]; force fields for silicas, aluminophosphates and zeolites [91Kra1]; anion coordination and sharing coefficient in zeolites and related materials, and observation that several Be-bearing tetrahedral frameworks have an interruption at a 3-connected vertex with a dangling hydroxyl [91Eng1; 91Eng2]; lattice energy and free energy minimization of zeolites [92Jac1]; stability and vibrational spectra of 3-ring zeolitic silica polymorphs [92deM1]; energy function for a H6Si2O7 molecule [94Boi1: see above]; interatomic potentials for aluminophosphates, obtained from the properties of berlinite, the AlPO4 analog of quartz [94Gal1]; computed lattice energies for silica, tested against experimental heat of formation [94Hen1]; prediction of enthalpy of formation from crystal structure [95Vie1]; structure-stability relationships for all-silica structures [95Tet1; 95deB1]; shape analysis of organic templates used in zeolite synthesis [96Boy1]; void fraction and nature of porous solids [96Gar1]; de novo design of templating agents [96Lew1]; structural characterization of zeolite-like structures based on Ga and Al germinates [98Bu2]; contrasting behavior of Ge and Si in structures containing cub [99O’Ke2]; computer modeling of silica clusters and rings aimed at understanding nucleation, growth and templating in hydrothermal synthesis [98Cat1]. Geometrical response to applied force. The geometrical response of a framework depends on the ‘inherent rigidity’. Some nets have flexible ‘hinges’; others are more resistant (many papers, especially on RHO; not listed here). The following chronologic list covers various concepts: self-limiting distortion by anti-rotation hinges in flexible but noncollapsible frameworks [92Bau1]; fluctuations in window size simulated by crystal dynamics [92Dee1]; 3-connected nets, with helical chains which twist upon heating or compression, as a basis for ‘auxetic’ materials [93Bau1]; deformation analysis of the octagonal prism opr in MER, PAU and RHO nets [94Bie1]; thermal expansion of zeolites and aluminophosphates [95Tsc1]; response to changes in pressure and temperature [96Kho1].

14

Three-dimensional nets: relation to minimal surfaces

The mathematics of minimal surfaces originally dealt only with regular assemblages of continuously curved surfaces. Various zeolites and mesophase molecular sieves have channels whose surfaces can be modeled by spheres lying at the vertices of an open 3D net. The simplest channels are cylindrical, as in aluminophosphate number 5 (AFI). Selected vertices from the AFI 3D net generate a 63-connected hexagonal 2D net wrapped on the cylinder. Several other zeolite nets, including FAU and MFI, have channels intersecting in 3D. The vertices lining the channels define 3D-polyhedra. The following list covers both continuous minimal surfaces and discontinuous ones related to zeolites and other chemical materials: literature review of 3-periodic minimal surfaces for congruent labyrinths, interface of two labyrinths congruent or incongruent and balance surface, graphs of most of the interpenetrating sphere-packings, branched and multiple catenoids, and infinite strips [87Fis1, 89Fis1, 89Fis2, 89Fis3, 88Koc1]; adaptation of chemical structures to curved surfaces, including periodic minimal surfaces, optimal form, fractal hierarchy, periodic equi-potential surfaces, periodic zero potential surfaces, commutable vs. noncommutable point configuration sphere packings, and many examples including sodalite, faujasite, defect clathrates and quartz [87von1]; infinite periodic minimal surfaces and topological characterisation of crystal structures, including the Kelvin/sodalite type [87Hyd1; 89Hyd2]; history of minimal surfaces and crystal structures, including LTA & P surface, FAU & F surface, cristobalite/diamond & D surface, sodalite & P/D adjoint surfaces, analcime & gyroid, gmelinite & H

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15 Challenges

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surface [88And1]; hyperbolic minimal surfaces and MFI structure type of genus 9 [91Hyd1]; symmetry restrictions related by the Bonnet transformation [89Lid1; 90Lid1]; various concepts related to curvature of hexagonal (graphite) sheet and diamond [91Mac1; 93Mac1; 93Mac2]; hypothetical minimal surfaces obtained from hyperbolic curvature of graphite net, related to P, D and G types [95Fog1]; derivation of periodic minimal surfaces using finite-element methods [94Mac1]; flexicrystallography [95Mac1]; architecture of pores in mesoporous MCM-48 [96Alf1; 97And1]; curved surfaces in crystal structures [96Kli1]; role of curvature in condensed matter (97Hyd1).

15

Challenges

To conclude the above survey of concepts and data, the last 40 years have seen an explosion in the mathematics of the topology of nets. Some concepts were invented in the preceding two millennia, but not all had been rigorously worked out. For example, Kepler left various drawings now seen to be the first flutters of new ideas on nets and polyhedra. As is common in science, new technology provides new observations that revitalize old mathematical ideas, and drive the invention of new ones. The first crystal structure determination of framework structures some 70 years ago became possible only after X-ray crystallography had moved to a high enough technical level. What will the new technologies of today and tomorrow permit? A major priority is to establish a unique classification of nets that can be searched automatically when a potentially new net is invented theoretically or found in a new structure. It might consist of a connectivity matrix in which the sequence of vertices is arranged by increasing distance from an agreed origin in the space group. This should be straightforward for a strictly tetrahedral net, but not for nets of mixed geometry. Systematic enumeration of topologic families of nets using various techniques was productive over the past 40 years [99Tre1], but there are many families that need systematic follow-up of earlier casual exploration. The combination of a chain with a 3-connected 2D net is in hand [99Han1; 99Han2; 99Han3], and updates should be made as new 2D nets are discovered. The systematic assembly of polyhedral and 1D subunits into 3D tetrahedral nets is at an early stage of development. Reconnaissance studies with selected polyhedral units containing 4- and 5-rings has yielded some fascinating 3D nets, including one with a 3D channel system with 14-ring windows in all directions. How to best systematize the search for new nets is not obvious at this time. Two subunits sharing a 5-ring (say) can be fitted together in various ways, depending on the symmetry of the units. How can a third unit be added, and so on, in a rigorous procedure that covers all possibilities? Can a mindless computer be programmed to dock modules together in a completely systematic way? Possibly, future development will proceed in many different ways, some old and some new. Perhaps the main era of discovery of new natural zeolites is over [92Tsi1], but further ones may turn up. Discovery of new synthetic zeolites and related minerals continues full blast in 1999, as demonstrated by the Proceedings of the 12th International Zeolite Conference [99Tre2]. Finally, there are more than 100 nets in known structures that contain some combination of 3-, 4-, 5and 6- connected vertices. The triangle-tetrahedral and tetrahedral-octahedral nets are particularly attractive for topological analysis. Prospects for giant pores, as exemplified by the open octahedraltetrahedral net of the iron phosphate mineral cacoxenite, are discussed in [99Fér1].

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16

16 Compilation of units

[Ref. p. 251

Compilation of units

16.1 General description This final section covers the 4,2-connected nets found in framework structures of natural and synthetic materials. Table 16.2.1 lists the information on the nets in the data-bases of the Consortium for Theoretical Frameworks. Most have near-tetrahedral geometry, but some have extra ligands, particularly aluminophosphates [86Ben1; 88Smi2]. Some nets with one or more 3-connected vertices and a dangling ligand, such as OH, are also listed. Table 16.2.1 lists all the tetrahedral and related nets of actual materials in the databases of the Consortium for Theoretical Frameworks on December 19, 1999. A systematic search of the principal chemical journals has been made since 1956, and all nets approved by the Structure Commission of the International Zeolite Association have been cross-checked. Although the Structure Commission does not consider dense nets, various compact nets in Table 1, including those in cristobalite, tridymite, and feldspar, are important to zeolite chemists. Tables 16.3.1, 16.4.1, and 16.5.1 list the polyhedral, 2D and 1D subunits arranged in mathematical sequence, and Tables 16.3.2, 16.4.2, and 16.5.2 present the same material in alphabetical sequence of the three-letter code. Tables 16.3.1 and 16.3.2 contain information describing the polyhedral units in Table 16.2.1; Tables 16.4.1 and 16.4.2, the 2D-nets; Tables 16.5.1 and 16.5.2, the 1D units. Tables 16.3.1, 16.4.1 and 16.5.1 are arranged in topological sequence. Tables 16.3.2, 16.4.2 and 16.5.2 are in alphabetical sequence. Table 16.2.2 gives a reference for each net not described by the Structure Commission of the International Zeolite Association, together with recent important papers on structures already coded by the Structure Commission. Figs. 16.3.1, 16.4.1, and 16.5.1 contain stereo drawings of the subunits in Tables 16.3.2, 16.4.2, and 16.5.2 arranged in alphabetical sequence. Each group of figures contains an introductory section describing some of the important features of the subunits.

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16.2 Systematic enumeration of subunits

27

16.2 Systematic enumeration of subunits Table 16.2.1 provides a guide to 4-connected 3D nets and their subunits. The first block of nets (a) is in alphabetical order of a three-letter code given in the first column. Codes in ordinary type were assigned to nets in zeolites by the Structure Commission of the International Zeolite Association. Codes in bold type were assigned by jvs to nets in various minerals & materials, including clathrates, not considered as zeolites by the IZA-SC, or not yet approved. They are useful for book-keeping, but have not been approved by any international agency. Please check with the web site of the IZA-SC for the latest assignment of codes. The second block of nets (b) was obtained from a variety of materials. They have interesting topological features, but were not listed in (a) for various reasons. The second column lists the rings in the 3D net that provide access from one volume unit to another volume unit. For example, FAU has a 6-ring between a truncated octahedron (code: tco) and a hexagonal prism (hpr), and a 12-ring between the large fau cages. A circuit that goes around an adjacent 4-ring and a 6-ring has order 8, but is not listed because it does not provide access between two volume units. The criterion of access between volume units was chosen because it is both mathematically significant and useful for understanding of molecular sorption of a zeolite based on the 3D net. The polyhedral units in column 5 are arranged topologically in Table 16.3.1, and alphabetically in Table 16.3.2. Each unit is assigned a unique three-letter code in lower case. In some 3D nets, certain polyhedral units can be assembled into columns or strings. Chains of various types also contribute to the 1D units in column 3, which are arranged topologically in Table 16.5.1, and alphabetically in Table 16.5.2. 2D-nets in column 4 are arranged topologically in Table 16.4.1 and alphabetically in Table 16.4.2. The catalog number of the Consortium for Theoretical Frameworks in the last column is merely an identification number lacking mathematical significance. There were 1322 entries on December 19, 1999. For convenience, each entry is followed by the name and chemical composition of materials based on that net. The mineral names of natural zeolites were chosen by the Nomenclature Subcommittee of the International Mineralogical Association [98Coo1]. Some nets are present in synthetic materials with many trade names. The selection of names merely reflects prominence in the scientific/patent literature, and has no commercial or legal significance. For example, FAU is represented by Linde Types X and Y, the first patented names.

Landolt-Börnstein New Series IV/14

Table 16.2.1 Subunits in alphabetical order of structure type codes. code (a)

name

rings

1D units commentary

Li-A(BW)

4,6,8

ACJ

AlPO-CJ2

4,6,8

ACO

ACP-1

4,8

AEI

AlPO-18

4,6,8

AEL

AlPO-11

4,6,10

c, kcb, kea, kei, ken, kew, sao ,zz synthetic LiAlSiO4.aq; also Ga, Rb, Cs & Tl analogs c, fhe, kek, krc synthetic AlPO4.organic c, kax synthetic (Co,Al)PO4 containing ethylene diamine and water kej, kfi, khg, khk synthetic AlPO4 containing tetraethylammonium and water afv, ape, bhs, c, kbg, kbn, keg

4,6,14

synthetic AlPO4; SAPO-11 & MCM-7 contain Si; various substituents for Al, including B afv, bhs, c, kbg, kbn, keg, ktv, vfi, vtn

4,6,8

synthetic AlPO4; transformation product of AlPO-54 (VFI) kea, keb, kfl, kyv, z

AEN

polyhedral units

catalog number

fee, hex

kdq, vvs, knn

fee ,fto

krb, kre

fee

cub, iet, ste, sti

complex

hpr, per

752

ael, hex

afi, bog, kah, kyw, loh, lov, odp

263

api, hex

afi, bog, etn, kah, ktw, kyw, loh, lov

733

brw, fsy, toa'

kyu

402

Main file of Zeolites and other materials assigned a three-letter code

ABW

AET

2D units

AlPO-8

AlPO-EN3

synthetic AlPO4 AlPO-EN3 has same (4:2)-net as AlPO-53B but extra ligands on Al; also UiO-12 GaPO-methylammoniumhydroxide has same (4:2)-net but extra ligand on Ga

4 725 46

Table 16.2.1 (continued). code

name

rings

AFC

AlPO-53C

4,6,8

AFG

afghanite

4,6

AFI

AlPO-5

AFN

AFO

AFR

4,6,12

1D units commentary atn, z, zz synthetic AlPO4 code submitted to the IZA-SC, but not approved because of high density afg, cna (Na,etc)8(Si,Al)12O24(Cl,etc)3.aq cancrinite mineral group, but not CAN topology of cancrinite afv, ave, bhs, c, kbg, kbn, keg

4,6,8

synthetic AlPO4; synthesized with many organics, especially tetrapropylammonium ion synthetic SAPO-5 contains Si various substituents for Al, including B & Co; As substitutes for P various other synthetics including MCM-6 & SSZ-24 kus, kut, odc

AlPO-41

4,6,10

synthetic AlPO-14 GaPO-14; extra ligands dangle from the Ga, distorting the geometry of the (4,2) net hex, uii afv, ape, bhs, c, kbg, kbn, keg

SAPO-40

synthetic AlPO4; synthesized with di-n-propylamine; SAPO-41 contains Si; various substituents for Al 4,6,8,12 bre, krx, lvi, ts, zkq

AlPO-14

synthetic (Si,Al,P)2O4

2D units

polyhedral units

catalog number

brw

kaa, kom, xib

1246

gml

can, lio

gml, hex

afi, apf, bog, kah, kyw, loh, lov

81

fee, gml

iet, klf, knu, krs, kry

399

afi, bog, kah, kyw, loh, lov, odp

736

afr, iet, ohc, sti, tti

871

gml, complex

187

Table 16.2.1 (continued). code

name

rings

AFS

MAPSO-46

1D units commentary 4,6,8,12 esc, krw

AFT

AlPO-52

4,6,8

synthetic Mg3Al11P13Si1O56 khx, khy synthetic AlPO4; synthesized with tetraethylammonium & tripropylamine szy, szz synthetic Si5Al23P20O96 afy, krv synthetic Co3Al5P8O32; also MAPO-50 z synthetic aluminophosphate with mixed tetrahedral/ square-planar vertices; the square-planar vertex can also be described as octahedral with two dangling ligands; the tetrahedral Cu and square-planar Pt atoms of [NMe4][CuPt(CN)4] have AHF topology ape, bhs, c, kbg, kbn, keg, ktv, vfi

AFX

SAPO-56

4,6,8

AFY

CoAPO-50

4,8,12

AHF

AlPO-H4

4,6,8

AHT

AlPO-H2

4,6,10

-AJT

AlPO-JDF-20

synthetic AlPO4; reversible transition with AlPO4-E 4,6,8,20 ts synthetic

2D units

polyhedral units

catalog number

gml

afe, afi, afo, afs, bph, iet, kno

413

gml

aft, cha, gme, hpr, kno

640

gml

aft, gme, hpr

gml

afe, cub, iet, sti

ahf',"

hes, kah

1031

ffs, hex

bog, kah, ktw, loh, lov, odp

418

jdf',"

-

984

1134 414

Table 16.2.1 (continued). code

name

rings

1D units commentary

2D units

ALS

AlAsO4.Me4NOH

4,6,8

kyg

fos, ,fsf, fsy, kyf, sfg hex, tth

ANA

analcime

4,6,8

ammonioleucite hsianghualite leucite pollucite wairakite

APC

AlPO-C

4,6,8

APD

AlPO-D

4,6,8

AST

AlAsO-I AlPO-16

4,6

octadecasil

synthetic AlAsO4.Me4NOH tetrahedral net abstracted from net with trigonal bipyramidal Al aff‘ kgv, ts NaAlSi2O6.aq analcime mineral group: the listed members below are valid: others are uncertain (NH4)AlSi2O6 Li2Ca3Be3Si3O12F2 KAlSi2O6 CsAlSi2O6.0.4aq CaAl2Si4O12.2aq various synthetics including AlPO-24, Na-B, Ca-D, Gaderivatives, cobalt aluminum phosphates, CsTiSi2O6.5 fee, fos, hex c, cc, key, kij synthetic AlPO4: reversible dehydration product of AlPOH3 which has extra ligands on Al apd, ,fee, bhs, c, kaz, kbg, keg, kep, khs, krg hex synthetic AlPO4: irreversible dehydration product of AlPO-C AlAsO4.ethanolamine krp, sod" ast, kfh, kgt, kht synthetic AlPO4 with encapsulated 1,4-ethylene piperidine and water synthetic SiO2 clathrasil

polyhedral units

catalog number 963

kds, krf

583

apc, kdm

634

kaa, kah, kdl, kdu, loh, lov

622

cub, iet, knw, kop, sti, trd

214

Table 16.2.1 (continued). code

name

rings

ATN

AlPO-39

4,6,8

ATO

MAPO-39 AlPO-31

4,6,12

ATS

MAPO-36

4,6,12

ATT

AlPO-12-TAMU

4,6,8

AlPO-33 ATV

AWO

AlPO-25

AlPO-21

GaPO-C4

4,6,8

4,6,8

1D units commentary ati, atn, zz synthetic AlPO4; synthesized with di-n-propylamine synthetic (HnMgnAl1-n)PO4 cnc, kay, z, zhp synthetic AlPO4; synthesized with di-n-propylamine; SAPO-31 contains Si; various substituents for Al kcn, nao, zz synthetic HMgAl11P12O48; also SAPO-36 c, cc, kdf, kei, ker, kfd, ss synthetic AlPO4; not to be confused with AlPO-12 made by Linde Corporation first synthesis using TMA, but first structure published on AlPO-12-TAMU (Texas A&M University) afv, bhs, c, ecc, kbg, kbn, keg

2D units

polyhedral units

fee, kyd"

kaa, koj, kom, ocn

93

gml, hex

kab, kah, kok, lau, zlv

738

gml, kuz"

kah, koh, oth

753

brw, fee

gsm, kdq

102

brw, hex

afi, bog, kah, kyw, loh, lov, oop

651

afs, kaq

401

epitaxial dehydration product of AlPO-21: [previous incorrect hypothetical structure coded as ATF was abandoned] brw, ffs, fsf c, cc synthetic AlPO4 with extra ligands on Al, originally coded as ATF (now abandoned); can contain various organics; transforms epitaxially to ATV Ga analog of AlPO-21

catalog number

Table 16.2.1 (continued). code

name

rings

AWW

AlPO-22

4,6,8

BAF

synthetic

4,6,12

BAN

banalsite

4,6,8

BAV

stronalsite bavenite

4,6,8

BCG

synthetic

4,6

BEA* Beta-A Beta-B Beta-C

Beta (-A, -B, -C) 4,5,6,12 bea, btg, kux 4,5,6,12 beb, btg, kux 4,5,6,12 bec, btg, kux

tschernichite BIK

1D units commentary kcx, kdg synthetic AlPO4; synthesized with DDO; some tetrahedral P encapsulated in the aww polyhedral unit c, cnc, z, zz BaFe2O4 bhs, bs, c, kbx, kuq Na2BaAl4Si4O16 Na2SrAl4Si4O16 ton, z Ca4Be2Al2Si9O26(OH)2 c BaCaGa4O8

bikitaite

5,6,8

synthetic NanAlnSi1-nO2; synthesized with tetraethylammonium ion BEA structure type is polytypic, and the -A, -B & -C polytypes are the simplest: the IZA-SC uses the star to show disorder Beta is highly siliceous whereas tschernichite has considerable Al; also Nu-2 CaAl2Si6O16.8aq; natural disordered mineral analog of synthetic disordered Beta c, hhz, kps, mao, s, scs, z LiAlSi2O6.aq

2D units

polyhedral units

catalog number

fee, lan"

aww, rpa, zlw

576

fos, hex

baf

470

fee

bog, kah, lov

bav"

lai

977

ftn, hex

fni, kah, oth

539

bab, bta bab, btb bab, bta, btb

bet, mtw, tes bet, mtw, tes bet, mtw, tes

577 578 579

bik, hex

knp, koq, pes

98

43

Table 16.2.1 (continued). code

name

BOG

boggsite

BNT

beryllonite

1D units commentary 4,5,6,10 brt, kri, p, pp, complex 12 Ca8Na3Al19Si77O192.70aq 4,6,8 afv, c, kaz, kbg, kbp, keg, kgo

BPH

trimerite BePO-H

NaBePO4 CaMn2(BeSiO4)3 4,6,8,12 bpi, krj

BPT

BeP2O6-II

synthetic Na7K7Be14P14O56.20aq 4,5,6,7,8 bre, c, kcc, kcd, s, scs, z

BRE

brewsterite

4,5,6,8

BRL

brewsterite-Ba brewsterite-Sr beryl bazzite cordierite indialite sekaninaite

rings

4,6,8,9

synthetic BeP2O6-phase II bre, brs, hel, heu, scs Ba2Al4Si12O32.10aq (Sr,Ba,Ca)2Al4Si12O32.10aq bre, kck, kez, kur, ts Be3Al2Si6O18 Be3(Sc,Al)2Si6O18 (Mg,Fe)2Al4Si5O18 [-ordered] (Mg,Fe)2Al4Si5O18 [-disordered] (Fe,Mg)2Al4Si5O18

2D units

polyhedral units

catalog number

fer, gml

bog, bru, eun, koi, lov, pes

636

apd, brw, hex

afi, fsi, kaa, kah, kaq

9

gml

afo, afi, afs, bpa, bph, iet

416

kyd", kye", toa'

complex

609

brw, complex”

bru, lov

591

qur”, toa’

ber

278

Table 16.2.1 (continued). code

name

rings

CAN

cancrinite

4,6,12

[afghanite [bystrite cancrinite cancrisilite davyne [franzinite [giuseppettite hydroxycancrinite [liottite

pitiglianoite quadridavyne [sacrofanite tiptopite [tounkite vishnevite

1D units commentary cnc, voi, zz warning: a clear distinction must be made between the cancrinite mineral group, composed of all ABC-6 members, and cancrinite sensu stricto, which has the AB stacking sequence (Na,etc)8(Si,Al)12O24(Cl,etc)3.aq belongs to cancrinite mineral group, but has AFG topology] "sulfide-cancrinite" belongs to cancrinite mineral group, but has LOS topology] Na6Ca2Al6Si6O24(CO3)2 Na7Al5Si7O24(CO3).3aq Na4K2Ca2Al6Si6O24(SO4)Cl2 (Na,Ca)7(Si,Al)12O24(SO4,etc).aq belongs to cancrinite mineral group but has FRA topology] K15Na40Ca7Al48Si48O192(SO4)11Cl2 belongs to cancrinite mineral group: may have 8-repeat in ABC-6 family] Na8Al6Si6O24(OH)2.2aq Ca11Na9K4Al18Si18O72(SO4)4(CO3)2Cl3(OH)4.2aq belongs to cancrinite mineral group: but has LIO topology] Na6K2Al6Si6O24(SO4).2aq (Na,K)6Ca2Al6Si6O24Cl4 Na6Ca2KAl6Si6O24(OH)3(SO4).etc. belongs to cancrinite mineral group: may have 14-repeat in ABC-6 family] K2Na2Li3Ca1Be6P6O24(OH)2.aq Na8Al6Si6O24(SO4)2Cl.aq belongs to cancrinite mineral group: may have 12-repeat in ABC-6 family] Na8Al6Si6O24(SO4)2.etc various synthetics, including Ge & Ga derivatives

2D units

polyhedral units

catalog number

gml ,kuz", kyd"

can, knr, kok

95

Table 16.2.1 (continued). code name CAS

name

rings

CsAlSi5O12

5,6,8

CBA

CaB2O4(IV)

3,6,7

CBU

CuB2O4

3,5,8

CFI

CIT-5

CGF

CoGaPO-5

CGS

MeGaPO-6

CHA

chabazite chabazite-Ca chabazite-Na chabazite-K willhendersonite

1D units commentary hhz, kce, kcg, kcl, kgf, z synthetic CsAlSi5O12 khn

synthetic CaB2O4(IV) s, scs, tff synthetic CuB2O4 4,5,6,14 hhz, hsr, hast ,hsu, z, zz synthetic SiO2 4,6,8,10 complex synthetic Co4Ga5P9O36.organic 4,6,8,10 kcb, zzj synthetic (Co/Zn)Ga3P4O12.organic synthetic TsG-1: K10Ga10Si22O64.5aq 4,6,8 kej, kgd, khc Ca2Al4Si8O24.12aq Na4Al4Si8O24.12aq K4Al4Si8O24.12aq KCaAl3Si3O12.5aq many synthetics including AlPO-34, SAPO-34, CoAPO44, MAPO-44, CoAPO-47, MAPO-47, K-G, Linde D, Linde R, LZ-218, ZK-14, MCM-2, ZYT-6, & Al-Co-phosphates

2D units

polyhedral units

catalog number

bik, hex

eun, kum, pes

242

kyr", complex

due, dum

853

complex

shf, tfe

1055

eug ,hex

eun, hsp, pes

1247

brw, complex zzu”

bog, lov

1224

kqr, zna

1290

gml

cha, hpr, zlx

83

981

Table 16.2.1 (continued). code

name

rings

-CHI

chiavennite

4,5,6,9, 10

CLA

clathrate A

CLB

clathrate B

CLC

clathrate C

CLD

clathrate D

CLE

clathrate E

-CLF

clathrate F

-CLO

cloverite

COE

coesite

CON

CIT-1

COO

SSZ-26,33A

1D units commentary c, z

CaMnBe2Si5O13(OH)2.2aq complex, not determined synthetic C20H44PBr.32aq clathrate synthetic Cs(Me4N)2(OH)3 clathrate 5,6 complex, not determined synthetic C23H41NO2.39.5aq clathrate 4,5,6 complex, not determined synthetic C3NH9.8H2O clathrate 5,6 complex, not determined synthetic C20H44NF.38aq clathrate 4,5,6,7 complex, not determined synthetic C4H11N.9.75aq clathrate 4,6,open see SOD, CTF net 108 synthetic C4H12NOH.5aq clathrate 4,5,6

4,6,8,20 clo synthetic GaPO4.quinuclidinium fluoride; [not a mineralname: refers to the clover-shaped channel] 4,6,8,9 bs, cc SiO2 10 4,5,6,10 f,olm,complex 12 synthetic H2B2Si54O112 ordered member of SSZ-33 family, type B SSZ-26 & SSZ-33 are disordered members with 15 & 30 % type A, 85 & 70 Type B resp. 4,5,6,10 f, olm, complex synthetic; polytype of SSZ-26,33B 12

2D units

polyhedral units

chi"

kah

981

clc"

cla, clb

966

cld"

clb, mla, red

967

kro"

hpr, mtn, red, stv

968

kro"

clb, mla, red

969

complex

cle,clf

970

-

toc minus 3 parallel edges of adjacent 6 cub, grc, iet, rpa, sti

971

kyd", usg'

complex

598

bab, fer, gml

bru, kab, kah, lau, mel, wwf, wwt

1006

bab, gml, mtt

bru, kab, kah, lau, mel, wwf, wwt

1005

col"

catalog number

982

Table 16.2.1 (continued). code

name

rings

CRI

cristobalite

6

carnegieite CZP

NaZnPO4.aq

4,8

DAC

dachiardite

4,5,6,8, 10

dachiardite-Ca

DDR

dachiardite-Na decadodecasil-3R

DFO

DAF-1

DFT

DAF-3

4,5,6,8

dodecasil-1H

(Ca,etc)2.5Al5Si19O48.13aq; svetlozarite is disordered variant (Na,etc)4Al4Si20O48.13aq kgr, kig

decadodecasil-3R SiO2.organics; synthetic clathrasil, part of polytypic series; also Sigma-1 4,6,8,10 kzf, kzg, kzh 12

4,6,8

DAF-2 DAF-3 DOH

1D units commentary fhe, kch, kga, kub, ton, z SiO2 NaAlSiO4 many synthetic analogues nzp, ts synthetic NaZnPO4.aq; Co substitutes for Zn has enantiomorphic net c, epi, frt, kad, kcq, kde, kqy, mod, s, z

4,5,6

synthetic Mg14Al52P66O264.organic.aq bhs, c, ecc, kbg, kbn synthetic CoPO4.0.5C2H10N2 synthetic ZnPO4.0.5C2H10N2, structure determined from Zn variety kfs, kfv, kgg, kgr SiO2.organic; synthetic clathrasil, part of polytypic series

2D units

polyhedral units

catalog number

hex, cri"

hes, lai

-

-

985

bik, dac, hex

dah, koa, ste, tes, zlt

248

kro"

det, dtr, kny, knz, kob, kol, red

588

gml

bog, cub, eni, evh, ftt, iet, kab, kno, kze, sti

964

fee ,hex

bog, kah, ktw, loh, lov, oop

39

kro"

doh, doo, knq, red

593

1

Table 16.2.1 (continued). code

name

DON

UTD-1F

EAB

TMA-E

ECA

bellbergite ECR-1

ECB

ECR-1

rings

1D units commentary 4,5,6,14 afv, c, cc, vtn

4,6,8

4,5,6,8, 12

4,5,6,8, 12

EDI

edingtonite

4,8

EMT

kalborsite Barrer-N Linde F K-F [ice-VI EMC-2

4,6,12

2D units

polyhedral units

catalog number

utd

afi, etn, kah, mel, wwf, wwt

1313

synthetic Si64O128.org/inorg related to UTC & UTI aeb, eba, ts synthetic (Me4N)2Na7Al9Si27O72.aq (K,Ba,Sr)2Sr2Ca2(Ca,Na)4Al18Si18O72.30aq c, csh, kbi, kcq, kde, kqy, kuw, mod, s, scs, ss, tix

gml

eab, gme, hpr, kno, 118 knx

bik, ecr

dah, gme, kaa, kno, 575 ste, tes

synthetic; model A; see below for model B; not considered by IZA-SC c, kbi, kcq, kqy, mod, s, ss

bik, ecr

gme, kaa, kno, mrd, ste

601

fee

des, krq, kzd

258

zzw”, complex

hpr, toc, wof, wou

204

synthetic; model B; see above for model A; not considered by IZA-SC fib, kco, ktf, s BaAl2Si3O10.4aq K6Al4Si6O20B(OH)4Cl K6Al5Si5O20Cl.4aq synthetic synthetic two interpenetrating EDI nets] khw, khz synthetic Na21Al21Si75O192 hexagonal polytype, related to cubic polytype FAU; incorrectly called hexagonal faujasite; intergrowths occur

Table 16.2.1 (continued). code

name

rings

EPI

epistilbite

4,5,6,8, 10

ERI

erionite-Na erionite-K erionite-Ca

ESV

EUO

FAU

ERS-7

EU-1

faujasite faujasite-Ca faujasite-Mg faujasite-Na

1D units commentary c, epi, ktu, mod, s, scs, z

2D units

polyhedral units

bik, dao, hex

ktg, tes

250

can, eri, hpr

119

(Ca,Na2)Al2Si4O12.4aq has orthorhombic relative with space group Fddd, CTF 251 4,6,8 gml kha, khb (Na,etc)9Al9Si27O72.28aq (K,etc)9Al9Si27O72.28aq (Ca,etc)4.5Al9Si27O72.28aq various synthetics, including LZ-220 & AlPO-17 (extra ligand on Al) & GaPO caution: polytypic intergrowth with OFF may occur, as in Linde T 4,5,6,8 mio, mip min” synthetic SiO2 analog of ESV in space group Bmmm with mil & mim cages 4,5,6,10 eue, ktx, nas, sss eon, eui

4,6,12

synthetic NanAlnSi1-nO2.aq various synthetics, including ZSM-50 & TPZ-3 zzw” (Ca,etc)2(Al4Si8O24.16aq (Mg,etc)2(Al4Si8O24.16aq (Na,etc)4(Al4Si8O24.16aq various synthetics, including Linde X & Y, SAPO-37, LZ210, EMC-1, CSZ-3, & Co-Al-phosphate cubic polytype, related to hexagonal polytype EMT; intergrowths occur

mil, mim

catalog number

1296 1297

eun, euo, kdw, koc, nna, non, pes, zly

573

fau, hpr, toc

203

Table 16.2.1 (continued). code

name

FER

ferrierite ferrierite-K ferrierite-Na ferrierite-Mg

[FRA

franzinite

FSP

feldspar

K-feldspar (sanidine, orthoclase, microcline) Na-feldspar (albite) reedmergnerite anorthite celsian buddingtonite

rings

1D units commentary 5,6,8,10 c, frt, kcq, kcz, kdz, kgb, kqy, mod, s, scs (K,etc)6Al6Si30O72.18aq (Na,etc)5Al5Si31O72.18aq (Mg,etc)3.5Al7Si29O72.18aq various synthetics, including Sr-D, ZSM-35, NU-23, FU9 & ISI-6 4,6 knb, knc member of ABC-6 group with ideal sequence ABCABCBACB; stacking faults, incomplete structure refinement] 4,6,8 bre, bs, cc, kbb, kcb mineral group comprised of following end-members, with subtle structural differences KAlSi3O8

NaAlSi3O8 NaBSi3O8 CaAl2Si2O8 BaAl2Si2O8 NH4AlSi3O8 various synthetics including simple Ga, Ge, Fe, Sr analogs and complex ammonium & potassium aluminum cobalt phosphates

2D units

polyhedral units

catalog number

bik ,fer ,hex frr, tes

600

gml

can, los, toc

201

fee

kaa, lov

26

Table 16.2.1 (continued). code

name

GIS

gismondine amicite garronite gismondine gobbinsite

GME

gmelinite gmelinite-Ca gmelinite-K gmelinite-Na

GOO

goosecreekite

GST

GaAsO4-2

1D units commentary 4,8 cc, fhe, ker, kfd, ts K4Na4Al8Si8O32.10aq NaCa2.5Al6Si10O32.14aq CaAl2Si2O8.4.5aq Na5Al5Si11O32.12aq various synthetics including MAPSO-43, N-B, Na-P1, Na-P2, &Al-Co-phosphates 4,6,8,12 cc, gmn, kgd, ts, tsn Ca4Al8Si16O48.22aq K8Al8Si16O48.22aq Na8Al8Si16O48.22aq various synthetics, including Sr-F caution: ideally AABB member of ABC-6 polytypic series, but stacking errors and polytypic intergrowths occur 4,6,8 bs, goo, ktz, s, ts CaAl2Si6O16.5aq 4,6,8,10 c, kee, keh, keq, kfa

HEU

heulandite

synthetic GaAsO4-2 4,5,8,10 hel, hen, heu, s

clinoptilolite-Ca clinoptilolite-K clinoptilolite-Na heulandite-Ca heulandite-K heulandite-Na heulandite-Sr

rings

(Ca,etc.)6Al6Si30O72.20aq (K,etc.)6Al6Si30O72.20aq (Na,etc.)6Al6Si30O72.20aq (Ca,etc.)4Al8Si28O72.24aq (K,etc.)7Al7Si29O72.24aq (Na,etc.)7Al7Si29O72.24aq (Sr,etc.)4Al8Si28O72.24aq; various synthetics, including LZ-219

2D units

polyhedral units

catalog number

fee

gsm

23

gml

gme, hpr, kno

82

brw

gos

608

fee, ffs, fos, fsy

iet, kdt, kog, kov, sti

748

brw, complex”

bru, knt, lov

602

Table 16.2.1 (continued). rings

1D units commentary 4,5,6,12 complex column, not coded

code

name

IFR

ITQ-4

ISV

ITQ-7

ITE

SSZ-42

3,4,6,8, 12

JBW

NaJ(BW)

4,6,8

KAG

KAlGeO4

4,6,10

KBG

KBGe2O6

4,6,7

KEA

keatite

5,6,7,8

KFI

ZK-5

4,6,8

complex synthetic SiO2 various synthetics including Mobil MCM-58 & Chevron SSZ-42 fhe, kei, kox, kua, kub, s, sao, ton, z, zz synthetic Na3Al3Si3O6.1.5aq nepheline hydrate I synthetic; not a mineral afv, c, ked, kfg, kuv synthetic KAlGeO4 kci, kft, z, zz synthetic KBGe2O6 c, s, scs synthetic SiO2: not a mineral khf, khg, khl, khm, khv

4,6,8

synthetic Na30Al30Si66O192.98aq various synthetics including Barrer-Robinson P & Q bs, c, for, kcw, kec, kem, kgh, kyc, lvi

KGE

K2Ge8O16O1

synthetic SiO2: calcined ITQ-4 4,5,6,11, f, znh, 12 complex to be coded

synthetic K2Ge8O16O1; the seventeenth O is a dangling ligand

2D units

polyhedral units

complex

kab, kah, lau, wwf, 1235 wwt

bab, nuj

gml

cub, iet, kah, kof, koh, lau, mel, nuh, sti itp, oot, xvi

brw, hex

hes, kdq, lai, vvs

fsv, hex

afi, kag, knn

538

bor, kuz"

kdp

749

kuy"

complex

596

fee, complex

grc, hpr, pau

205

brw ,fee, fst, iet, kah, kal, kam, hex ohc, sti

catalog number

1312

1248

96

720

Table 16.2.1 (continued). code

name

rings

KPO

kaliophilite-O1

4,6,8

1D units commentary c, kbp, ken, kgq

KZP

KZnPO4

4,6,8

KZS

K2ZnSi4O10

3,4,5,6, 7,8,9

LAU

laumontite

4,6,10

leonhardite LCP

leucophosphite

4,8

spheniscidite tinsleyite AlPO-15

-LEI

leifite

4,5,6,7

2D units

polyhedral units

catalog number

brw, fee, hex, kuu

kaq, kdi, kdj, kdq, knv

536

synthetic KAlSiO4, not a mineral afv, c, kaz, kbg, kbp, keg, kgo, z

brw, hex

afi, kaa, kah, kaq, ten

10

synthetic KZnPO4 bs, kcb, kef

kya", kyb"

tfs

751

ael, lam", qur"

bog, kah, lau, lov

633

iet, kjr, kra, krz

400

ber

978

synthetic K2ZnSi4O10 synthetic K2BeSi4O10 bre, bs, lum, the, ts

Ca4Al8Si16O48.18aq partial dehydrate of laumontite, occurring as mineral synthetic (Co0.33Ga0.67)PO4.0.33(C5H5NH) fee, vvv fhe, kys, kyt, z Kfe2(PO4)2OH.2aq; extra ligands dangle from the Fe distorting the geometry of the (4,2) net (NH4,K)(Fe,Al)2(PO4)2OH.2aq KAl2(PO4)2OH.2aq synthetic NH4Al2(PO4)2OH.2aq; extra ligands dangle from the Al distorting the geometry of the (4,2) net synthetic Ga analog of AlPO-15 lef", lei kck, z Na4H2Be2Al4Si14O37(OH)2.NaF.2aq

Table 16.2.1 (continued). code

name

LEV

levyne levyne-Ca levyne-Na

LIO

liottite

LOS losod bystrite LOV

lovdarite

LTA

Linde Type A

LTL

Linde Type L perlialite

LTN

Linde Type N

rings

1D units commentary 4,6,8 kfz, kgd, kho (Ca,etc)3Al6Si12O36.17aq Na6Al6Si12O36.17aq various synthetics, including LZ-132 & -133, ZK-20, MAPO-35 & SAPO-35, SSZ-17, & Nu-3 4,6 kgx, kgy, kgz Ca11Na9K4Al18Si18O72(SO4)4(CO3)2Cl3(OH)4.2aq ABCBCB member of cancrinite mineral group, see CAN 4,6 ldo, lso synthetic Na12Al12Si12O48.18aq "sulfide-cancrinite" ABCB member of cancrinite mineral group, see CAN 3,4,6,8,9 bre, bs, kad, kbb, kbf, kbi, vpv K4Na12Be8Si28O72.18aq 4,6,8 kfm, kfu, kfx, khj, khp, ost synthetic; ideally Na12Al12Si12O48.27aq, but commonly has higher Si/Al various other synthetics, including Ga-Ge-analog, Alpha, N-A, ZK-4, ZK-21, ZK-22, LZ-215, SAPO-42 4,6,8,12 bre, kbi, lel, ofr, ss synthetic K6Na3Al9Si27O72.21aq K9Na(Ca,Sr)Al12Si24O72.15aq various synthetics, including Ba-G, ECR-2, LZ-212 & Ga analog 4,6,8 kic, kie, kih synthetic NaAlSiO4.0.3aq synthetic Shepelev NaZ-21

2D units

polyhedral units

catalog number

gml, lvy"

hpr, lev

189

gml

can, lio, los

124

gml

can, los

110

fee, lod

kaa, lov, sfi, ste

572

feo tth ,zzv” cub, grc, iet, sti, toc

202

ltl, complex

can, hpr, kaa, lil, ste

631

complex

can, grc, hpr, ltn, toc

635

Table 16.2.1 (continued). code

name

-MAR

maricopaite

MAZ

mazzite

MEI

ZSM-18

MEL

ZSM-11

rings

1D units commentary 5,6,8,12 c, kcq, kqy, mdn, mod, p, s, scs

4,5,6,8, 12

Pb7Ca2Al12Si36(O,OH)100.32aq interrupted net with some edges removed from MOR c, csh, mzz, scs, ss, tix

Mg2.5K2Ca1.5Al10Si26O72.30aq various synthetics, including Omega, LZ-202, ZSM-4 & Ga analog 3,4,5,7,1 kuc, mig 2 synthetic NanAlnSi1-nO2.0.4aq 4,5,6,8, kgk, kgl, khi, p, pet 10

MEP

melanophlogite

5,6

MER

Zintl phase merlinoite

4,8

synthetic NanAlnSi1-nO2 ordered member of pentasil family various synthetics, including silicalite Type-2 (SiO2) & boralite-D, which are not minerals kfz, kgi, kgm, kgw, khu SiO2.(encapsulated species): natural mineral & synthetic analogs metal8Si46 cc, chf, kue, kuf, mer, ts K5Ca2Al9Si23O64.22aq various synthetics, including Linde W, Barrer-Baynham K-M & Ba analog, & Al-Co-phosphates

2D units

polyhedral units

catalog number

bik, hex, mar'

tes

987

ltl, kyq"

gme, koi, kno, maz 632

gml

iet, meg, mei

fer ,mln

kaa, kah, kdr, kod, 589 koe, kuh, mel, pen, tes

kud"

mla, red

fee, fos ,vvv

opr, pau, ste

726

595

17

Table 16.2.1 (continued). code

name

rings

MFI

ZSM-5

4,5,6,8, 10

MFS

mutinaite ZSM-57

MGN

moganite

4,6,8

MON

montesommaite

4,5,8

MOR

mordenite

4,5,6,8, 12

MSO

MCM-61

4,6

4,5,6,8, 10

1D units commentary kgl, kri, kui, kuj, p, pet

2D units

polyhedral units

catalog number

fer, mfv

eun, kah, kns, knt, kof, kuh, mel, pen, pes, tes

590

synthetic NanAlnSi1-nO2 series ordered member of pentasil family structure first published for Al-free endmember synthetic silicalite Type-1 (SiO2), which is not a mineral many chemical analogs and trade names, including LZ-105 natural mineral containing substantial Al bik, fer c, kcq, kcr, kcs, kcz, kdb, kdh, kpa, s, scs synthetic H1.5Al1.5Si34.5O72 bre, kcb, kcc, kcd, keo, s, ts, z SiO2, can be envisaged as a regular unit-cell twin of quartz bs, c, kcy, kda, s, scs K9Al9Si23O64.10aq c, csh, kcq, kde, kqy, mdn, mod, p, s, scs (Na,etc)8Al8Si40O96.28aq various synthetics, including Barrer-White Na-D, largeport mordenite, Koizumi-Roy Ca-Q, Breck-Skeels LZ211 & Zeolon 100 (not a mineral) maricopaite has MOR topology except for missing edges; see -MAR zni, complex synthetic K2Al2Si28O60.18-crown-6-ether

kKdk kdo, pes, tes

641

qur", toa'

kds

282

bik ,fee

euo, kdw, kdy

729

bik, hex, mor

dah, mrd, tes

594

hex, complex

ber, hpr, kab, kah, lau, znl

1311

Table 16.2.1 (continued). code

name

rings

MTN

ZSM-39

5,6

MTT

Zintl phase ZSM-23

5,6,10

MTW

ZSM-12

MWW

MCM-22

[ITQ-1 NAT

natrolite gonnardite mesolite natrolite scolecite tetranatrolite

4,5,6,8, 12

1D units commentary kgr, kif synthetic SiO2.organic: a silica clathrasil various other synthetics, including dodecasil-3C & holdstite (not a mineral) metalxSi136 hhz, kcg, kcj, kcm, ton, z synthetic NanAlnSi1-nO2 various other synthetics, including KZ-1, ISI-4 & EU-13 f, hhz, kce, kcf, kcg, kfq, kug, ton, z, zzi

synthetic NanAlnSi1-nO2 various other synthetics, including CZH-5, Nu-13, Theta3, TPZ-12 4,5,6,10 complex 12 synthetic Na3H2B5Si67O144; also PSH-3 & SSZ-25 synthetic SiO2; synthesized with trimethyladamantammonium ion; incomplete MWW net] 4,8,9 fib, kct, kfy (Na,Ca)7Al9Si11O40.12aq Na16Ca16Al48Si72O240.64aq Na2Al2Si3O10.2aq CaAl2Si3O10.3aq may not be distinct from gonnardite various synthetic phases, including Ga, Ge & Rb analogs

2D units

polyhedral units

catalog number

kro"

koo, mtn, red

592

hex, mtt

hes, kdx, lai, pes

604

btb, hex

eun, hes, kot, lai, mtw, pes

605

eui, kro", complex

doo, hpr, kah, kzd, mel, complex

fee

des, kzd

1101

260

Table 16.2.1 (continued). code

name

NES

NU-87

NON

gottardiite nonasil

NU-86 (A)

NU-86 (B)

rings

1D units commentary 4,5,6,10 eue, ktx, kul, sss 12 synthetic H4Al4Si64O136.naq Na3Mg3Ca5Al19Si117O272.93aq 4,5,6 ktx, nas, noa synthetic SiO2.organic clathrasil; also ZSM-51 4,5,6,10 f, kyh, kyi, kyl, kym 11,12

2D units

polyhedral units

eui, kuk, toa'

eun, euo, kdw, non, pes

763

eui, nos

nna, nns, non

574

nus, nux, complex

bet, kah, kof, mel, mtw, nug, nuh, tes, wwf, wwt

873

bet, kah, kof, mel, mtw, nug, nuh, tes, wwf, wwt bet, cub, iet, kah, kof, mel, mtw, nug, nuh, sti, tes, wwt can, gme, hpr, kno

874

synthetic silica; polytypic series with (B) & (C), which are also ideal end-members; IZASC code under discussion 4,5,6,10 f, kyh, kyi, kyl, kym nui, nux, 11,12 complex

NU-86 (C)

4,5,6,10 f, kyh, kyj, kyk, kym 11,12

nut, nux, complex

OFF

4,6,8,12 kgd, off, ofr, ss, tix

gml, kyn", ooo

offretite offretite-Ca offretite-K offretite-Mg

OSI

UiO-6

4,6,12

(Ca,etc)2.5Al5Si13O36.16aq (K,etc)5Al5Si13O36.16aq (Mg,etc)2.5Al5Si13O36.16aq various synthetics, including TMA-O & LZ-217: caution, polytypic intergrowth with ERI may occur, as in Linde T kay, nao, osi, z, zz synthetic AlPO4

krp

kah, lau, oth

catalog number

875

106

1153

Table 16.2.1 (continued). rings

code name OSU

osumilite

-PAR

armenite brannockite chayesite darapiosite eifelite emeleusite merrihueite milarite poudretteite roedderite sogdianite sugilite tuhualite yagiite zektzerite parthéite

PAU

paulingite paulingite-Ca paulingite-K

1D units commentary 4,6,9 kdc, kfo, kfz, kgs (K,Na)(Mg,Fe)2(Al,Fe)3(Si,Al)12O30.aq BaCa2Al6Si9O30.2aq KLi3Sn2Si12O30 K(Mg,Fe)(Mg,Fe)2Fe3+Si12O30 (K,Na)3Li(Mn,Zn)2ZrSi12O30 KNa3Mg4Si12O30 Na4Li2Fe2Si12O30 (K,Na)2(Fe,Mg)5Si12O30 KCa2AlBe2Si12O30.aq KNa2B3Si12O30 (Na,K)2(Mg,Fe)5Si12O30 (K,Na)2Li2(Li,Fe,Al)2ZrSi12O30 KNa2(Fe,Mn,Al)2Li3Si12O30 (Na,K)Fe2+Fe3+Si6O15 (Na,K)3Mg4Al6(Si,Al)24O30 LiNa(Zr,Ti,Hf)Si6O15 4,6,8,10 complex Ca2Al4Si4O15(OH)2.4aq 4,6,8 khr, kia, kib, ts (Ca,etc)5Al10Si32 O84.34aq (K,etc)10Al10Si32O84.44aq synthetic ECR-18

2D units

polyhedral units

catalog number

sod"

ber, hpr, osu

279

fee

-

973

complex

grc, kos, opr, oto, pau, phi, plg

406

Table 16.2.1 (continued). code

name

rings

PET

petalite

5,6,7,8

PHI

philippsite

4,8

harmotome phillipsite-Ca phillipsite-K phillipsite-Na

PRC

paracelsian

4,6,8

danburite hurlbutite slawsonite QTZ

quartz

6,8

berlinite RBS

Rb6Si10O23

6,12

RHO

Rho

4,6,8

pahasapaite

1D units commentary z LiAlSi4O10 cc, kcp, kfe, kge, kgn, kgu, ts (Ba,etc)5Al5Si11O32.12aq (Ca,etc)2.5Al5Si11O32.12aq (K,etc)5Al5Si11O32.12aq (Na,etc)5Al5Si11O32.12aq various synthetics including ZK-19, ammonium-Al-Co-phosphate c, cc, kbl, kes, kev, kfc BaAl2Si2O8 CaB2Si2O8 CaBe2P2O8 (Sr,Ca)Al2Si2O8 synthetic Cu(pyrimidine)2[BF4]: connectivity of Cu qhe, the, z SiO2 AlPO4 various synthetic analogues c synthetic Rb6Si10O23; some dangling ligands kun, roh, ts, wne synthetic (Na,Cs)12Al12Si36O96.44aq (Ca,etc)11Li8Be24P24O96.38aq various synthetic analogs including ECR-10, LZ-214 & beryllophosphate-R

2D units

polyhedral units

catalog number

zzt", toa'

complex

597

fee, kyo", vvv

kor, oto, phi

24

& brw, fee, ffs, afs, vvn fos, hex

6

qua", qur", toa'

-

90

rbs'

-

980

rho

grc, opr

206

Table 16.2.1 (continued). code

name

-RON

roggianite

RSN

RUB-17

RTE

RUB-3

rings

1D units commentary 3,4,6,10 kay, ts 12 Ca14NaBe5Al15Si28O90(OH)14(OH)2.34aq the original code -ROG was for an incomplete structure 3,4,5,6, bs, c, csh, kbb, kbf, kbv, kua, s, scs, tof 8,9 synthetic K4Na12Zn8Si28O72.18aq 4,5,6,8 rtf, wwv

RTH

RUB-13

4,5,6,8

RUT

RUB-10

4,5,6,8 4,5,6,8

SAL

Sr21Al36O75

3,6,8

SAO

STA-1

4,6,12

synthetic SiO2 ordered type A decasil: other ordered relatives are types B &C Type B relative of RTE is net 1183 Type C relative of RTE is net 1184 [RUB-4 is a disordered A and C decasil] rti, rtj, rtk synthetic B2Si30O64.organic in twin relation to net 1132 (RTH) is net 1182 (ITQ-3 silica) rti, complex complex synthetic B4Si32O72.organic the synthetic Sr21Al36O75 ts, complex columns not coded synthetic Mg5Al23P28O112.org.aq

2D units

polyhedral units

catalog number

uii

lau

fee lod, vps

euo, lov, sfi

1133

fee, kye”, zzy ,zzz”

rte, tte

1131

974

1183 1184 complex

cle, rth

1132

complex nkm"

cle, itq rob, rwb, tte

1182 1030

qua"

-

1193

gml, krp

afs, aww, iet, lau, ohc, sti

1245

Table 16.2.1 (continued). rings

1D units commentary 4,7,8 bs (Ca,Na)9Al4Si6O26F 4,6,8 zzl synthetic Mg-substituted AlPO4 ABC-6 family, (AABA)(BBCB)(CCAC)-rhombohedral 4,6,8,12 complex, to be coded

2D units

polyhedral units

fee

-

975

gml

can, hpr, niw

192

fee, complex

iet, knu, koj, ocn, opr, sti, uce

1250

kah, mel

1300

code

name

-SAR

sarcolite

SAT

STA-2

SBE

UCSB-8Co

SBF

UCSB-15GaGe

SBH

UCSB-3/ACP-3

SBI

UCSB-5

SBN

UCSB-9

SBS

UCSB-6GaCo

synthetic (Co/Mn/Mg/Zn)xAl1-xPO4 4,5,6,10 zzo complex synthetic (Ga/Al)GeO4.piperazine 4,6,8 brw, fee bhs, c, chf, kba, kbi, s synthetic ZnAsO4.organic, (Co,Al)PO4.organic & GaGeO4.organic -USS (ussingite) can be converted into net 38 (UCSB-3 & ACP-3) by replacing hydrogen bonded OH by 2connected O 4,6,8,10 complex fee, complex synthetic Co6Al6P12O48.(OH)2.aq.amine 4,5,6,8 complex fee, shn Ga2Ge3O10.amine 4,6,8,12 zzn ltl

UCSB-10GaZn

synthetic CoxGa1-xPO4 & various (Mg/Mn/Co/Mn/Mg)(Ga/Al) phosphates 4,6,8,12 complex, to be coded

SBT

synthetic ZnGaP2O8 & various Co/Mg/Zn Al phosphates

ltl, complex

lov, ste

catalog number

38

complex

1302

kzd, complex

1303

can, eab, hpr, iet, sti,ucs, znf

1249

can, hpr, iet, knu, sti, complex

1251

Table 16.2.1 (continued). code

name

SBU

UCSB-4

SBV

UCSB-7

SCP

scapolite

SFF

marialite meionite SSZ-44

SGT

Sigma-2

SOD

sodalite beryllosodalite bicchulite danalite hauyne helvite hydroxysodalite kamaishilite nosean sodalite (continued on p. 65)

rings

1D units commentary 4,5,6,8 complex synthetic CoAl2P3O12.amine 4,6,12 complex spiral synthetic family (K/Na/EDA)(Ga/Zn)(Ge/As).n aq complex structure best described as 3,4-square 3D spiral wrapped on isometric minimal surface 4,5,8 alh, bs, c, csh, kad, kea, s, scs mineral group with following ideal endmembers Na4Al3Si9O24Cl Ca4Al6Si6O24CO3 4,5,6,10 wai, complex to be coded synthetic SiO2 4,5,6 kfj, kfn, kfr, kid synthetic SiO2.organic clathrasil; also ZSM-58 4,6 kdd, kfb, kgj, ts Na4BeAlSi4O12Cl Ca2Al2SiO6(OH)2 (Fe,Mg,Zn)4Be3Si3O12S (Na,Ca)4-8Al6Si6O24(SO4,Cl)1-2 Mn4Be3Si3O12S various Ca2Al2SiO6(OH)2: ?distortion of bicchulite? Na8Al6Si6O24SO4 Na4Al3Si3O12Cl

2D units

polyhedral units

catalog number

hex, tth

complex

1301

complex

complex

1299

bik, fee

bru, lov, ste

100

wag

eun, nuh, pes, wah

1294

fex, kyp"

kon, sgt, sgw

580

gml, qur", sod"

toc

108

Table 16.2.1 (continued).

SRA

Sr15Al24O48(OH)6

STF

SSZ-35

STC

steacyite

STI

ASU-7 stilbite

STT

barrerite stellerite stilbite-Ca stilbite-Na SSZ-23

1D units commentary N(CH3)4Si2(Si0.5Al0.5)O6 Na4BeAlSi4O12Cl many synthetics too numerous for complete listing: includes AlPO4-20, G, TMA-sodalite, halozeotype CZX-1, & Al/Ga-Co phosphates expanded & interrupted variants have been synthesized (Table 16.2.2) 3,6,7,8 the, complex synthetic Sr15Al24O48(OH)6 4,5,6,10 wam, complex column not coded synthetic SiO2: also ITQ-9 4,6,12 ast, bre, kfz, khe, sta K(Na,Ca)2ThSi8O20 various possible mineral analogs need further evaluation synthetic GeO2.organic.water 4,5,6,8, hen, heu, kea, kfz, stl 10 Na2Al2Si7O18.6aq CaAl2Si7O18.7aq Ca4(Na,K)Al9Si27O72.28aq (Na,etc)9Al9Si27O72.28aq 4,5,6,7,9 mrt, mru

SVY

svyatoslavite

4,6,8

code

name tsaregorodtsevite tugtupite

metavariscite

rings

synthetic SiO2 atn, c, kay, kaz, thr, z CaAl2Si2O8 AlPO4.2aq; Al has 2 water ligands in addition to the 4 oxygens linked to tetrahedral P; dehydrates to AlPO4-A

2D units

polyhedral units

catalog number

qua", sal"

zlz

complex

eun, nuh, pes, wan

1295

zzx”

cub, iet, lau, sti, sty

291

brw

bru, iet, kaa, kuo, sti

603

complex

bet, eun, mrr, mrs, pes

1298

fee, hex

kaa, kom, lau

619

3

Table 16.2.1 (continued). code

name

rings

SZF

SUZ-4

4,5,6,8,10

TER

terranovaite

4,5,6,8, 10

THO

thomsonite

4,8

TON

Theta-1

5,6,10

TRI

tridymite tridymite kalsilite nepheline panunzite trikalsilite yoshiokaite

6

TSC

tschörtnerite

4,6,8

1D units commentary kcr, kng, kpk, kpp, s, szf synthetic kri,p,complex NaCaAl3Si17O40.7aq fib, kco, kcu, kfy, s Ca2NaAl5Si5O20.6aq various synthetics including Al/Ga-Co/Zn-phosphates hhz, kcg, kcj, tno, ton, z NanAlnSi1-nO2.0.1aq various synthetics including KZ-2, NU-10, Nu-15, ZSM22, ISI-1 afv, c, kbg, keg, lao, z SiO2 KAlSiO4 (Na,K)AlSiO4 (K,Na)AlSiO4 [= tetrakalsilite] (K,Na)AlSiO4 Ca5Al10Si6O32 many synthetic analogues zzk ~ K3Ca4Al12Si12O48.Cu3(OH)8.6aq

2D units

polyhedral units

catalog number

brw, fer

hpr, kdk, son, ygw

928

fer, uii, complex

bog, bru, eun, lov, pes, complex

1152

fee

des, krr, kzd

259

fer, hex

hes, kdx, lai, pes

606

hex

afi, kah, kyw

complex

grc, hpr, opr, toc, vsr

2

207

Table 16.2.1 (continued).

UTD-1-C

1D units commentary 4,6,8 c, columns of ulm cages not yet coded synthetic (FePO4)4F2.3aq.C6H14N2; extra ligands are removed from Fe to obtain the 4,2-connected tetrahedral net 4,6,8,10 c, s Na2AlSi3O8OH; interrupted, extra OH ligand on Al USS (ussingite) can be converted into net 38 (UCSB-3 & ACP-3) by replacing hydrogen bonded OH by 2connected O 4,5,6,14 bhs, c, thr, tts, ute

UTD-1-I

synthetic SiO2-organic; C-centered model that may contribute to XRPD data related to DON 4,5,6,14 afv, bhs, c, tts, vtn

code

name

ULT

ULM-12

-USS

ussingite

UTC

UTI

VAR

variscite

mansfieldite scorodite strengite yanomamite

rings

4,6

synthetic SiO2-organic; body-centered model that fits experimental X-ray powder diffraction data fairly well related c, kaz, kbp, ket AlPO4.2aq; Al has two aq as extra ligands; dehydrates to AlPO4-B AlAsO4.2aq do. FeAsO4.2aq do. FePO4.2aq do. TlAsO4.2aq do. synthetic MCM-3 containing Al, P & Si synthetic mono-CaGa2O4 synthetic InPO4.2aq

2D units

polyhedral units

catalog number

ult"

ulm

1244

usg'

-

hex, utd

bog, kah, lau, lov, oth

1161

hex, utd

afi, bog, etn, kah, lov, mel

1160

fee, fsy, hex

fsi, kaa, kaq

979

5

Table 16.2.1 (continued). code

name

rings

VET

VPI-8

5,6,7,8, 12

VFI

VPI-5

VNI

VPI-9

VSV

VPI-7

-WAD

gaultite wadalite

WEI

weinebeneite

-WEN

wenkite

YUG

yugawaralite

4,6,18

1D units commentary hhz, veu, complex synthetic SiO2 afv, bhs, c, kbg, keg, ktv, vfi, yee

2D units

polyhedral units

catalog number

vet”, complex

pes

1135

eoo, hex

afi, bog, kah, ktw, kup, kyw, loh, lov

synthetic AlPO4.organic; extra ligands on Al in assynthesized material various synthetics, including AlPO-54, MCM-9 & H1 3,4,5,8 bs, s, complex fee, complex synthetic Rb44K4Zn24Si96O240.48aq 3,4,5,8,9 bs, c, csh, kbv, s, scs, vpe, vpv fee, vps synthetic Na26H6Zn16Si56O144.44aq Na4Zn2Si7O18.5aq 8 interrupted net related to garnet; actually, the net in 12CaO.7Al2O3 has historical priority fee, sfe", 3,4,6,8, sfa, sfb, sfc, sfd CaBe3P2O8(OH)2.4aq sff", sfj" 10 4,6,8,10 ofr, ss wek, Ba4Ca6Al8Si12O39(OH)2(SO4)3.naq complex 4,5,8,10 bs, kcv, kel, kex, the, ts, z biz, fee, yug" CaAl2Si6O16.4aq synthetic Sr-Q

znb, znc, znd

520

1158

euo, kaj, sfi

860

-

986

sfg, sfi

962

can, hpr

976

kdn, ygr, ygw

630

Table 16.2.1 (continued). code

name

rings

ZON

ZAPO-M1

4,6,8

1D units commentary bre, bs, lvi, s synthetic Zn8Al24P32O128.organic synthetic AlPO4 synthetic GaPO4 Polytype A of ZON is net 1180 Polytype B of ZON is net 1181 z,complex

UiO-7 DAB-2

ZSK

K2ZnSi2O6

3,7,8

ZST

ZSM-10

synthetic K2ZnSi2O6 4,6,8,12 kbi, lel, off, ofr, ss, too synthetic ZSM-10, correct model A incorrect model B (subunits not entered in Tables) 4,6,8,12 kbi, kow, off, oon, ss, tix

(b)

2D units

polyhedral units

catalog number

brw, complex

iet, ohc, sti, zne

1179

zsk”, complex

complex

1180 1181 1157

zsm, complex

can, hpr, kaa, kno, lil, pau, ste

zsm, complex

dpr, eni, gme, kaa, kno, ocn, opr, ste

1177

1178

Miscellaneous materials, not given three-letter code; subunits may be of general interest

name

rings

NbO [AlPO-12 reconnected

6,8 4,6,8

?Ba2Fe3AlO8-domain 4,6 ?tvedalite

4,5,10

1D units commentary apt, ken, keu, kew, kyy, kyz, kza, thr one of the linkages must be reconnected to obtain CTF 398 topology] afv, c, kba, kbg, lao, z ts, zz interrupted net, proposed structure

2D units

polyhedral units

catalog number

cri" fee, fos fsy, hex

kzb, lau, vvs

91 398

ftn hex tve"

afi, fni, kah, oth -

956 989

Table 16.2.1 (continued). name

rings

?kaliophilite model

4,6

cubic-metaboric acid Zintl NaGaSn2

3,4,8 5,8

ULM-3 & SMB-6

Polymorph B

have pentahedral & octahedral Ga/Al because of Al-Al bridges. Stripped tetrahedral net assigned CTF 1213 after removal of dangles, 2-connected tetrahedra & 1 pentahedron except for 1-connected O(23). Stripped net assigned CTF 1220 4,8 (mixed square-planar & tetrahedral net). Also Cd(CN)2.2/3H2O Disordered SiO2: 8 possible models proposed of which the following four fit best with X-ray diffraction 4,5,6,12 ave, bhs, c, thr, tts brw, btb, hex 4,5,6,12 afv, ave, bhs, c, tts btb, hex

Polymorph C

4,5,6,12 ave, bhs, c, thr, tts

bta, hex

Polymorph D

4,5,6,12 afv, ave, bhs, c, tts

bta, hex

ice-XII

7,8

stn”

ULM-5

[NMe]4[CuPt(CN)4] SSZ-31 = NCL-1 Polymorph A

1D units commentary afv, kba, kbg, lao, z model fits cell dimensions, but no structure analysis c, zzm, complex 3-helix, 4-helix, 6-helix

z’, 5-double-helix hypothetical relative of ice-XII

2D units

polyhedral units

catalog number

eig, hex

afi, fni, kah, oth

1001

hbo" lvy", stn", zin"

ste, trc complex

1043 1066 1213 1220

1263

apf, bog, kaa, kah, kof, lau, lov, mel afi, apf, bog, kah, kof, lov, mel apf, bog, kaa, kah, kof, lau, lov, mel afi, apf, bog, kah, kof, lov, mel -

1282 1284 1283 1285 1292 1293

Table 16.2.1 (continued). (c)

New structures under analysis

name dual of beta-Sn Mu-2

DOH/MTN

LZ-276 & 277, Phi ISV ITQ-7 MTF MCM-35 Al(PO4).C24H91N16.17H2O SSZ-45 ASU-7 ND-1 ULM-6 MCM-61 ZSM-48

rings

1D units 2D units commentary 3,5,6,10 zmb, zmc 4,8, [12] Ga32P32O120(OH)16F6.organic.12aq: complex interrupted net based on Ga4P4(OH)2 cubes 1:1 DOH/MTN polytype of 10.67 choline hydroxide.tetran-propylammonium fluoride.30 water supertetrahedral sulfides: fragments of net 1 (analogous to diamond and blende) sharing some vertices to generate interrupted nets; under topologic analysis faulted ABC-hpr synthetic large-pore silica with 3D channel system; similar to Beta but contains cub synthetic silica; 1D-channel spanned by 8-ring synthetic 1,2-connected-tetrahedral net synthetic silica; 1D-channel spanned by 12-ring; related to TON synthetic Ge10O20.diethylamine.H2O synthetic Zn3(PO4)2(PO3OH).organic.H2O synthetic Al4(PO4)4F2.organic.H2O synthetic clathrate; (Si,Al)90O180K6.organic synthetic; structure uncertain; may be some polytypic combination of fer 2D net linked up (U) & down (D)

polyhedral units

catalog number

zma cub, iet, sti

1304 1305

1308

1312 1316 1315 1318 1319 1320 1321 1322 ????

62

16.2 Systematic enumeration of subunits

[Ref. p. 251

Table 16.2.2 References to nets in table 16.2.1. Covers nets not listed by the Structure Commission of the International Zeolite Association and recent structure determinations of nets listed by the IZA-SC. Updated to December 31, 1999. code

name

catalog number

reference

ACJ AEN AEN AFC AFX AHF -AJT ALS ANA ATN BAF BAN -BAV BCG BNT BPT BRL CBA CBU CGS CGS CLA CLB CLC CLD CLE -CLF COE COO CRI ECA ECB ESV FRA FRA FSP GST ISV ITE KAG

AlPO-CJ2 AlPO-53B AlPO-53B AlPO-53C SAPO-56 AlPO-H4 AlPO-JDF-20 AlAsO4.org CsTiSi2O6.5 Al7MgHP8O32 BaFe2O4 banalsite bavenite BaCaGa4O8 beryllonite BeP2O6-II beryl CaB2O4(IV) CuB2O4 MeGaPO-6 TSG-1 clathrate clathrate clathrate clathrate clathrate clathrate coesite SSZ-26,33A cristobalite ECR-1-A ECR-1-B ERS-7 franzinite franzinite feldspar GaAsO4-2 ITQ-7 SSZ-42 KAlGeO4

725 402 402 1246 1134 1031 984 963 593 93 470 43 977 539 9 609 278 853 1055 1290 1290 966 967 968 969 970 971 598 1005 1 575 601 1296 201 201 26 748 1312 1248 538

93Fér1 99Kir1 99Kon1 99Kir1 99Wil1 90Gab1 93Jon1 91Li1 97McC1 99Bau2 71Mit1 73Hag1 66Can1 99Kub1 73Giu1 77Ave1 95Art1 91Ros1 71Mar1 99Cow1 99Lee1 82Sol1 62Bon1 70McM1 93Fei1 67McM1 66McM1 87Smy1 93Lob1 85Plu1 87Leo1 87Leo1 99Mil1 77Mer1 79Rin1 88Smi2 89Che1 99Vil1 99Che1 87Bar1

Landolt-Börnstein New Series IV/14

Ref. p. 261]

16.2 Systematic enumeration of subunits

63

Table 16.2.2 (continued). code

name

catalog number

reference

KBG KEA KGE KPO KZP KZS LCP -LEI -MAR MEP MEP MGN MTF MTF MTN [MWW} NU-86(A) NU-86(B) NU-86(C) OSU PET PRC QTZ RBS SAL -SAR SAT SCP SFF SOD [SOD expanded [SOD interrupted SRA STC STC STF STF SVY TRI ULT -USS UTC UTI VAR VAR -WAD

KBGe2O6 keatite K2Ge8O16O1 kaliophilite-O1 KZnPO4 K2ZnSi4O10 leucophosphite leifite maricopaite Zintl Metal8Si46 Cs8In8Ge38 moganite MCM-35 UTM-1 Zintl MetalxSi136 ITQ-1

osumilite petalite paracelsian quartz Rb6Si10O23 Sr21Al36O75 sarcolite STA-2 scapolite SSZ-44 hauyne

749 596 720 536 10 751 400 978 987 595 595 282 1317 1317 592 1101 873 874 875 279 597 6 90 980 1193 975 192 100 1294 108

Sr15Al24O48(OH)6 steacyite GeO2.organic.water ITQ-9 SSZ-35 svyatoslavite tridymite ULM-12 ussingite UTD-1-C UTD-1-I variscite variscite wadalite

619 291 291 1295 1295 3 2 1244 979 1161 1160 5 5 986

86Kla1 59Shr1 73Fay1 83Mer1 89Yak1 90Koh1 72Moo1 74Cod1 94Rou1 98Ren1 99Men1 92Mie1 99Bar1 99Ple1 98Ren1 99Njo1 93Sha1 93Sha1 93Sha1 88Arm1 82Tag1 85Chi1 82Lag1 73Sch1 78Nev1 87Giu1 97Lob1 75Lin1 99Wag1 97Sap1 95Fen1] 95Fen1] 82Den1 72Ric1 98Li1 98Vil1 99Wag1 73Tak1 77Bau1 96Cav1 74Ros1 97Lob1 97Lob1 94Bot1 98Tan1 93Tsu1

Landolt-Börnstein New Series IV/14

64

16.2 Systematic enumeration of subunits

[Ref. p. 251

Table 16.2.2 (continued). code

name

catalog number

reference

ZSK ZST

K2ZnSi2O6 ZSM-10 A B ZnAsO4.org (Al0.15Co0.85)PO4.org A B C D

1157 1177 1178 38 38 1282 1284 1283 1285 91 398 956 989 1043 1066 1182 1213 1220

95Hog1 96Hig1 96Hig1 98Bu1 99Xu1 97Lo2 97Lo2 97Lo2 97Lo2 75Wel1 84Par2 79Har1 92Lar1 75Wel1 96Vau1 97Cam1 96Loi1 94Loi1

[NMe]4[CuPt(CN)4] & Cd(CN)2.2/3aq ice-XII

1263

91Nis1

1292

98O’Ke3

supertetrahedral sulfides supertetrahedral sulfides UCSB-4 UCSB-5 UCSB-9 Mu-2 DOH/MTN Al9(PO4)12.C24H91N16.17H2O SSZ-48 ASU-7 ND-1 ULM-6 MCM-61 ZSM-48

1301 1302 1303 1305 1308 1315 1318 1319 1320 1321 1322 ???

99Li1 99Li2 99Bu1 99Bu1 98Bu3 98Rei1 99Uda1 99Xu2 99Wag2 99O’Ke2 99Yan1 99Sim1 99Sha1 85Sch1

UCSB-3 ACP-3 SSZ-31 = NCL-1

NbO AlPO-12 reconnect ?Ba2Fe3AlO8-domain ?tvedalite cubic-metaboric acid Zintl NaGaSn2 ITQ-3 silica ULM-3 & SMB-6 ULM-5

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

65

16.3 Polyhedral units The following commentary is designed to help non-mathematicians understand the subtleties of the topologic features of the polyhedra described in Tables 16.3.1 and 16.3.2 and Fig. 16.3.1. The first analyses of 3D nets yielded polyhedra whose vertices are at the intersection of three edges and three faces (e.g. Fig. 2, the polyhedra in FAU, LTA and SOD). For two decades, this feature was retained in analyses on nets in zeolites, but became too restrictive. In five weeks during summer 1998, I had to write a review of the topology of zeolites and related materials for a special issue of Chemical Reviews [88Smi1]. I noticed that some of the cages were best described by polyhedra in which some vertices were 3-connected , but others were 2-connected (Figs. 3 and 4 therein). As time became available during the next three years, a systematic analysis began to yield many 3,2-connected polyhedral units: e.g. afi, afs, apf, baf & ber on the first two pages of the present Fig. 16.3.1. Subsequent study demonstrated that the original focus on 3-connected closed polyhedra was quite inadequate, and that other polyhedral units should be considered. Further, many simple units could be combined into larger units. The Series Editor, Reinhard X. Fischer, kindly supplied some concepts that are presented here in somewhat different words for initial discussion. -

A polyhedral unit is a ‘body’ consisting of n-membered rings of vertices which represent positions of T (= tetrahedral) atoms.

-

Polyhedral units may be open or closed. Closed polyhedral units consist of faces of n-membered rings enclosing voids or cavities.

-

Polyhedral units with all rings containing up to 6 vertices are cages, as proposed by Friedrich Liebau. [This is based on the observation that 3-, 4- and 5-rings of oxygen atoms in zeolites are impermeable to almost all molecules under most conditions. Even 6-rings are impermeable to most molecules.]

-

As an example, Fischer suggests that vvs might not be considered as a polyhedral unit because it consists of a kdq with an additional pair of 4-rings that spans an 8-ring. However, I prefer to list vvs as well as kdq, partly because vvs is a more elegant unit with an edge-sharing double-4 at both east and west. Furthermore, during synthesis the possibility of kdq evolving into vvs should be considered.

In general, I agree that mathematical rigor is desirable in selection and definition of polyhedral units. In practice, as of today, I take a very broad view that any unit that might be useful to a synthesis chemist should be listed here. Hence, the units used here may prove to be an inconsistent bouillabaise, that will be adjusted into a more logical set after further thinking. The polyhedral unit afi (page 103) is described as the 1,3,5-open hexagonal prism (labeled hpr; page 111). The vertical edges of hpr can be numbered 1 through 6, and alternate ones at positions 1, 3 and 5 have been removed to generate the afi unit. The resulting 2-connected vertices are marked with a filled circle to distinguish them from the 3-connected vertices at the 2, 4 and 6 edges. Whereas hpr is described as 4662, with six congruent square faces and two congruent hexagonal faces, its derivative afi becomes 6362, with three vertical hexagonal faces and two horizontal hexagonal faces. Of course, the three vertices marked by filled circles are not restricted in position in contrast to the corresponding vertices in hpr. In general, 2-connected vertices tend to be further apart than 3-connected ones. The drawing of afi shows the typical positions for the vertices in the nets listed in Tables 16.3.1 and 16.3.2.

Landolt-Börnstein New Series IV/14

66

16.3 Polyhedral units

[Ref. p. 251

It is very important for an understanding of the topological relations among polyhedra to distinguish between the two topologic types of 6-rings in afi, and not combine them into 65. Even so, specifying the different types of faces does not always provide a unique characterization, as shown by the pairs lov/krb, koh/kyu, kam/kdq and lau/bog. Addition of the vertex symbol [82Smi1] to the face symbol does provide a distinction. Thus lov (page 126) has two types of congruent vertices, four at 4.6.6 and four at 4.6 in contrast to krb (page 122) with four distinct pairs of vertices. Simply for brevity, just the face symbol is given in Tables 16.3.1 and 16.3.2, and arbitrary labels -a and -b are used to distinguish members of the listed pairs. The afi unit can also be described as the hexa-stellated trigonal prism. The (regular-) trigonal prism has two triangular faces connected by three square faces. This generates six congruent edges. Stellation of an edge means that it is replaced by two edges connected by a 2-connected vertex. The six vertices of the trigonal prism remain 3-connected. Replacement of a single edge by three edges and two 2-connected vertices generates a handle, as exemplified by bog (page 105), which is the 1,3-handle cube (cub). The bog unit transforms into the bru unit (1,3-stellated cube) by replacing each of the two handles with a stellation. The afs unit (page 103) is described as the 1,2-open hexagonal prism in Tables 2 and 3, but can also be derived from a cube. Double-stellation is exemplified by can (page 105), the 1,2-3,4-5,6-double-stellated hexagonal prism, and the double-handle is illustrated by aww (page 104), the 1,2-double-handle hexagonal prism. The new vertices in both operations become 3-connected because of the cross-linkage of the double units. Although aww is perhaps most easily seen as the derivative of a hexagonal prism, it can also be derived from a cube by conversion of two pairs of adjacent edges. Systematic enumeration of all the ways of converting one or more edges of a cube, tetrahedron and hexagonal prism by opening, stellation or creation of a handle was done by Koen Andries [94And 1]. This pioneering study should be extended systematically to other polyhedra. Vertex truncation is illustrated by cla (page 106), the 2-vertex-truncated symmetrical trapezohedron. The square faces were obtained by slicing off the top and bottom of a regular trapezohedron consisting of eight trapeziums meeting at two vertices equidistant from a zig-zag waist. The shape of the original trapeziums, and the extent of the truncation, can be adjusted to generate the near-regular pentagons in the drawing. Some polyhedra are not easy to describe. The polyhedral unit clb (page 106) has point symmetry 2 6 m. 6 is the international crystallographic code for a triad rotation axis perpendicular to a mirror plane = 3/m: commonly it is printed as 6 with an overbar. The drawing is deliberately arranged with the triad axis north-south in the plane of the paper, and the mirror plane east-west near-perpendicular to the paper. Each of the two vertices on the triad axis lies at the intersection of three pentagonal faces and three edges. The north and south vertices on the triad axis are linked by the sequence pentagon/hexagon/pentagon alternating three times with edge/pentagon/pentagon/edge, or briefly 5,6,5/e,5,5,e. Obviously, an infinite series of polyhedra having two vertices suspended by polygons and edges should be enumerated. The cle polyhedral unit (page 106) is the dual (= reciprocal polyhedron) of the triangulated dodecahedron, otherwise denoted bisdisphenoid ([82Smi1], Fig. 3.16c therein). This means that each vertex of one polyhedron is related to a face of its reciprocal polyhedron, and vice versa: i.e., the line segment joining the vertex to the centroid of one polyhedron is perpendicular to the face of the other. Some of the polyhedral units are composed of two or more simpler polyhedral units. Thus dah (page 106) is described as ste (page 134) sharing each of its two octagons with zlt (page 139). An important unanswered mathematical question is how many of such multiple units should be listed. The only practical answer that I can give at this time is that a multiple unit is described if it appears useful in characterizing a cage, window or channel important for adsorption or diffusion. Because the dah unit characterizes a valuable cage in the 3D nets DAC, MOR and 575, it was included. I rechecked all 3D nets specifically in June 1999 to reach near-consistency of choice. However, I readily confess that the choice between inclusion or rejection of a particular multiple unit needs to be put on an objective basis programmable by an electronic computer.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

67

The tetra-stellated cube, coded as eun (page 108), can be expressed as two pes (page 131) sharing a hexagon. Because it is related so neatly to a cube, it is important to use eun as well as pes. The tri-edge-stellated tetrahedron, iet (page 111), can also be described as the “capped hexagon”, where capping means that the uppermost vertex is the cap to alternate vertices of the crinkled hexagon. The kah unit (page 112) can be described as two vertices linked in trigonal geometry by three handles. Removal of three of the twelve edges of the cube can generate the connectivity of kah, and appropriate deformation of T-T-T angles produces the neat geometry of kah. The easiest way to see these relations is to build a model with plastic stars and tubes. Pairs of polyhedral units may be related by a sigma transformation (i.e. a mirror operation), [73Sho1]. An example is kdq (page 114) which is related to its parent hes (page 110) by extra vertices and edges. Again, most students prefer to build star-tube models in order to visualize the mirror relationship. I have not had time yet to systematically examine the present polyhedral units for sigma relationships, nor have I examined them systematically to generate new units of theoretical interest. The units gme (page 110) and plg (page 132) are related by a rotation of the upper and lower parts that enforces 2-connected vertices in plg. The units pau (page 131) and rpa (page 132) also have a rotation relationship, but this results here in only a change of polygons in the waist. No systematic search has been made yet for other pairs of rotated relatives. To conclude this section, many of the new topologic types of polyhedral units are based on evolutions of the fully regular Platonic solids and the semiregular Archimedean and Catalan polyhedra ([82Smi1], Table 3.1 therein on p. 78), by various geometrical changes. These processes include the use of trestles and arches not described above, but expressed in Tables 16.3.1 and 16.3.2 for several polyhedra and corresponding drawings in Figure 16.3.1 (trestle, kaa, kaj, kum, kuo; arch, kjr). The scene is set for further mathematical advances that should lead to systematic enumeration of several further classes of polyhedra following the general procedure in [94And1]. It is truly satisfying that topologic study of polyhedra taken from nets in actual crystal structures is leading to advances in pure mathematics, and that theoretical polyhedra have subsequently been used to derive new 3D nets, some of which have turned up in new structures. The present set of polyhedra is strictly limited to those abstracted from structures of known materials. Many more polyhedra have been found in theoretical 3D nets generated by enumerations based on algorithms, and are being prepared for publication in peer-reviewed journals.

Landolt-Börnstein New Series IV/14

68

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.1 Polyhedral subunits in increasing order of face-symbol. The face symbol lists the order (number of edges = number of vertices) of each topologically-distinct set of faces, with the multiplicity (number of faces) as superscript. The face need not be a planar or regular polygon. Each face symbol is arranged in increasing order of polygon type, and then in decreasing order of multiplicity. The label is a convenient three-letter acronym given in lower case. The point group of symmetry uses the International Crystallographic System; inverse axes are written with a bar. The description has two parts: a brief topological statement of essential features, and a list of all the 3D nets from observed structures (Table 16.2.1) that contain the unit. Nets not assigned a three-letter code by the IZA-SC are denoted with the chronological sequence number of the Consortium for Theoretical Frameworks. Table 16.3.2 contains identical entries arranged in alphabetical sequence of label. Updated December 19, 1999. face symbol

label

point group

description

3456

tfs

1

31415251 314353 3151

znc mei shf

m 3m1 2

3153

znb

3m1

hexagonal base linked to an apex from the 1,2,4 vertices, net 751; not drawn euo and zna sharing 5, VNI 1-vertex-truncated cube, MEI spiro-3,5 = 3- & 5- rings sharing vertex, net 1055 tri-stellated tetrahedron, VNI

2

3 324161 324162 325282

sfi oot itp tfe

42 m mm2 mm2 2

spiro-5, LOV, RSN, VSV, WEI 1-open trigonal prism, ITE 1-handle trigonal prism, ITE edge-shared-3-5-5-3 suspended by handle between 8 rings, net 1055

3276-a

dum

32

six-edge stellated, 1,1'-vertex-truncated rhombohedron, net 853

3276-b

due

32

six-edge stellated, 1,1'-vertex-truncated trigonal trapezohedron, net 853

34426282102

sfg

2/m

1,5-open,4,4',8,8'-vertex-truncated octagonal prism, WEI, net 963

3484

znd

42 m

two-handle derivative of all-vertextruncated tetrahedron, VNI

3692

zlz

32

three sti sharing three vertices around 3, net 619

384264104

zma

42 m

vertex-truncated aww, net 1304

trc

4/m 3 m

vertex-truncated cube, net 1043

1 1 1 1

8 6

38

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

69

Table 16.3.1 (continued). face symbol

label

point group

description

4444445555566667799 mrr 4141414162626261616161 kdi

1 m

4141416262616181 4152

kdj euo

m mm2

41525252

nuh

mm2

41525262 41525281101101

koc kuh

mm2 m

415262

mel

mm2

41526262

kof

mm2

415282

kdk

mm2

41545252

non

mm2

415481 4161618281 41626181

kon kdl kaq

4mm m m

41628281 4164121 418281102 4241414161618281

kal knw kor kdm

mm2 4mm mm2 m

42414152525252626261101101

wah

m

4241415252526261616181 424141626282 4241416282 424152515161 42415252

rob vvn tti mrs bet

m mm2 mm2 m mm2

complex 19-hedron, STT complex, knv sharing 8 with two kaq, net 536 complex, kzc sharing 8 with kaq, net 536 4-ring with handle bridging opposite vertices, EUO, MON, NES, RSN, VSV 1,3-stellated pentagonal prism, ISV, NU-86, nets 873-875 1,2,3-stellated pentagonal prism, EUO 1',3',4",6"-stellated-5'5"-handle hexagonal prism, MEL, MFI 1,3-open pentagonal prism, CON, DON, ISV, MEL, MFI, MWW, nets 873-5, 1005, 1160, 1282-1285, 1300 1,3-handle pentagonal prism, MFI, ISV, nets 873-5, 1282-1285 1,3-open,2-handle pentagonal prism, MFS, SZF koc with two vertices bridging twoconnected nodes, EUO, NES, NON 1',2',3',4'-stellated cube, SGT 1,3,5,6-open octagonal prism, APD 1,2,4-open hexagonal prism, AFS, nets 5, 9, 10, 536 1,2,4,7-open octagonal prism, net 720 1',2',3',4'-handle cube, AST 1,2,3,4,2',4'-handle cube, PHI 1,5-open,2,3-double-handle octagonal prism, APC 10-prism altered to upper-45666665, lower-6545545545, SFF convex half of rwb cage, RUT kal sharing 8 with afs, net 6 1,4-open octagonal prism, AFR complex 8-hedron, STT 1-stellated pentagonal prism, BEA*, STT, nets 577-9, 873-5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Landolt-Börnstein New Series IV/14

70

16.3 Polyhedral units

Table 16.3.1 (continued). face symbol

[Ref. p. 251

label

point group

description

2 1 2 1

4456

wwt

mm2

42415262

wwf

mm2

424162

gos

2

42416281

afs

mm2

424241616181

kqr

m

4242416181

zna

m

4242418281 42424241416261616182 4242424164626281 4242426262818181 4242546261 42425462626182 424261

oto ulm ltn zne mim mil sti

mm2 m mm2 m mm2 mm2 mm2

424262-a 424262-b

ohc kyf

mm2 2

42426261-a

baf

mm2

42426261-b

kre

2

424264626282 42426482

kzb vvs

2/m 2/m

42526262-a

hsp

mm2

1-open pentagonal prism, CON, DON, IFR, net 1005, part of bet, BEA*, STT, nets 577-9, 873-5, part of wwf, CON, DON, IFR, nets 873, 874, 1005 1-handle pentagonal prism, CON, DON, IFR, nets 873, 874, 1005 1,5,6'-edge-stellated trigonal prism, GOO 1,2-open hexagonal prism, AFS, AWO, BPH, SAO, net 6 1-handle hexagonal prism, handle replaces edge of 6 not 4, CGS 1-open hexagonal prism, edge removed from 6 not 4, CGS 1,2-open octagonal prism, PAU, PHI complex, net 1244 five-double-handle cube, LTN complex, ZON complex, ESV, net 1297 complex, ESV, net 1297 1-open cube, SBE, SBS, SBT, STI, ZON, net 748 part of cub, ACO, AFY, AST, -CLO, DFO, ISV, LTA, nets 291, 875, 1305, part of ohc, AFR, SAO, ZON, net 720 1-handle cube, AFR, SAO, ZON, net 720 one 4 of ppr replaced by diagonal edge, net 963 1-open hexagonal prism, net 470 part of hpr extra edge joining a vertex from the two 4 of lau, net 725 kzc/kaa/kzc sharing 8, net 398 1,2-4,5-double-stellated,1",2',4',5"stellated hexagonal prism, = two kdq sharing 8, ABW, JBW, net 398 1,2-double-stellated,3,4,5-stellated pentagonal prism, CFI

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

71

Table 16.3.1 (continued). face symbol

label

point group

description

2 2 2 2

4 5 6 6 -b

nug

mm2

425281-a 425281-b 4254

kdn koi bru

mm2 mm2 mmm

42545464 42545482

stv dah

mmm mmm

425461101

knq

mm2

425462

mtw

mmm

4254626182 42546282

kdo mrd

mm2 mmm

425482-a

ygw

2/m

1-stellated,4-handle hexagonal prism, nets 873- 875 1-open,2,4-stellated cube, YUG 1',1"-handle trigonal prism, BOG, MAZ 1,3-stellated cube, BOG, BRE, CON, HEU, STI, TER, nets 100, 1005 beta-tetrakaidecahedron, net 968 ste sharing 8 with two zlt, DAC, MOR, net 575 1',2',4',5'-stellated hexagonal prism, DOH 1,4-stellated hexagonal prism, BEA*, MTW, nets 577-9, 873-5 kqc sharing 8 with two kdk, MFS 1,2,5,6-stellated octagonal prism, MOR, net 601 7,9,10',12'-stellated hexagonal prism, SZF, YUG

4258

cla

82 m

4258545484

ktg

mmm

4261616172 4262-a

kdp lov

m mmm

Landolt-Börnstein New Series IV/14

2-vertex-truncated symmetrical trapezohedron, net 966 4.[5-5/5-5/5-5/5-5](8-8).4 with 2-connected vertex at outer 5/5, EPI five-stellated mei, net 749 1,3-open cube, part of bog, AEL, AET, AFI, AFO, AHT, ATV, BOG, CGF, DFO, DFT, LAU, TER, VFI, nets 43, 1160, 1161, 1282-1285, part of bru, BOG, BRE, CON, HEU, STI, TER, nets 100, 1005, part of cub, ACO, AFY, AST, -CLO, DFO, LTA, ISV, nets 291, 875, 1305, part of kdu, APD, part of ktw, AET, AHT, DFT, VFI, part of loh, AEL, AET, AFI, AFO, AHT, APD, ATV, DFT, VFI, part of ohc, AFR, SAO, ZON, net 720 part of sti, DFO, SBE, SBS, SBT, STI, ZON, nets 720, 748

72

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.1 (continued). face symbol

label

point group

description

2 2

4 6 -b

krb

2

426261

loh

mm2

426262

oth

mm2

42626262-a 42626262-b

koh kyu

mm2 2

42626282 426282-a 426282-b

knv kam kdq

mm2 mm2 mm2

4262101

knn

mm2

4264-a

lau

4/m m m

4264-b

bog

mmm

426461141 426462

kop kab

mm2 mmm

4264626161

ten

mm2

42646262

fsi

2/m

4264102

ftt

mmm

two-stellated,one-handle tetrahedron, net 725 1-open,3-handle cube, AEL, AET, AFI, AFO, AHT, APD, ATV, DFT, VFI 1,3-open hexagonal prism, ATS, OSI, nets 539, 956, 1001, 1161 1,3-handle hexagonal prism, ATS, ISV 6 sharing edges with projecting 4 at opposite ends, AEN 1,2,3-handle hexagonal prism, net 536 1,2,3-handle cube, net 720 1,3-open,2-handle hexagonal prism, = sigma of hes, ABW, ATT, JBW, net 536 1,2,3-open hexagonal prism, ABW, net 538 1,2,3,4-stellated cube = 1,4-open hexagonal prism, ATO, CON, IFR, ISV, LAU, MSO, OSI, -RON, SAO, nets 3, 291, 398, 1005, 1161, 1282, 1283 1,3-handle cube, AEL, AET, AFI, AFO, AHT, ATV, BOG, CGF, DFO, DFT, LAU, TER, VFI, nets 43, 1160, 1161, 1282-1285 1',2',4',5'-handle hexagonal prism, AST 1,4-handle hexagonal prism, ATO, CON, DFO, IFR, MSO, net 1005 kaa sharing 8 with two mirror-related kaq, net 10 kaa sharing 8 with two inversion-related kaq, nets 5, 9 1,3,6,8-open-decagonal prism, = 1',2',4',5',1",2",4",5"-stellated hexagonal prism, DFO

4264104

kdr

42m

4282-a

kds

mmm

4282-b 4284-a

kdt ste

mm2 4/m m m

gsm with stellation of all eight inclined edges of 4, MEL 1,3-open,2,4-stellated cube, ANA, net 282 1,3-open,2-handle cube, net 748 1,2,3,4-handle cube = 1,2,5,6-open octagonal prism, ACO, DAC, LOV, LTL, MER, nets 38, 100, 575, 601, 1043, 1177

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

73

Table 16.3.1 (continued). face symbol

label

point group

description

2 4

kjr

mmm

1,3-stellated,2,4-arch cube, net 400

3

kzd

62 m

43434363616183

lev

3m1

43436183121

zlx

3m1

43535353616173

clf

3m1

43535361

det

3m1

43536191

knz

3m1

1,2,3-open trigonal bipyramid, EDI, MWW, NAT, THO, net 1303 in AABCCABBC-6 sequence, 446/84, LEV three edges of 6 in can replaced by handle, CHA opposing rotated-6 with edge-7/5-edgeedge/edge-5-4, clathrate, net 970 opposing vertex & 6 with edge-5/5-4, DDR 1',3',5'-stellated hexagonal prism , DDR

45

sgt

62 m

435653536183

dtr

3m1

435663

doo

62 m

4361

iet

3m1

436361121

knx

3m1

osu

62 m

kra

3m1

kno

62 m

4 8 -b 4

3 6

3 3 2

469

4383 3 3

4 8 12

2

Landolt-Börnstein New Series IV/14

1,2,3-vertex-truncated trigonal bipyramid, SGT opposing vertex & 6 connected e-8-e/ 5-4-5-e, DDR opposing vertices connected edge-6-edge/5-4-5, DOH, MWW tri-edge-stellated tetrahedron, "capped hexagon", net 400, part of afo, AFS,BPH, part of cub, ACO, AFY, AST, -CLO, DFO, ISV, LTA, nets 291, 875, 1305, part of krs, AFN, part of mei, MEI, part of ohc, AFR, SAO, ZON, net 720 part of sti, DFO, SBE, SBS, SBT, STI, ZON, nets 720, 748 1',3',5'-handle hexagonal prism, EAB 1,4,7-open nonagonal prism, = 1',3',5',1",3",5"-stellated hexagonal prism, net 279 three-handle cube, net 400 1',3',5',1",3",5"-handle hexagonal prism, = 1,2,5,6,9,10-open dodecagonal prism, net 1177, 1178 part of aft, AFT, AFX, part of gme, EAB, GME, MAZ, OFF, nets 575, 601 part of bph, AFS, BPH, part of evh, DFO

74

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.1 (continued). face symbol

label

point group

description

des

42m

444241828281

phi

mm2

44424242626284 44424254545464626262

apc rwb

2/m 2/m

444254546262828281

itq

mm2

444254646282

rte

2/m

44425454648282

rth

2/m

44426182

di-edge-stellated tetrahedron, EDI, NAT, THO, part of hes and pes 1,2-3,4-double-handle,5,8-open octagonal prism, PAU, PHI two kdm sharing 8, APC two rob with inversion at center of shared 8, 4-connected & concave at 8, RUT two 8 suspended by 48/554/66/554/48/554/66/554, net 1182 two 8-rings suspended by 45646546/65464564, RTE two 8-rings suspended by 45648465/84654564 & four 5 in waist, RTH one-handle afo, AFN

4

4

krs

mm2

4 2 4

446

aww

42m

1,2-double-handle hexagonal prism, = di-double-handle cube, AWW, SAO

444284-a

gsm

42m

444284-b 444288

krz kry

2/m 4/m m m

444442414162828282

per

mm2

444442646282122

afr

mm2

4444446282 444482

klf kuo

2/m mmm

1,2-double-handle,3,8-open octagonal prism, = sigma of kdq, ATT, GIS two kra sharing 4, net 400 eight-stellated truncated octahedron, AFN mono-double-handle,tetra-doublestellated truncated octahedron, AEI two 12 linked by (8/edge-4/6-4/4-6/64/edge-4)2, AFR two krs sharing 8, AFN sti-kaa-sti sharing 6, = two-trestle hexagonal prism, STI

4454

cle

42m

4454545462102

wan

2/m

445462

tte

mmm

445484

ygr

mmm

4461101

koj

mm2

dual of triangulated dodecahedron = didouble-stellated cube, RTH, nets 970, 1182 10-rings suspended by upper665455456/lower-545666545, STF 1,2-4,5-double-stellated hexagonal prism, RTE, RUT two mirror-related kdn joined by two edges & two handles, YUG 1',4'-open hexagonal prism, ATN,SBE

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

75

Table 16.3.1 (continued). face symbol 4 2 1

4 6 6 10

1

label

point group

description

knr

mm2

1',4'-handle hexagonal prism, CAN

4 2 2 1

4666

ktw

mm2

bog/loh sharing 6, AET, AHT, DFT, VFI

4 2 2

4 6 8 -a

xvi

mmm

1,5-open octagonal prism, ITE

4 2 2

4 6 8 -b

knu

2/m

446462

xib

mmm

446482161 446482 4464122

zlw kdu sty

4mm mmm 4/m m m

4482 4482102

kov kog

2/m mmm

1',4"-handle hexagonal prism, AFN, SBE, SBT 1,4-stellated,2,3-5,6-double-stellated hexagonal prism, net 1246 1',3',5',7'-handle octagonal prism, AWW kaa sharing 6 with two loh, APD 1,4,7,10-open dodecagonal prism, = eight-stellated octagonal prism, net 291 1',4"-open hexagonal prism, net 748 1',1",4',4"-handle hexagonal prism, = 1,2,6,7-open decagonal prism, net 748

46

cub

4/m 3 m

cube, ACO, AFY, AST, -CLO, DFO, LTA, ISV, nets 291, 875, 1305

46434366122

bph

62 m

1,2-5,6-9,10-double-handle dodecagonal prism, AFS, BPH, composite of kno and three afs

464356122

maz

62 m

1,2-5,6-9,10-double-stellated dodecagonal prism, MAZ, composite of kno and three koi

46436283

gme

62 m

1,2-3,4-5,6-double-handle hexagonal prism, = rotated truncated octahedron, AFT, AFX, EAB, GME, MAZ, OFF, nets 575, 601, 1178

46436286123

znf

62 m

twelve-stellated eab, SBS

6 3 6 2 3

44668

eab

62 m

ABCCBA-6 sequence, 484/646, EAB, SBS

4643123122

ucs

62 m

six-handle gme, SBS

aft

62 m

in ABBCCBBA-6 sequence, 44844/848, sigma(h)-eri, AFT, AFX

wof

62 m

tri-double-handle relative of gme, EMT

6 6 3 2 6 3

444688

46464362123 6 6 6 3 3

4 4 5 7 7 12

2

meg

6

complex, MEI

6 6 2 6

4468

cha

32

in ABBCCA-6 sequence, 448/844, AFT, CHA

4646636286

eri

62 m

in ABBCBBA-6 sequence, 44644/88, ERI

Landolt-Börnstein New Series IV/14

76

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.1 (continued). face symbol

label

point group

description

4 4 6 6 8 -a

niw

32

4662

hpr

6/m m m

466283

krf

32

in ABBCBCCA-6 sequence, 4468/8644, SAT hexagonal prism, AEI, AFT, AFX, CHA, EAB, EMT, ERI, FAU, GME, KFI, LEV, LTL, LTN, MSO, MWW, OFF, SAT, SBS, SBT, SZF, TSC, -WEN, nets 279, 968, 1177 hexa-edge-stellated vertex-truncated trigonal prism, ANA

466286

plg

32

rotated-gme, = hexa-stellated truncated octahedron, PAU

4663

afo

62 m

double-capped afi, = two iet joined by three edges, AFS, BPH

466362

can

62 m

1,2-3,4-5,6-double-stellated hexagonal prism, AFG, CAN, ERI, LIO, LOS, LTL, LTN, OFF, SAT, SBS, SBT, -WEN, nets 201, 1177

466663

kag

62 m

three 1,2,3-open hexagonal prism sharing 2 edges per pair, net 538

46666362

los

62 m

in ABCBA-6 sequence, 464/66, LIO, LOS, net 201

4666666362

lio

62 m

in ABCBCBA-6 sequence, 4664/666, AFG, LIO

4668

toc

4/m 3 m

truncated octahedron, EMT, FAU, LTA, LTN, SOD, TSC, net 201

46612 466126662

trd znl

4/m 3 m 6/m m m

truncated rhombic dodecahedron, AST top and bottom 6 joined by vertical 6-4 6/edge-6-edge, MSO

46103122

kze

62 m

48448482

pau

4/m m m

48448882124

uce

4/m m m

486482

ocn

4/m m m

1-2,3-4,5-6 double-stellated, 2',4',6',2",4",6"- handle hexagonal prism, DFO 1,2-3,4-5,6-7,8-double-handle octagonal prism, KFI, MER, PAU, net 1177 sixteeen-edge-stellated grc, = top & bottom-8 with alternating 848/4.12.4 sides, SBE 1,2-3,4-5,6-7,8-double-stellated octagonal prism, tetragonal analog of can, ATN, SBE, net 1178

6 6 6 2 6

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

77

Table 16.3.1 (continued). face symbol

label

point group

description

468

rpa

82 m

4882

opr

8/m m m

rotated relative of pau, = tetragonal analog of toc, AWW, -CLO octagonal prism, MER, PAU, RHO, SBE, TSC, net 1178

wou

62 m

hexa-double-handle relative of eab, EMT

fau

43m

4124661286122

evh

6/m m m

hexa-double-handle relative of truncated octahedron, FAU edge- & vertex-truncated hexagonal prism, 6-fold analog of eab, DFO

412468383122

bpa

62 m

hexa-double-handle relative of gme, BPH

4 4 8 12 -a

afe

32

4124686122-b

lil

6/m m m

41266122

eni

6/m m m

hexa-double-handle truncated octahedron, rotated relative of bpa, AFS, AFY 1,2-3,4-5,6-7,8-9,10-11,12-doublehandle dodecagonal prism, LTL, net 1177 1,2-3,4-5,6-7,8-9,10-11,12-doublestellated dodecagonal prism, DFO, net 1178

4126886

grc

4/m 3 m

great rhombicuboctahedron, -CLO, KFI, LTA, LTN, PAU, RHO, TSC

4246881286

vsr

4/m 3 m

52525261 525281

kod zlt

mm2 mm2

526181 5262

koa pes

mm2 mm2

52626161 526281

kot koe

m mm2

528281

kaj

mm2

536391121

koo

3m

vertex-truncated small rhombicuboctahedron, TSC 1,2,4-stellated pentagonal prism, MEL 1-stellated,3,4-open pentagonal prism, DAC, net 248 1,3,4-open pentagonal prism, DAC penta edge-stellated tetrahedron, BIK, BOG, CAS, CFI, EUO, MFI, MFS, MTT, MTW,NES, SFF, STF, STT, TER, TON, VET, part of hes, part of eun 1,1',4',2",3"-stellated cube, MTW 1,1',1"-handle trigonal prism = 1,2open,4-handle pentagonal prism, MEL 1,2-handle,3,5-open pentagonal prism = one-trestle,two-handle cube, VSV 1',2',3',4',5',6',1",3",5"-stellated hexagonal prism, MTN

8 8 2

412464366123122 12 6 4

4 4 6 12

12 6 6

4

2

Landolt-Börnstein New Series IV/14

78

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.1 (continued). face symbol

label

point group

description

3 1

59

knt

3m

tri-handle tetrahedron, HEU, MFI

4

5 -a

tes

42m

tetra-edge-stellated tetrahedron, BEA*, DAC, EPI, FER, MEL, MFI, MFS, MOR, nets 575, 577-9, 873-5, 987

54-b

kdw

42m

54525264646462102

zly

mm2

di-handle tetrahedron, = two euo sharing 4, EUO, MON, NES complex, related to nns, EUO

55

pen

42m

54546864 5462

nns eun

mmm 2/m

546262 5462102

kdx kns

mm2 2/m

5464

nna

222

4 4

58 5551101

kdy kob

42m 5m

two kaj sharing 8, MON 1',2',3',4',5'-stellated pentagonal prism, DDR

565663

clb

62 m

5661121

kny

6mm

5,6,5/edge,5,5,edge joining opposing vertices, nets 966, 967, 969 1',2',3',4',5',6'-stellated hexagonal prism, DDR

5692

kol

32

1',3',5',2",4",6"-stellated hexagonal prism, DDR

58546464

sgw

42m

586262

son

mmm

58646282 512

frr red

mmm 53m

complex with two strings of four 6 sharing opposite edges, SGT ber with four outer vertices, sigma-1 of stf, SZF fff sharing 8 with four koa, FER regular dodecahedron, DDR, DOH, MEP, MTN, nets 967, 968, 969

51262

mla

12 2 m

4 4

di-stellated cle, = 1,2,4-stellated pentagonal prism with one outer vertex, MEL, MFI complex, NON two pes sharing 6, = tetra-stellated cube, BOG, CAS, CFI, EUO, MFI, MTW, NES, SFF, STF, STT, TER hes sharing 6 with two pes, MTT, TON 1',2',3',5',2",4",5",6"-stellated hexagonal prism, MFI tetra-stellated cle, EUO, NON

edge-5/5-edge joining opposing 6, MEP, nets 967, 969

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

79

Table 16.3.1 (continued). face symbol

label

point group

description

12 4

5 6

mtn

43m

5126662

doh

6/m m m

6261101 626261 62626182

knp zlv kum

mm2 2 mm2

6282

kaa

mmm

opposing edges related by inverse-tetrad linked by 5-edge-6/6-edge-5, MTN, net 968 opposing 6 joined by edge-6-edge/5edge-5, DOH 1,3,4,5-open hexagonal prism, BIK 1-open,4',1"-handle cube, ATO, net 738 1,2,3,4-stellated,(5,6)-horizontal-trestle hexagonal prism, CAS 1,2,4,5-open hpr, = di-trestle cube, APD, ATN, LOV, LTL, MEL, STI, nets 3, 5, 9, 10, 26, 575, 601, 1177, 1178, 1246, 1282-1285

63

kah

62 m

two vertices joined by three handles, = tri-open cube, AEL, AET, AFI, AFO, AHT, APD, ATO, ATS, ATV, -CHI, CON, DFT, DON, IFR, ISV, LAU, MEL, MFI, MSO, MWW, OSI, VFI, nets 2, 9, 10, 43, 539, 720, 873-5, 956, 1001, 1005, 1031, 1160, 1161, 1282-5, 1300

6362

afi

62 m

1,3,5-open hexagonal prism, = hexastellated trigonal prism, AEL, AET, AFI, AFO, ATV, DON, VFI, nets 2, 9, 10, 538, 956, 1001, 1160, 1284, 1285 part of afo, AFS, BPH

64

hes

43m

646282

koq

2/m

6482-a

oop

4/m m m

hexa-stellated tetrahedron, JBW, MTT, MTW, TON, nets 1, 1031 1,2,4,5,1',5',2",4"-stellated hexagonal prism, BIK 1,3,5,7-open octagonal prism, = octastellated cube, = tetragonal analog of afi, ATV, DFT

6482-b

kom

42m

1',3',2",4"-handle cube, = two kaa sharing 8, ATN, nets 3, 1246

65102

odp

10 2 m

1,3,5,7,9-open decagonal prism, = 5-fold analog of afi, AEL, AFO, AHT

66

lai

32

6662-a

ber

6/m m m

hes sharing 6, = sigma-1 of vvs , JBW, MTT, MTW, TON, nets 1, 977 hexa-stellated hexagonal prism, = 6-fold analog of stp, MSO, nets 278, 279, 978

Landolt-Börnstein New Series IV/14

80

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.1 (continued). face symbol

label

point group

description

6 6 -b

kyw

62 m

1,3,5-handle hexagonal prism, AEL, AET, AFI, AFO, ATV, VFI, net 2

666363

fni

62 m

66122-a

apf

6/m m m

tri-open kag, = three knp sharing two edges per pair, nets 539, 956, 1001 1,3,5,7,9,11-open dodecagonal prism, = 6-fold analog of afi, AFI, nets 1282-1285

66122-b

kok

32

1',3',5',2",4",6"-handle hexagonal prism, hexagonal analog of kom, ATO, CAN

67142

etn

14 2 m

1,3,5,7,9,11,13-open 14-prism, = 7-fold analog of afi, AET, DON, net 1160

69182

kup

18 2 m

8482-a

krr

mmm

1,3,5,7,9,11,13,15,17-open 18-prism, = 9-fold analog of afi, VFI 1,3,5,7-open,2,6-handle octagonal prism, THO

8482-b

krq

42m

84122

kos

4/m m m

6 2

1,2,3,4-stellated,2',4',1",3"-handle cube, EDI 1',2',3',4',1",2",3",4"-handle cube, PAU

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

81

Table 16.3.2 Polyhedral subunits in alphabetical order. face symbol

label

point group

description

4 4 8 12 -a

afe

32

hexa-double-handle truncated octahedron, rotated relative of bpa, AFS, AFY

6362

afi

62 m

1,3,5-open hexagonal prism, = hexastellated trigonal prism, AEL, AET, AFI, AFO, ATV, DON, VFI, nets 2, 9, 10, 538, 956, 1001, 1160, 1284, 1285 part of afo, AFS, BPH

4663

afo

62 m

444442646282122

afr

mm2

double-capped afi, = two iet joined by three edges, AFS, BPH two 12 linked by (8/edge-4/6-4/4-6/64/edge-4)2, AFR

444442646282122

afr

mm2

42416281

afs

mm2

464643628683

aft

62 m

in ABBCCBBA-6 sequence, 44844/848, sigma(h)-eri, AFT, AFX

44424242626284 66122-a

apc apf

2/m 6/m m m

two kdm sharing 8, APC 1,3,5,7,9,11-open dodecagonal prism, = 6-fold analog of afi, AFI, nets 1282-1285

444264

aww

42m

1,2-double-handle hexagonal prism, = di-double-handle cube, AWW, SAO

42426261-a

baf

mm2

1-open hexagonal prism, net 470 part of hpr

6662-a

ber

6/m m m

42415252

bet

mm2

hexa-stellated hexagonal prism, = 6-fold analog of stp, MSO, nets 278, 279, 978 1-stellated pentagonal prism, BEA*, STT, nets 577-9, 873-5

4264-b

bog

mmm

1,3-handle cube, AEL, AET, AFI, AFO, AHT, ATV, BOG, CGF, DFO, DFT, LAU, TER, VFI, nets 43, 1160, 1161, 1282-1285

bpa

62 m

hexa-double-handle relative of gme, BPH

bph

62 m

1,2-5,6-9,10-double-handle dodecagonal prism, AFS, BPH, composite of kno and three afs

bru

mmm

1,3-stellated cube, BOG, BRE, CON, HEU, STI, TER, nets 100, 1005

12 6 6

2

412468383122 6 3 3 6

4 4 4 6 12

4254

Landolt-Börnstein New Series IV/14

2

two 12 linked by (8/edge-4/6-4/4-6/6 -4/edge-4)2, AFR 1,2-open hexagonal prism, AFS, AWO, BPH, SAO, net 6

82

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.2 (continued). face symbol

label

point group

description

6 3 2

466

can

62 m

1,2-3,4-5,6-double-stellated hexagonal prism, AFG, CAN, ERI, LIO, LOS, LTL, LTN, OFF, SAT, SBS, SBT, -WEN, nets 201, 1177

46466286

cha

32

in ABBCCA-6 sequence, 448/844, AFT, CHA

4258

cla

82 m

2-vertex-truncated symmetrical trapezohedron, net 966

565663

clb

62 m

5,6,5/edge,5,5,edge joining opposing vertices, nets 966, 967, 969

4454

cle

42m

43535353616173

clf

3m1

46

cub

4/m 3 m

dual of triangulated dodecahedron = didouble-stellated cube, RTH, nets 970, 1182 opposing rotated-6 with edge-7/5-edgeedge/edge-5-4, clathrate, net 970 cube, ACO, AFY, AST, -CLO, DFO,

42545482

dah

mmm

44

des

42m

43535361

det

3m1

5126662

doh

6/m m m

435663

doo

62 m

435653536183

dtr

3m1

3276-b

due

32

six-edge stellated, 1,1'-vertex-truncated trigonal trapezohedron, net 853

3276-a

dum

32

six-edge stellated, 1,1'-vertex-truncated rhombohedron, net 853

4643666283

eab

62 m

41266122

eni

6/m m m

ABCCBA-6 sequence, 484/646, EAB, SBS 1,2-3,4-5,6-7,8-9,10-11,12-double-stellated dodecagonal prism, DFO, net 1178

4646636286

eri

62 m

in ABBCBBA-6 sequence, 44644/88, ERI

etn

14 2 m

1,3,5,7,9,11,13-open 14-prism, = 7-fold analog of afi, AET, DON, net 1160

7

6 14

2

LTA, ISV, nets 291, 875, 1305 ste sharing 8 with two zlt, DAC, MOR, net 575 di-edge-stellated tetrahedron, EDI, NAT, THO, part of hes and pes opposing vertex & 6 with edge-5/5-4, DDR opposing 6 joined by edge-6-edge/5edge-5, DOH opposing vertices connected edge-6-edge/5-4-5, DOH, MWW opposing vertex & 6 connected edge-8-edge/5-4-5-edge, DDR

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

Table 16.3.2 (continued). face symbol

83

label

point group

description

4 2

56

eun

2/m

4152

euo

mm2

4124661286122

evh

6/m m m

two pes sharing 6, = tetra-stellated cube, BOG, CAS, CFI, EUO, MFI, MTW, NES, SFF, STF, STT, TER 4-ring with handle bridging opposite vertices, EUO, MON, NES, RSN, VSV edge- & vertex-truncated hexagonal prism, 6-fold analog of eab, DFO

4124664124

fau

43m

hexa-double-handle relative of truncated octahedron, FAU

666363

fni

62 m

58646282 42646262

frr fsi

mmm 2/m

4264102

ftt

mmm

tri-open kag, = three knp sharing two edges per pair, nets 539, 956, 1001 fff sharing 8 with four koa, FER kaa sharing 8 with two inversion-related kaq, nets 5, 9 1,3,6,8-open-decagonal prism, = 1',2',4',5',1",2",4",5"-stellated hexagonal prism, DFO

46436283

gme

62 m

424162

1,2-3,4-5,6-double-handle hexagonal prism, = rotated truncated octahedron, AFT, AFX, EAB, GME, MAZ, OFF, nets 575, 601, 1178 1,5,6'-edge-stellated trigonal prism,

gos

2

12 8 6

4 68

grc

4/m 3 m

great rhombicuboctahedron, -CLO, KFI, LTA, LTN, PAU, RHO, TSC

444284-a

gsm

42m

1,2-double-handle,3,8-open octagonal prism, = sigma of kdq, ATT, GIS

64

hes

43m

4662

hpr

6/m m m

42526262-a

hsp

mm2

hexa-stellated tetrahedron, JBW, MTT, MTW, TON, nets 1, 1031 hexagonal prism, AEI, AFT, AFX, CHA EAB, EMT, ERI, FAU, GME, KFI, LEV, LTL, LTN, MSO, MWW, OFF, SAT, SBS, SBT, SZF, TSC, -WEN, nets 279, 968, 1177 1,2-double-stellated,3,4,5-stellated pentagonal prism, CFI

Landolt-Börnstein New Series IV/14

84

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.2 (continued). face symbol

label

point group

description

46

iet

3m1

324162 444254546262828281

itp itq

mm2 mm2

6282

kaa

mmm

426462

kab

mmm

tri-edge-stellated tetrahedron, "capped hexagon", net 400, part of afo, AFS, BPH, part of cub, ACO, AFY, AST, -CLO, DFO, ISV, LTA, nets 291, 875, 1305, part of krs, AFN, part of mei, MEI, part of ohc, AFR, SAO, ZON, net 720 part of sti, DFO, SBE, SBS, SBT, STI, ZON, nets 720, 748 1-handle trigonal prism, ITE two 8 suspended by 48/554/66/554/48/554/66/554, net 1182 1,2,4,5-open hpr, = di-trestle cube, APD, ATN, LOV, LTL, MEL, STI, nets 3,5, 9, 10, 26, 575, 601, 1177, 1177, 1246, 1282-1285 1,4-handle hexagonal prism, ATO, CON, DFO, IFR, MSO, net 1005

466663

kag

62 m

three 1,2,3-open hexagonal prism sharing 2 edges per pair, net 538

63

kah

62 m

528281

kaj

mm2

41628281 426282-a 41626181

kal kam kaq

mm2 mm2 m

4141414162626261616161

kdi

m

4141416262616181 415282

kdj kdk

m mm2

4161618281 4241414161618281

kdl kdm

m m

two vertices joined by three handles, = tri-open cube, AEL, AET, AFI, AFO, AHT, APD, ATO, ATS, ATV, -CHI, CON, DFT, DON, IFR, ISV, LAU, MEL, MFI, MSO, MWW, OSI, VFI, nets 2, 9, 10, 43, 539, 720, 873-5, 956, 1001, 1005, 1031, 1160, 1161, 1282-5, 1300 1,2-handle,3,5-open pentagonal prism = one-trestle,two-handle cube, VSV 1,2,4,7-open octagonal prism, net 720 1,2,3-handle cube, net 720 1,2,4-open hexagonal prism, AWO, nets 5, 9, 10, 536 complex, knv sharing 8 with two kaq, net 536 complex, kzc sharing 8 with kaq, net 536 1,3-open,2-handle pentagonal prism, MFS, SZF 1,3,5,6-open octagonal prism, APD 1,5-open,2,3-double-handle octagonal prism, APC

3 1

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

85

Table 16.3.2 (continued). face symbol

label

point group

description

2 2 1

4 5 8 -a 4254626182 4261616172 426282-b

kdn kdo kdp kdq

mm2 mm2 m mm2

1-open,2,4-stellated cube, YUG kqc sharing 8 with two kdk, MFS five-stellated mei, net 749 1,3-open,2-handle hexagonal prism, = sigma of hes, ABW, ATT, JBW, net 536

4264104

kdr

42m

4282-a 4282-b 446482

kds kdt kdu

mmm mm2 mmm

gsm with stellation of all eight inclined edges of 4, MEL 1,3-open,2,4-stellated cube, ANA, net 282 1,3-open,2-handle cube, net 748 kaa sharing 6 with two loh, APD

54-b

kdw

42m

546262

kdx

mm2

4 4

58 4284-b 4444446282 4262101

kdy kjr klf knn

42m mmm 2/m mm2

two kaj sharing 8, MON 1,3-stellated,2,4-arch cube, net 400 two krs sharing 8, AFN 1,2,3-open hexagonal prism, ABW, net 538

4383122

kno

62 m

6261101 425461101 446261101 5462102

knp knq knr kns

mm2 mm2 mm2 2/m

5391 446282-b

knt knu

3m 2/m

42626282 4164121 436361121 5661121

knv knw knx kny

mm2 4mm 3m1 6mm

1',3',5',1",3",5"-handle hexagonal prism, = 1,2,5,6,9,10-open dodecagonal prism, nets 1177, 1178 part of aft, AFT, AFX, part of gme, EAB, GME, MAZ, OFF, nets 575, 601 part of bph, AFS, BPH, part of evh, DFO 1,3,4,5-open hexagonal prism, BIK 1',2',4',5'-stellated hexagonal prism, 1',4'-handle hexagonal prism, CAN 1',2',3',5',2",4",5",6"-stellated hexagonal prism, MFI tri-handle tetrahedron, HEU, MFI 1',4"-handle hexagonal prism, AFN, SBE SBT 1,2,3-handle hexagonal prism, net 536 1',2',3',4'-handle cube, AST 1',3',5'-handle hexagonal prism, EAB 1',2',3',4',5',6'-stellated hexagonal prism, DDR

Landolt-Börnstein New Series IV/14

di-handle tetrahedron, = two euo sharing 4, EUO, MON, NES hes sharing 6 with two pes, MTT, TON

86

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.2 (continued). face symbol

label

point group

description

3 3 1 1

4569 526181 5551101

knz koa kob

3m1 mm2 5m

41525262 52525261 526281

koc kod koe

mm2 mm2 mm2

41526262

kof

mm2

4482102

kog

mmm

42626262-a 425281-b 4461101

koh koi koj

mm2 mm2 mm2

1',3',5'-stellated hexagonal prism , DDR 1,3,4-open pentagonal prism, DAC 1',2',3',4',5'-stellated pentagonal prism, DDR 1,2,3-stellated pentagonal prism, EUO 1,2,4-stellated pentagonal prism, MEL 1,1',1"-handle trigonal prism = 1,2-open,4-handle pentagonal prism, MEL 1,3-handle pentagonal prism, MFI, ISV, nets 873-5, 1282-1285 1',1",4',4"-handle hexagonal prism, = 1,2,6,7-open decagonal prism, net 748 1,3-handle hexagonal prism, ATS, ISV 1',1"-handle trigonal prism, BOG, MAZ 1',4'-open hexagonal prism, ATN, SBE

66122-b

kok

32

1',3',5',2",4",6"-handle hexagonal prism, hexagonal analog of kom, ATO, CAN

5692

kol

32

1',3',5',2",4",6"-stellated hexagonal prism, DDR

6482-b

kom

42m

415481 536391121

kon koo

4mm 3m

426461141 646282

kop koq

mm2 2/m

418281102 4 2 8 12 52626161 4482 424241616181

kor kos kot kov kqr

mm2 4/m m m m 2/m m

4383 4262-b

kra krb

3m1 2

42426261-b

kre

2

466283

krf

32

1',3',2",4"-handle cube, = two kaa sharing 8, ATN, nets 3, 1246 1',2',3',4'-stellated cube, SGT 1',2',3',4',5',6',1",3",5"-stellated hexagonal prism, MTN 1',2',4',5'-handle hexagonal prism, AST 1,2,4,5,1',5',2",4"-stellated hexagonal prism, BIK 1,2,3,4,2',4'-handle cube, PHI 1',2',3',4',1",2",3",4"-handle cube, PAU 1,1',4',2",3"-stellated cube, MTW 1',4"-open hexagonal prism, net 748 1-handle hexagonal prism, handle replaces edge of 6 not 4, CGS three-handle cube, net 400 two-stellated,one-handle tetrahedron, net 725 extra edge joining a vertex from the two 4 of lau, net 725 hexa-edge-stellated vertex-truncated trigonal prism, ANA

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

87

Table 16.3.2 (continued). face symbol

label

point group

description

krq

42m

8 8 -a

krr

mmm

44426182 444288 444284-b 4258545484

krs kry krz ktg

mm2 4/m m m 2/m mmm

44626261 41525281101101

ktw kuh

mm2 m

62626182

kum

mm2

444482

kuo

mmm

1,2,3,4-stellated,2',4',1",3"-handle cube, EDI 1,3,5,7-open,2,6-handle octagonal prism, THO one-handle afo, AFN eight-stellated truncated octahedron, AFN two kra sharing 4, net 400 4.[5-5/5-5/5-5/5-5](8-8).4 with 2-connected vertex at outer 5/5, EPI bog/loh sharing 6, AET, AHT, DFT, VFI 1',3',4",6"-stellated-5'5"-handle hexagonal prism, MEL, MFI 1,2,3,4-stellated,(5,6)-horizontal-trestle hexagonal prism, CAS sti-kaa-sti sharing 6, = two-trestle hexagonal prism, STI

69182

kup

18 2 m

424262-b

kyf

2

42626262-b

kyu

2

6662-b

kyw

62 m

424264626282

kzb

2/m

kzd

62 m

1,2,3-open trigonal bipyramid, EDI, MWW, NAT, THO, net 1303

46103122

kze

62 m

66

lai

32

1-2,3-4,5-6 double-stellated, 2',4',6',2",4",6"- handle hexagonal prism, DFO hes sharing 6, = sigma-1 of vvs , JBW,

4264-a

lau

4/m m m

43434363616183

lev

3m1

4124686122-b

lil

6/m m m

4666666362

lio

62 m

4 2

8 8 -b 4 2

4

3

Landolt-Börnstein New Series IV/14

1,3,5,7,9,11,13,15,17-open 18-prism, = 9-fold analog of afi, VFI one 4 of pentagonal prism replaced by diagonal edge, net 963 6 sharing edges with projecting 4 at opposite ends, AEN 1,3,5-handle hexagonal prism, AEL, AET, AFI, AFO, ATV, VFI, net 2 kzc/kaa/kzc sharing 8, net 398

MTT, MTW, TON, nets 1, 977 1,2,3,4-stellated cube = 1,4-open hexagonal prism, ATO, CON, IFR, ISV, LAU, MSO, OSI, -RON, SAO, nets 3, 291, 398, 1005, 1161, 1282, 1283 in AABCCABBC-6 sequence, 446/84, LEV 1,2-3,4-5,6-7,8-9,10-11,12-double-handle dodecagonal prism, LTL, net 1177 in ABCBCBA-6 sequence, 4664/666, AFG, LIO

88

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.2 (continued). face symbol

label

point group

description

2 2 1

466

loh

mm2

1-open,3-handle cube, AEL, AET, AFI, AFO, AHT, APD, ATV, DFT, VFI

46666362

los

62 m

4262-a

lov

mmm

ltn

mm2

in ABCBA-6 sequence, 464/66, LIO, LOS, net 201 1,3-open cube, part of bog, AEL, AET, AFI, AFO, AHT, ATV, BOG, CGF, DFO, DFT, LAU, TER, VFI, nets 43, 1160, 1161, 1282-1285, part of bru, BOG, BRE, CON, HEU, STI, TER, nets 100, 1005, part of cub, ACO, AFY, AST, -CLO, DFO, LTA, ISV, nets 291, 875, 1305, part of kdu, APD, part of ktw, AET, AHT, DFT, VFI, part of loh, AEL, AET, AFI, AFO, AHT, APD, ATV, DFT, VFI, part of ohc, AFR, SAO, ZON, net 720 part of sti, DFO, SBE, SBS, SBT, STI, ZON, nets 720, 748 five-double-handle cube, LTN

maz

62 m

1,2-5,6-9,10-double-stellated dodecagonal prism, MAZ, composite of kno and three koi

4646567373122 314353 415262

meg mei mel

6 3m1 mm2

42425462626182 4242546261

mil mim

mm2 mm2

complex, MEI 1-vertex-truncated cube, MEI 1,3-open pentagonal prism, CON, DON, ISV, MEL, MFI, MWW, nets 873-5, 1005,1160, 1282-1285, 1300 complex, ESV, net 1297 complex, ESV, net 1297

51262

mla

12 2 m

42546282

mrd

mmm

4242424164626281 6 3 6

4 4 5 12

2

edge-5/5-edge joining opposing 6, MEP, nets 967, 969 1,2,5,6-stellated octagonal prism, MOR, net 601

41414141414151515151516161616171719191 mrr 424152515161 mrs

1 m

complex 19-hedron, STT complex 8-hedron, STT

51264

mtn

43m

425462

mtw

mmm

opposing edges related by inverse-tetrad linked by 5-edge-6/6-edge-5, MTN, net 968 1,4-stellated hexagonal prism, BEA*, MTW, nets 577-9, 873-5

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

89

Table 16.3.2 (continued). face symbol

label

point group

description

4 4 6 6 8 -a

niw

32

5464 54546864 41545252

nna nns non

222 mmm mm2

42526262-b

nug

mm2

41525252

nuh

mm2

486482

ocn

4/m m m

in ABBCBCCA-6 sequence, 4468/8644, SAT tetra-stellated cle, EUO, NON complex, NON koc with two vertices bridging two-connected nodes, EUO, NES, NON 1-stellated,4-handle hexagonal prism, nets 873- 875 1,3-stellated pentagonal prism, ISV, SFF, STF, nets 873-875 1,2-3,4-5,6-7,8-double-stellated octagonal prism, tetragonal analog of can, ATN, SBE, net 1178

65102

odp

10 2 m

424262-a 6482-a

ohc oop

mm2 4/m m m

324161

oot

mm2

4882

opr

8/m m m

436392

osu

62 m

426262

oth

mm2

4242418281 48448482

oto pau

mm2 4/m m m

5454

pen

42m

444442414162828282

per

mm2

5262

pes

mm2

6 6 6 2 6

Landolt-Börnstein New Series IV/14

1,3,5,7,9-open decagonal prism, = 5-fold analog of afi, AEL, AFO, AHT 1-handle cube, AFR, SAO, ZON, net 720 1,3,5,7-open octagonal prism, = octa-stellated cube, = tetragonal analog of afi, ATV, DFT 1-open trigonal prism, ITE TSC, net 1178 octagonal prism, MER, PAU, RHO, SBE, TSC, net 1178 1,4,7-open nonagonal prism, = 1',3',5',1",3",5"-stellated hexagonal prism, net 279 1,3-open hexagonal prism, ATS, OSI, nets 539, 956, 1001, 1161 1,2-open octagonal prism, PAU, PHI 1,2-3,4-5,6-7,8-double-handle octagonal prism, KFI, MER, PAU, net 1177 di-stellated cle, = 1,2,4-stellated pentagonal prism with one outer vertex, MEL, MFI mono-double-handle,tetra-double-stellated truncated octahedron, AEI penta edge-stellated tetrahedron, BIK, BOG, CAS, CFI, EUO, MFI, MFS, MTT, MTW, NES, SFF, STF, STT, TER, TON, VET, part of hes, part of eun

90

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.2 (continued). face symbol

label

point group

description

444888

phi

mm2

1,2-3,4-double-handle,5,8-open octagonal prism, PAU, PHI

466286

plg

32

512

red

53m

4241415252526261616181

rob

m

rotated-gme, = hexa-stellated truncated octahedron, PAU regular dodecahedron, DDR, DOH, MEP, MTN, nets 967, 968, 969 convex half of rwb cage, RUT

468

rpa

82 m

444254646282

rte

2/m

44425454648282

rth

2/m

44424254545464626262

rwb

2/m

34426282102

sfg

2/m

32

sfi

42m

spiro-5, LOV, RSN, VSV, WEI

45

sgt

62 m

1,2,3-vertex-truncated trigonal bipyramid, SGT

58546464

sgw

42m

3151

shf

2

586262

son

mmm

4284-a

ste

4/m m m

424261

sti

mm2

42545464 4464122

stv sty

mmm 4/m m m

4264626161

ten

mm2

complex with two strings of four 6 sharing opposite edges, SGT spiro-3,5 = 3- & 5- rings sharing vertex, net 1055 ber with four outer vertices, sigma-1 of stf, SZF 1,2,3,4-handle cube = 1,2,5,6-open octagonal prism, ACO, DAC, LOV, LTL, MER, nets 38, 100, 575, 601, 1043, 1177 1-open cube, SBE, SBS, SBT, STI, ZON, net 748 part of cub, ACO, AFY, AST, -CLO, DFO, ISV, LTA, nets 291, 875, 1305, part of ohc, AFR, SAO, ZON, net 720 beta-tetrakaidecahedron, net 968 1,4,7,10-open dodecagonal prism, = eight-stellated octagonal prism, net 291 kaa sharing 8 with two mirror-related kaq, net 10

4 2 1 2 2 1

8 8 2

3 6

rotated relative of pau, = tetragonal analog of toc, AWW, -CLO two 8-rings suspended by 45646546/65464564, RTE two 8-rings suspended by 45648465/84654564 & four 5 in waist, RTH two rob with inversion at center of shared 8, 4-connected & concave at 8, RUT 1,5-open,4,4',8,8'-vertex-truncated octagonal prism, WEI, net 963

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

91

Table 16.3.2 (continued). face symbol

label

point group

description

5 -a

tes

42m

325282

tfe

2

31415161

tfs

1

tetra-edge-stellated tetrahedron, BEA*, DAC, EPI, FER, MEL, MFI, MFS, MOR, nets 575, 577-9, 873-5, 987 edge-shared-3-5-5-3 suspended by handle between 8 rings, net 1055 hexagonal base linked to an apex from the 1,2,4 vertices, net 751; not drawn

4668

toc

4/m 3 m

truncated octahedron, EMT, FAU, LTA, LTN, SOD, TSC, net 201

3886

trc

4/m 3 m

vertex-truncated cube, net 1043

46 445462

trd tte

4/m 3 m mmm

4241416282 48448882124

tti uce

mm2 4/m m m

truncated rhombic dodecahedron, AST 1,2-4,5-double-stellated hexagonal prism, RTE, RUT 1,4-open octagonal prism, AFR sixteeen-edge-stellated grc, = top & bottom-8 with alternating 848/4.12.4 sides, SBE

4643123122 42424241416261616182

ucs ulm

62 m m

six-handle gme, SBS complex, net 1244

4246881286

vsr

4/m 3 m

424141626282 42426482

vvn vvs

mm2 2/m

42414152525252626261101101

wah

m

4454545462102

wan

2/m

vertex-truncated small rhombicuboctahedron, TSC kal sharing 8 with afs, net 6 1,2-4,5-double-stellated,1",2',4',5"-stellated hexagonal prism, = two kdq sharing 8, ABW, JBW, net 398 10-prism altered to upper-45666665, lower-6545545545, SFF 10-rings suspended by upper-665455456/lower-545666545, STF

wof

62 m

tri-double-handle relative of gme, EMT

wou wwf

62 m mm2

42415261

wwt

mm2

446462

xib

mmm

hexa-double-handle relative of eab, EMT 1-handle pentagonal prism, CON, DON, IFR, nets 873, 874, 1005 1-open pentagonal prism, CON, DON, IFR, net 1005, part of bet, BEA*, STT, nets 577-9, 873-5, part of wwf, CON, DON, IFR, nets 873, 874, 1005 1,4-stellated,2,3-5,6-double-stellated hexagonal prism, net 1246

4

6 12

46464362123 12 6 3 6

3

4 4 4 6 12 12 42415262

Landolt-Börnstein New Series IV/14

2

92

16.3 Polyhedral units

[Ref. p. 251

Table 16.3.2 (continued). face symbol

label

point group

description

4 2 2

4 6 8 -a 445484

xvi ygr

mmm mmm

425482-a

ygw

2/m

525281

zlt

mm2

626261 446482161 43436183121

zlv zlw zlx

2 4mm 3m1

54525264646462102

zly

mm2

1,5-open octagonal prism, ITE two mirror-related kdn joined by two edges & two handles, YUG 7,9,10',12'-stellated hexagonal prism, SZF, YUG 1-stellated,3,4-open pentagonal prism, DAC, net 248 1-open,4',1"-handle cube, ATO, net 738 1',3',5',7'-handle octagonal prism, AWW three edges of 6 in can replaced by handle, CHA complex, related to nns, EUO

39

zlz

32

three sti sharing three vertices around 3, net 619

384264104 4242416181

zma zna

42m m

3153 31415251

znb znc

3m1 m

vertex-truncated aww, net 1304 1-open hexagonal prism, edge removed from 6 not 4, CGS tri-stellated tetrahedron, VNI euo and zna sharing 5, VNI

3484

znd

42m

zne

m

znf znl

62 m 6/m m m

6 2

4242426262818181 6 3 2 6

4 4 6 8 12 466126662

3

two-handle derivative of all-vertex-truncated tetrahedron, VNI complex, ZON twelve-stellated eab, SBS top and bottom 6 joined by vertical 6-4 6/edge-6-edge, MSO

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

afe

afi

afo

afr

afs

aft

Fig. 16.3.1 Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

93

94

16.3 Polyhedral units

apc

apf

aww

baf

ber

bet

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

bog

bpa

bph

bru

can

cha

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

95

96

16.3 Polyhedral units

cla

clb

cle

clf

cub

dah

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

des

det

doh

doo

dtr

due

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

97

98

16.3 Polyhedral units

dum

eab

eni

eri

etn

eun

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

euo

evh

fau

fni

frr

fsi

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

99

100

16.3 Polyhedral units

ftt

gme

gos

grc

gsm

hes

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

hpr

hsp

iet

itp

itq

kaa

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

101

102

16.3 Polyhedral units

kab

kag

kah

kaj

kal

kam

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

kaq

kdi

kdj

kdk

kdl

kdm

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

103

104

16.3 Polyhedral units

kdn

kdo

kdp

kdq

kdr

kds

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

kdt

kdu

kdw

kdx

kdy

kjr

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

105

106

16.3 Polyhedral units

klf

knn

kno

knp

knq

knr

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

kns

knt

knu

knv

knw

knx

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

107

108

16.3 Polyhedral units

kny

knz

koa

kob

koc

kod

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

koe

kof

kog

koh

koi

koj

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

109

110

16.3 Polyhedral units

kok

kol

kom

kon

koo

kop

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

koq

kor

kos

kot

kov

kqr

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

111

112

16.3 Polyhedral units

kra

krb

kre

krf

krq

krr

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

krs

kry

krz

ktg

ktw

kuh

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

113

114

16.3 Polyhedral units

kum

kuo

kup

kyf

kyu

kyw

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

kzb

kzd

kze

lai

lau

lev

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

115

116

16.3 Polyhedral units

lil

loh

ltn

[Ref. p. 251

lio

los

lov

maz

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

meg

mei

mel

mil

mim

mla

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

117

118

16.3 Polyhedral units

mrd

mrr

mrs

mtn

mtw

niw

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

nna

nns

non

nug

nuh

ocn

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

119

120

16.3 Polyhedral units

odp

ohc

oop

oot

opr

osu

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

oth

oto

pau

pen

per

pes

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

121

122

16.3 Polyhedral units

phi

plg

red

rob

rpa

rte

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

rth

rwb

sfg

sfi

sgt

sgw

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

123

124

16.3 Polyhedral units

shf

son

ste

sti

stv

sty

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

ten

tes

tfe

toc

trc

trd

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

125

126

16.3 Polyhedral units

tte

tti

uce

ucs

ulm

vsr

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

vvn

vvs

wah

wan

wof

wou

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

127

128

16.3 Polyhedral units

wwf

wwt

xib

xvi

ygr

ygw

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.3 Polyhedral units

zlt

zlv

zlw

zlx

zly

zlz

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

129

130

16.3 Polyhedral units

zma

zna

znb

znc

znd

zne

[Ref. p. 251

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

znf

16.3 Polyhedral units

znl

Fig. 16.3.1 (continued). Drawings of polyhedral subunits in alphabetical order.

Landolt-Börnstein New Series IV/14

131

132

16.4 2D nets

[Ref. p. 251

16.4 2D nets The following commentary is designed to help non-mathematicians understand the subtleties of the topologic features of the units described in Tables 16.4.1 and 16.4.2 and Fig. 16.4.1. The present compilation of 2D nets is still growing rapidly as new structures are found in zeolites and related materials, and as new theoretical ones are invented. Indeed, a multiple infinity of units can be invented as new algorithms become worked out. The drawings on page 155 illustrate all the features of 2D nets. Net ael is composed entirely of 3connected vertices joined by edges to generate 4-, 6- and 10-rings. These are three types of vertices: type 1 is at the contact between a 4-, a 6- and a 10-ring. Type 2 also is at the meeting of a 4-, 6- and 10-ring, but is topologically different when vertices at a further distance are considered. Type 3 is at the junction of two 6-rings and one 10-ring. Types 1 and 2 are half as frequent as type 3 because they lie on a mirror line. Hence the formula above the drawing. The plane group symmetry is c 2 m m , with the c specifying the centering of the unit cell shown by four angle marks [82Smi1, Chapter 2]. Using a length of ~ 3.1 Å for each edge, the cell dimensions a and b are as listed below the drawing. For net aff', the ' specifies that 2-connected vertices are present. The vertex symbol shows that the 3connected vertex of type 1 at the intersection of one 4-ring and two (superscript) 12-rings has twice the multiplicity of the 2-connected vertex of type 2 at the intersection of two 12-rings. 2-connected vertices are marked with a dot, except when shown by a number in the circle for the prototype. Because of the 4fold symmetry, only a (= b) is specified. Net ahf'," displays 3-connected (types 1, 2 and 3), 2-connected (type 4) and 4-connected (type 5) vertices.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

133

Table 16.4.1 3-, 4- and mixed 3, n-connected 2D nets in 3D nets in inceasing order of the vertex symbol. The 2D nets are assembled into groups and subgroups. The Vertex Symbol describes the faces meeting at a vertex: thus 482 means that a square (4) meets two (superscript 2) octagons (8). The 3-connected nets in (a) are arranged in sequence of increasing number of topologically distinct vertices and increasing size of circuits. Vertices with the same triplet are arranged in decreasing multiplicity. The 4-connected vertices in (b) are denoted ", and are similarly arranged by size of circuit and multiplicity. The nets in (c) contain a mixture of 2- ('), 3- (unmarked) and 4- (") vertices arranged in order of connectivity, circuit size and multiplicity. Zeolite 3D nets approved by the IZA-SC are specified by the three-letter code. Other materials from Table 16.2.1 are specified by the number in the CTF Catalog. To facilitate invention of 3D nets, selected 2D nets occurring in theoretical 3D nets from the CTF Catalog are listed. The theoretical nets in the last column were assembled partly from the literature, (e.g. 81Grü1) and partly from new 3D nets invented by combination of subunits using algorithms. Table 16.4.2 has identical entries arranged alphabetically by label. Updated to December 19, 1999. vertex symbol

label

occurrence

3.122 4.82

ttw fee

4.6.12

gml

theoretical nets, e.g. 64 ABW, ACO, AFN, APC, APD, ATN, ATT, AWW, DFT, EDI, GIS, KFI, LOV, MER, MON, NAT, -PAR, PHI, RSN, RTE, SBE, THO, VNI, VSV, WEI, YUG, nets 3, 5, 6, 26, 38, 43, 100, 398, 400, 536, 720, 725, 748, 975, 1302, 1303 AFG, AFI, AFN, AFR, AFS, AFT, AFX, AFY, ATO, ATS, BOG, BPH, CAN, CHA, CON, DFO, EAB, ERI, GME, ITE, LEV, LIO, LOS, MEI, OFF, SAO, SAT, SOD, nets 201, 1005

63

hex

(392)3(93)1 (429)1(4.9.12)2 (4210)1(4.102)2 (4212)1(4.8.12)2 (4212)1(4.122)1-a

shn tfn ffs tth fos

(4212)1(4.122)1-b (4216)1(4.8.16)1

twy rho

(a) Three-connected nets.

Landolt-Börnstein New Series IV/14

ABW, AEL, AET, AFI, AFO, AHT, APC, APD, ATO, ATV, BIK, CAS, DAC, DFT, EPI, FER, JBW, MOR, MSO, MTT, MTW, TON, VFI, nets 1, 2, 3, 5, 6, 9, 10, 398, 470, 536, 538, 539, 720, 956, 963, 987, 1001, 1160, 1161, 1282, 1283, 1284,1285, 1301 net 1303 theoretical net 314 AHT, AWO, nets 6, 748 nets 963, 1301 APC, MER, nets 6, 398, 470, 748, 963 theoretical net 320 RHO

134

16.4 2D nets

[Ref. p. 251

Table 16.4.1 (continued). vertex symbol

label

occurrence

(4 18)1(4.6.18)2 (4224)1(4.6.24)1 (468)1(4.8.12)1 (468)1(628)1 (468)2(682)1

eoo tsv ltl fsy brw

(472)2(73)1 (528)2(582)1

bor bik

(4210)1(4.6.10)2(6210)2 (4210)1(4.6.10)2(6.102)1 (4210)1(4.8.10)2(4.8.10)2 (4210)2(4.8.10)2(8210)1 (4212)1(4.6.12)1(6212)1 (4212)2(4.6.12)2(6.122)1 (4212)1(4.8.12)1(4.8.12)1 (4214)2(4214)1(4.142)2 (4224)1(4224)1(4.8.24)1 (458)2(582)2(582)1 (4.5.18)6(53)1(5218)3 (462)1(4.6.12)2(6212)1 (468)3(482)3(83)1 (468)1(482)1(628)1 (468)2(482)2(682)1 (468)1(63)1(628)1 (468)2(63)1(628)2 (4.6.10)2(4.6.10)2(6210)1 (53)1(5212)3(5212)3 (526)1(568)2(582)1 (527)1(572)2(572)1 (562)2(568)2(582)1 (4210)1(4212)2(4.10.12)2(4.10.12)2 (4210)1(4.6.10)1(4.102)1(6210)1 (4210)1(4.6.10)2(63)1(6210)2 (4212)2(4.5.12)2(582)1(5.8.12)2 (4212)1(468)1(4.6.12)1(6.8.12)1 (4212)1(4.6.12)1(63)1(6212)1

twl fsv ttv ooo fix toh vvv uun voe fef eus krp wek fto ffv apd feo ael eui nos vnv urg ftf fsf ftn uiv vss bsh

VFI theoretical net 317 LTL, MAZ, SBS, SBT AEN, nets 5, 398, 748, 963 AEN, ATT, ATV, AWO, BRE, CGF, GOO, HEU, JBW, STI, SZF, ZON, nets 6, 9, 10, 38, 536, 720, 1246, 1282 net 749 BIK, CAS, DAC, EPI, FER, MFS, MON, MOR, nets 100, 575, 601, 987 theoretical net 12 net 538 theoretical net 227 OFF theoretical net 546 theoretical net 303 MER, PHI, net 400 theoretical net 449 theoretical net 518 theoretical net 584 theoretical net 704 AST, OSI, SAO -WEN net 725 theoretical net 447 APD, net 9 theoretical net 584 AEL, LAU EUO, MWW, NES, NON NON theoretical net 253 theoretical net 438 theoretical net 535 AWO, net 963 net 956 theoretical net 435 theoretical net 567 theoretical net 786

2

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

135

Table 16.4.1 (continued). vertex symbol

label

occurrence

(4 14)2(4 14)1(4.6.14)2(6.14 )1 (4214)2(4214)1(4.8.14)2(4.8.14)2 (4214)2(4214)2(4.8.14)2(8214)1 (4214)1(4.6.14)2(4.6.14)2(4.6.14)2 (4214)2(4.6.14)2(4.6.14)2(6214)1 (4216)1(4216)1(4.8.16)1(4.8.16)1 (458)1(4.5.12)1(5212)1(5.8.12)1 (458)2(482)2(582)2(582)1 (4.5.10)1(4.5.10)1(5210)1(5.102)1-parallel-4-clino (4.5.10)1(4.5.10)1(5210)1(5.102)1-parallel-4-ortho (4.5.10)1(4.5.10)1(5210)1(5.102)1-zigzag-4 (4.5.10)2(562)1(5.6.10)2(6210)1 (4.5.12)2(4.6.12)2(562)1(5.6.12)2

fvo ftu urr xfn eop uen mor urv dac dao sxn fes bab

(462)1(4.6.10)2(4.102)1(6210)1 (467)2(478)2(627)2(782)1 (467)2(478)2(672)1(782)1 (468)1(468)1(482)1(682)1 (468)2(482)1(482)1(682)1 (468)2(482)2(482)2(682)1 (468)1(482)1(63)1(628)1 (468)2(63)2(63)1(628)2 (4.6.10)2(5210)1(562)1(5.6.10)2 (472)3(472)3(672)3(73)1 (478)2(478)2(478)2(782)1 (526)1(5210)2(5210)1(5.6.10)2

uii uhe sse bso tva stg fst fss kuk lei uhi fer

(527)1(572)1(572)1(572)1 (528)2(528)1(528)1(582)2 (528)1 (562)1(568)2(628)1 (562)2(568)2(582)1(63)1 (389)2(392)1(469)2(692)1(829)1 (389)2(392)1(529)2(592)1(829)1 (4210)2(468)2(4.8.10)2(682)1(8210)1 (4210)1(4.6.10)2(63)2(63)1(6210)2 (4214)1(462)2(4.6.14)2(4.6.14)2(6214)2 (4220)1(4220)1(4220)1(4.8.20)1(4.8.20)1 (4.5.14)2(53)1(5214)2(5214)2(5214)1 (4.5.14)2(4.5.14)2(526)1(5214)1(5.6.14)2 (468)2(468)2(63)1(628)2(682)1

een biz fex fis lod vps knh eig api vei eug utd tvt

theoretical net 571 theoretical net 534 theoretical net 433 theoretical net 649 theoretical net 956 theoretical net 429 MOR theoretical net 437 DAC EPI theoretical net 613 theoretical net 586 BEA*-A, B, C = nets 577, 578, 579, ISV AFO, -RON, TER, net 736 theoretical net 437 theoretical net 342 theoretical net 787 theoretical net 342 theoretical net 638 net 720 theoretical net 349 NES net 978 theoretical net 435 BOG, CON, FER, MEL, MFI, MFS, SZF, TER, TON theoretical net 252 YUG SGT theoretical net 342 LOV, RSN RSN, VSV theoretical net 900 net 1001 AET theoretical net 518 CFI DON, nets 1160, 1161 theoretical net 272

2

2

Landolt-Börnstein New Series IV/14

2

136

16.4 2D nets

[Ref. p. 251

Table 16.4.1 (continued). vertex symbol 2

2

2

2

(468)2(48 )2(48 )1(48 )1(68 )1 (4.6.10)1(4.6.10)1(4.6.10)1(4.6.10)1(6210)1 (472)2(478)2(528)2(582)1(782)1 (482)1(482)1(526)1(5.6.8)2(582)1 (4210)1(468)1(4.8.10)1(4.8.10)1(4.102)1(628)1 (4.5.10)1(4.5.10)1(562)1(5.6.10)1(5.6.10)1(6210)1 (468)1(468)1(4.6.12)1(482)1(4.8.2)1(4.8.12)1 (4.6.10)2(526)1(5210)1(568)2(568)2(5.6.10)2 (4210)1(4.6.8)1(4.6.10)1(4.8.10)1(4.8.10)1(628)1(6210)1 (4.5.10)2(562)1(5.6.10)2(63)2(63)1(63)1(6210)1 (4.5.12)1(4.5.12)1(526)1(5212)1(5212)1(5.6.12)1 (5.6.12)1- zigzag (4.5.12)1(4.5.12)1(526)1(5212)1(5212)1(5.6.12)1 (5.6.12)1- parallel (467)2(478)2(562)2(568)2(582)1(627)2(782)1 (526)2(5210)2(5210)2(5210)1(5210)1(5.6.10)2(5.6.10)2 (528)2(528)2(528)2(528)1(528)1(582)2(582)2 (4.5.11)2(4.6.11)2(4.6.11)2(5211)1(562)1(562)1(5.6.11)2 (5.6.11)2 (4.5.12)2(4.6.10)2(4.6.12)2(5210)1(562)1(562)1(5.6.10)2 (5.6.12)2 (472)1(472)1(562)1(567)2(567)2(627)2(672)1(672)1 (458)1(4.5.12)1(468)1(468)1(4.6.12)1(568)1(5.6.12)1(582)1 (628)1(628)1 (458)1(4.5.12)1(468)1(468)1(468)1(4.8.12)1(4.8.12)1 (4.8.12)1(5212)1(5.8.12)1 (4.5.11)1(4.5.11)1(4.6.11)1(4.6.11)1(526)1(5211)1(5211)1 (5211)1(562)1(5.6.11)1(5.6.11)1(5.6.11)1(5.6.11)1 (4.5.12)1(4.5.12)1(526)1(526)1(5210)1(5210)1(5210)1 (5212)1 (5212)1(5.6.10)1(5.6.10)1(5.6.12)1(5.6.12)1 (4.5.12)2(4.5.12)2(526)2(526)2(5210)2(5210)2(5210)1 (5210)1(5212)2(5212)2(5.6.10)2(5.6.10)2(5.6.12)2(5.6.12)2

label

occurrence

zzy wag svn avt kuu fev zsm eon fui mfv bta

smt mtt sft nuj

RTE SFF theoretical net 611 theoretical net 659 net 536 theoretical net 587 nets 1177, 1178 EUO theoretical net 545 MFI BEA*-A & -C, nets 1283, 1285 MTW, BEA*-B & -C, nets 1282, 1284 not used yet in theoretical net MTT, net 604 theoretical net 615 ISV

nut

net 875

fvv mln

theoretical net 342 MEL

ecr

nets 575, 601

nux

nets 873, 874, 875

nui

net 874

nus

net 873

btb

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

137

Table 16.4.1 (continued). vertex symbol

label

occurrence

3262" 3426" 44" (3262)"1(3426)"4 (336)"1(3248)"2(3468)"2 (c) mixed 3, n-connected nets 1)

qua" lvy" cri" zzp" sfj"

nets 90, 619, 1193 LEV nets 1, 91 theoretical net 625 WEI

(3425)"1(3452)"2 (3172)2(373)1" (3182)1(3.4.82)"1 (31102)4(32102)"1 (426)1(4262)"1 (428)1(438)"1 (4212)1(4312)"1 (452)1(453)"1-a (452)1(453)"1-b (462)2(4262)"1 (462)4(64)"1 (482)2(82)'1 (53)2(54)"1 (526)4(5262)"1 (83)2(82)'1

zin" zsk" hbo" sfe" kyd" kyo" urn" yug" kuy" qur" sod" aff' stn" zzt" toa'

(103)1(102)'1 (123)2(122)'3 (368)2(3262)"1(3468)"2-a (372)1(372)1(373)"1 (426)1(63)1(4262)"1 (428)2(468)4(6282)"1 (452)1(562)1(4526)"1 (456)1(526)1(4526)"1 (4.5.10)2(4.5.10)2(425.10)"1 (462)1(462)1(4262)"1-a (462)1(462)1(4262)"1-b (372)3(672)3(73)1(3272)"3 (426)1(428)2(436)"2(4282)"1 (428)1(428)1(438)"1(438)"1 (428)1(468)1(468)1(4268)"1 (459)2(459)2(492)2(4259)"2 (462)8(63)4(436)"4(64)"1

usg' rbs' sff" kyr" kuz" zzx" kyp" kye" tve" lan" lam" lef" kyn" zzu" ult" chi" wnv"

net 1066 net 1157 net 1043 WEI ATN, CAN, nets 598, 609 PHI theoretical net 431 YUG net 596 LAU, SOD, nets 90, 278, 282 AST, SOD, net 279 ANA nets 1066, 1292 net 597 AEN, NES, nets 90, 278, 282, 597, 609 nets 598, 979 net 980 WEI net 853 ATS, CAN, net 749 net 291 SGT RTE, net 609 net 989 AWW LAU net 978 OFF CGS ULT -CHI theoretical net 217

(b) 4-connected nets

Landolt-Börnstein New Series IV/14

138

16.4 2D nets

[Ref. p. 251

Table 16.4.1 (continued). vertex symbol 3

3

2

4

(5 )3(5 )1(5 6)3(5 )"3

label

occurrence

kro"

DDR, DOH, MTN, MWW, nets 968, 969 VET net 987 net 751 net 751 EMT, FAU net 1030 net 1031 net 619 MAZ theoretical net 423 net 984

(53)4(5212)4(5212)4(5212)4(54)"1 vet" 2 2 2 (5 12)1(5 12)1(5.12 )1(5.12)'1 mar' (345)1(456)1(3456)"1(3562)"1(4562)"1 kya" 2 2 2 2 (349)1(39 )1(49 )1(49 )1(349 )"1 kyb" (426)1(4212)1(462)1(436)"1(426.12)"1 zzw" 2 2 2 3 2 2 (45 )2(5 6)2(5 6)2(45 )2"(5 6 )"1 nkm" (462)2(628)2(682)2(82)'1(4262)"1 ahf '," 2 2 2 2 2 2 (368)1(368)1(3 6 )"1(3 6 )"1(3 68)"1(36 8)"1 sal" (425)1(425)1(428)2(4252)"2(4252)"2(4282)"1 kyq" (426)1(63)1(63)1(63)1(63)1(4262)"1 fye" 2 2 2 2 2 (4 20)4(4 20)4(4.6.20)4(6.20)'4(4 6.20)"2(6 20 )"1 jdf '," (452)1(458)1(458)1(528)1(4548)"1(453)"1 min" 2 2 2 2 2 2 4 (45 )2(458)4(5 8)2(4 5 )"2(4 8 )"1(5 )"1 mnl" (452)4(526)4(526)2(526)2(536)"4(54)"1 clc" 2 2 3 3 3 4 (46 )2(46 )2(6 )4(6 )2(6 )2(6 )"1 bav" (462)2(468)2(468)2(63)1(436)"2(4262)"1 wen" 3 2 2 2 2 (4 )1(46 )1(468)2(468)2(468)2(4 6 )"1(4 68)"2 zzv" (4220)1(462)2(462)2(462)2(468)2(4.6.20)2(4.6.20)2 col" (4262)"2(4262)"1 (53)4(53)2(53)2(53)2(526)4(526)2(54)"4(54)"2(54)"1 kud" 2 2 2 (4 5)1(458)1(468)1(468)1(468)1(48 )1(48 )1(568)1 (682)1(4256)"1(4548)"1 zzz" 3 3 (5 )2(5 )2(53)2(53)2(53)2(53)2(53)2(53)1(53)1(53)1 cld" (53)1(526)2(526)2(526)2(526)2(526)2(526)1(526)1 (54)"2(54)"2 (54)"2(54)"2(54)"2(54)"1(54)"1(54)"1(54)"1 1)

ESV theoretical net 1297 net 966 net 977 theoretical net 217 LTA -CLO MEP RTE net 967

Arranged with 3-connected vertices first, then 2-connected, and 4-connected last. Grouped in ascending order of types of vertices. ' and " denote 2- and 4-connected vertices, respectively.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

139

Table 16.4.2 3-, 4- and mixed 3, n-connected 2D nets in 3D nets in alphabetical order. vertex symbol

label

occurrence

(4.6.10)2(4.6.10)2(6 10)1 (482)2(82)'1 (462)2(628)2(682)2(82)'1(4262)"1 (468)1(63)1(628)1 (4214)1(462)2(4.6.14)2(4.6.14)2(6214)2 (482)1(482)1(526)1(5.6.8)2(582)1 (4.5.12)2(4.6.12)2(562)1(5.6.12)2

ael aff' ahf'," apd api avt bab

(462)2(462)2(63)4(63)2(63)2(64)"1 (528)2(582)1

bav" bik

(528)2(528)1(528)1(582)2 (472)2(73)1 (468)2(682)1

biz bor brw

AEL, LAU ANA net 1031 APD, net 9 AET theoretical net 659 BEA*-A, B, C = nets 577, 578, 579, ISV net 977 BIK, CAS, DAC, EPI, FER, MFS, MON, MOR, nets 100, 575, 601, 987 YUG net 749 AEN, ATT, ATV, AWO, BRE, CGF, GOO, HEU, JBW, STI, SZF, ZON, nets 6, 9, 10, 38, 536, 720, 1246, 1282 theoretical net 786 theoretical net 787 BEA*-A & -C, nets 1283, 1285 MTW, BEA*-B & -C, nets 1282, 1284 -CHI net 966 net 967

2

(4212)1(4.6.12)1(63)1(6212)1 bsh (468)1(468)1(482)1(682)1 bso 2 2 2 (4.5.12)1(4.5.12)1(5 6)1(5 12)1(5 12)1(5.6.12)1 bta (5.6.12)1- zigzag (4.5.12)1(4.5.12)1(526)1(5212)1(5212)1(5.6.12)1 btb (5.6.12)1- parallel (459)2(459)2(492)2(4259)"2 chi" (452)4(526)4(526)2(526)2(536)"4(54)"1 clc" 3 3 3 3 3 3 3 3 3 3 (5 )2(5 )2(5 )2(5 )2(5 )2(5 )2(5 )2(5 )1(5 )1(5 )1 cld" (53)1(526)2(526)2(526)2(526)2(526)2(526)1(526)1 (54)"2(54)"2 (54)"2(54)"2(54)"2(54)"1(54)"1(54)"1(54)"1 (4220)1(462)2(462)2(462)2(468)2(4.6.20)2(4.6.20)2 (4262)"2(4262)"1 44" (4.5.10)1(4.5.10)1(5210)1(5.102)1-parallel-4-clino (4.5.10)1(4.5.10)1(5210)1(5.102)1-parallel-4-ortho (458)1(4.5.12)1(468)1(468)1(468)1(4.8.12)1(4.8.12)1 (4.8.12)1(5212)1(5.8.12)1 2 (5 7)1(572)1(572)1(572)1 (4210)1(4.6.10)2(63)2(63)1(6210)2 (4.6.10)2(526)1(5210)1(568)2(568)2(5.6.10)2 (4218)1(4.6.18)2 (4214)2(4.6.14)2(4.6.14)2(6214)1

Landolt-Börnstein New Series IV/14

col"

-CLO

cri" dac dao ecr

nets 1, 91 DAC EPI nets 575, 601

een eig eon eoo eop

theoretical net 252 net 1001 EUO VFI theoretical net 956

140

16.4 2D nets

[Ref. p. 251

Table 16.4.2 (continued). vertex symbol

label

occurrence

(4.5.14)2(5 )1(5 14)2(5 14)2(5 14)1 (53)1(5212)3(5212)3 (4.5.18)6(53)1(5218)3 4.82

eug eui eus fee

(458)2(582)2(582)1 (468)2(63)1(628)2 (526)1(5210)2(5210)1(5.6.10)2

fef feo fer

(4.5.10)2(562)1(5.6.10)2(6210)1 (4.5.10)1(4.5.10)1(562)1(5.6.10)1(5.6.10)1(6210)1 (528)1 (562)1(568)2(628)1 (4210)1(4.102)2 (468)2(482)2(682)1 (562)2(568)2(582)1(63)1 (4212)1(4.6.12)1(6212)1 (4212)1(4.122)1-a (4210)1(4.6.10)1(4.102)1(6210)1

fes fev fex ffs ffv fis fix fos fsf

(468)2(63)2(63)1(628)2 (468)1(482)1(63)1(628)1 (4210)1(4.6.10)2(6.102)1 (468)1(628)1 (4210)1(4212)2(4.10.12)2(4.10.12)2 (4210)1(4.6.10)2(63)1(6210)2 (468)1(482)1(628)1 (4214)2(4214)1(4.8.14)2(4.8.14)2 (4210)1(4.6.8)1(4.6.10)1(4.8.10)1(4.8.10)1(628)1(6210)1 (4214)2(4214)1(4.6.14)2(6.142)1 (472)1(472)1(562)1(567)2(567)2(627)2(672)1(672)1 (426)1(63)1(63)1(63)1(63)1(4262)"1

fss fst fsv fsy ftf ftn fto ftu fui fvo fvv fye"

CFI EUO, MWW, NES, NON theoretical net 704 ABW, ACO, AFN, APC, APD, ATN, ATT, AWW, DFT, EDI, GIS, KFI, LOV, MER, MON, NAT, -PAR, PHI, RSN, RTE, SBE, THO, VNI, VSV, WEI, YUG, nets 3, 5, 6, 26, 38, 43, 100, 398, 400, 536, 720, 725, 748, 975, 1302, 1303 theoretical net 584 theoretical net 584 BOG, CON, FER, MEL, MFI, MFS, SZF, TER, TON theoretical net 586 theoretical net 587 SGT AHT, AWO, nets 6, 748 theoretical net 447 theoretical net 342 theoretical net 546 APC, MER, nets 6, 398, 470, AWO, net 963 748, 963 theoretical net 349 net 720 net 538 AEN, nets 5, 398, 748, 963 theoretical net 535 net 956 net 725 theoretical net 534 theoretical net 545 theoretical net 571 theoretical net 342 theoretical net 423

3

2

2

2

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

141

Table 16.4.2 (continued). vertex symbol

label

occurrence

4.6.12

gml

AFG, AFI, AFN, AFR, AFS, AFT, AFX, AFY, ATO, ATS, BOG, BPH, CAN, CHA, CON, DFO, EAB, ERI, GME, ITE, LEV, LIO, LOS, MEI, OFF, SAO, SAT, SOD, nets 201, 1005

(3182)1(3.4.82)"1 63

hbo" hex

(4220)4(4220)4(4.6.20)4(6.20)'4(426.20)"2(62202)"1 (4210)2(468)2(4.8.10)2(682)1(8210)1 (53)3(53)1(526)3(54)"3

jdf '," knh kro"

(462)1(4.6.12)2(6212)1 (53)4(53)2(53)2(53)2(526)4(526)2(54)"4(54)"2(54)"1 (4.6.10)2(5210)1(562)1(5.6.10)2 (4210)1(468)1(4.8.10)1(4.8.10)1(4.102)1(628)1 (452)1(453)"1-b (426)1(63)1(4262)"1 (345)1(456)1(3456)"1(3562)"1(4562)"1 (349)1(392)1(492)1(492)1(3492)"1 (426)1(4262)"1 (456)1(526)1(4526)"1 (426)1(428)2(436)"2(4282)"1 (428)1(438)"1 (452)1(562)1(4526)"1 (425)1(425)1(428)2(4252)"2(4252)"2(4282)"1 (372)1(372)1(373)"1 (462)1(462)1(4262)"1-b (462)1(462)1(4262)"1-a (372)3(672)3(73)1(3272)"3 (472)3(472)3(672)3(73)1

krp kud" kuk kuu kuy" kuz" kya" kyb" kyd" kye" kyn" kyo" kyp" kyq" kyr" lam" lan" lef" lei

net 1043 ABW, AEL, AET, AFI, AFO, AHT, APC, APD, ATO, ATV, BIK, CAS, DAC, DFT, EPI, FER, JBW, MOR, MSO, MTT, MTW, TON, VFI, nets 1, 2, 3, 5, 6, 9, 10, 398, 470, 536, 538, 539, 720, 956, 963, 987, 1001, 1160, 1161, 1282, 1283, 1284,1285, 1301 net 984 theoretical net 900 DDR, DOH, MTN, MWW, nets 968, 969 AST, OSI, SAO MEP NES net 536 net 596 ATS, CAN, net 749 net 751 net 751 ATN, CAN, nets 598, 609 RTE, net 609 OFF PHI SGT MAZ net 853 LAU AWW net 978 net 978

Landolt-Börnstein New Series IV/14

142

16.4 2D nets

[Ref. p. 251

Table 16.4.2 (continued). vertex symbol 2

2

2

(389)2(39 )1(469)2(69 )1(8 9)1 (468)1(4.8.12)1 3426" (5212)1(5212)1(5.122)1(5.12)'1 (4.5.10)2(562)1(5.6.10)2(63)2(63)1(63)1(6210)1 (452)1(458)1(458)1(528)1(4548)"1(453)"1 (458)1(4.5.12)1(468)1(468)1(4.6.12)1(568)1(5.6.12)1(582)1 (628)1(628)1 (452)2(458)4(528)2(4252)"2(4282)"1(54)"1 (458)1(4.5.12)1(5212)1(5.8.12)1 (526)2(5210)2(5210)2(5210)1(5210)1(5.6.10)2(5.6.10)2 (452)2(526)2(526)2(453)2"(5262)"1 (526)1(568)2(582)1 (4.5.12)1(4.5.12)1(526)1(526)1(5210)1(5210)1(5210)1 (5212)1 (5212)1(5.6.10)1(5.6.10)1(5.6.12)1(5.6.12)1 (4.5.11)2(4.6.11)2(4.6.11)2(5211)1(562)1(562)1(5.6.11)2 (5.6.11)2 (4.5.12)2(4.5.12)2(526)2(526)2(5210)2(5210)2(5210)1 (5210)1(5212)2(5212)2(5.6.10)2(5.6.10)2(5.6.12)2(5.6.12)2 (4.5.12)2(4.6.10)2(4.6.12)2(5210)1(562)1(562)1(5.6.10)2 (5.6.12)2 (4.5.11)1(4.5.11)1(4.6.11)1(4.6.11)1(526)1(5211)1(5211)1 (5211)1(562)1(5.6.11)1(5.6.11)1(5.6.11)1(5.6.11)1 2 (4 10)2(4.8.10)2(8210)1 3262" (462)2(4262)"1 (123)2(122)'3 (4216)1(4.8.16)1 (368)1(368)1(3262)"1(3262)"1(3268)"1(3628)"1 (31102)4(32102)"1 (368)2(3262)"1(3468)"2-a (336)"1(3248)"2(3468)"2 (528)2(528)2(528)2(528)1(528)1(582)2(582)2 (392)3(93)1 (467)2(478)2(562)2(568)2(582)1(627)2(782)1 (462)4(64)"1 (467)2(478)2(672)1(782)1 (468)2(482)2(482)2(682)1 (53)2(54)"1 (472)2(478)2(528)2(582)1(782)1

label

occurrence

lod ltl lvy" mar' mfv min" mln

LOV, RSN LTL, MAZ, SBS, SBT LEV net 987 MFI

mnl" mor mtt nkm" nos nui

theoretical net 1297 MOR MTT, net 604 net 1030 NON net 874

nuj

ISV

nus

net 873

nut

net 875

nux

nets 873, 874, 875

ooo qua" qur" rbs' rho sal" sfe" sff" sfj" sft shn smt sod" sse stg stn" svn

OFF nets 90, 619, 1193 LAU, SOD, nets 90, 278, 282 net 980 RHO net 619 WEI WEI WEI theoretical net 615 net 1303 not used yet in theoretical net AST, SOD, net 279 theoretical net 342 theoretical net 638 nets 1066, 1292 theoretical net 611

ESV MEL

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

143

Table 16.4.2 (continued). vertex symbol

label

occurrence

(4.5.10)1(4.5.10)1(5 10)1(5.10 )1-zigzag-4 (429)1(4.9.12)2 (83)2(82)'1

sxn tfn toa'

(4212)2(4.6.12)2(6.122)1 (4224)1(4.6.24)1 (4212)1(4.8.12)2 (4210)1(4.8.10)2(4.8.10)2 3.122 (468)2(482)1(482)1(682)1 (4.5.10)2(4.5.10)2(425.10)"1 (468)2(468)2(63)1(628)2(682)1 (4210)1(4.6.10)2(6210)2 (4212)1(4.122)1-b (4216)1(4216)1(4.8.16)1(4.8.16)1 (467)2(478)2(627)2(782)1 (478)2(478)2(478)2(782)1 (462)1(4.6.10)2(4.102)1(6210)1 (4212)2(4.5.12)2(582)1(5.8.12)2 (428)1(468)1(468)1(4268)"1 (562)2(568)2(582)1 (4212)1(4312)"1 (4214)2(4214)2(4.8.14)2(8214)1 (458)2(482)2(582)2(582)1 (103)1(102)'1 (4.5.14)2(4.5.14)2(526)1(5214)1(5.6.14)2 (4214)2(4214)1(4.142)2 (4220)1(4220)1(4220)1(4.8.20)1(4.8.20)1 (53)4(5212)4(5212)4(5212)4(54)"1 (527)1(572)2(572)1 (4224)1(4224)1(4.8.24)1 (389)2(392)1(529)2(592)1(829)1 (4212)1(468)1(4.6.12)1(6.8.12)1 (4212)1(4.8.12)1(4.8.12)1 (4.6.10)1(4.6.10)1(4.6.10)1(4.6.10)1(6210)1 (468)3(482)3(83)1 (462)2(468)2(468)2(63)1(436)"2(4262)"1 (462)8(63)4(436)"4(64)"1

toh tsv tth ttv ttw tva tve" tvt twl twy uen uhe uhi uii uiv ult" urg urn" urr urv usg' utd uun vei vet" vnv voe vps vss vvv wag wek wen" wnv"

theoretical net 613 theoretical net 314 AEN, NES, nets 90, 278, 282, 597, 609 theoretical net 303 theoretical net 317 nets 963, 1301 theoretical net 227 theoretical nets, e.g. 64 theoretical net 342 net 989 theoretical net 272 theoretical net 12 theoretical net 320 theoretical net 429 theoretical net 437 theoretical net 435 AFO, -RON, TER, net 736 theoretical net 435 ULT theoretical net 438 theoretical net 431 theoretical net 433 theoretical net 437 nets 598, 979 DON, nets 1160, 1161 theoretical net 449 theoretical net 518 VET theoretical net 253 theoretical net 518 RSN, VSV theoretical net 567 MER, PHI, net 400 SFF -WEN theoretical net 217 theoretical net 217

2

Landolt-Börnstein New Series IV/14

2

144

16.4 2D nets

[Ref. p. 251

Table 16.4.2 (continued). vertex symbol 2

(4 14)1(4.6.14)2(4.6.14)2(4.6.14)2 (452)1(453)"1-a (3425)"1(3452)"2 (3172)2(373)1" (468)1(468)1(4.6.12)1(482)1(4.8.2)1(4.8.12)1 (3262)"1(3426)"4 (526)4(5262)"1 (428)1(428)1(438)"1(438)"1 (43)1(462)1(468)2(468)2(468)2(4262)"1(4268)"2 (426)1(4212)1(462)1(436)"1(426.12)"1 (428)2(468)4(6282)"1 (468)2(482)2(482)1(482)1(682)1 (425)1(458)1(468)1(468)1(468)1(482)1(482)1(568)1 (682)1(4256)"1(4548)"1

label

occurrence

xfn yug" zin" zsk" zsm zzp" zzt" zzu" zzv" zzw" zzx" zzy zzz"

theoretical net 649 YUG net 1066 net 1157 nets 1177, 1178 theoretical net 625 net 597 CGS LTA EMT, FAU net 291 RTE RTE

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

ael

aff '

ahf ',"

apd

api Fig. 16.4.1 Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

145

146

16.4 2D nets

avt

bab

bav"

bik

biz

bor

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

brw

bsh

bso

bta

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

147

148

btb

16.4 2D nets

[Ref. p. 251

chi"

clc" Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

cld"

col"

cri"

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

149

150

16.4 2D nets

dac

dao

ecr

een

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

eig

eon

eoo

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

151

152

eop

16.4 2D nets

[Ref. p. 251

eug

eui

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

eus

fee

fef

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

153

154

16.4 2D nets

feo

fer

fes

fev

fex

ffs

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

ffv

fis

fix

fos

fsf

fss

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

155

156

16.4 2D nets

fst

fsv

fsy

ftf

ftn

fto

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

ftu

fui

fvo

fvv

fye"

gml

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

157

158

16.4 2D nets

hbo"

hex

[Ref. p. 251

jdf ',"

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

knh

kro"

krp

kud"

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

159

160

16.4 2D nets

kuk

kuu

kuy"

kuz"

kya"

kyb"

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

kyd"

kye"

kyn"

kyo"

kyp"

kyq"

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

161

162

16.4 2D nets

kyr"

lam"

lan"

lef"

lei

lod

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

lvy"

ltl

mar'

mfv

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

163

164

16.4 2D nets

min"

mln

mnl"

mor

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

mtt

nos

nkm"

nui

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

165

166

16.4 2D nets

nuj

[Ref. p. 251

nut

nus

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

nux

ooo

qua"

qur"

rbs'

rho

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

167

168

16.4 2D nets

sal"

sfe"

sff"

sfj"

sft

shn

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

smt

sod"

sse

stg

stn"

svn

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

169

170

16.4 2D nets

sxn

tfn

toa'

toh

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

tsv

tth

ttv

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

171

172

16.4 2D nets

ttw

tva

tve"

tvt

twl

twy

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

uen

uhi

uiv

16.4 2D nets

uhe

uii

ult"

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

173

174

16.4 2D nets

urg

urn"

urr

urv

usg'

utd

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

uun

vei

vet"

vnv

voe

vps

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

175

176

16.4 2D nets

vss

vvv

wag

wek

wen"

wnv"

[Ref. p. 251

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

yug"

xfn

zin"

zsk"

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

177

178

16.4 2D nets

[Ref. p. 251

zsm

zzp"

zzt"

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.4 2D nets

zzu"

zzv"

zzw"

zzx"

zzy

zzz"

Fig. 16.4.1 (continued). Drawings of 2D nets in alphabetical order.

Landolt-Börnstein New Series IV/14

179

180

16.5 1D units

[Ref. p. 251

16.5 1D units The following commentary is designed to help non-mathematicians understand the subtleties of the topologic features of the units described in Tables 16.5.1 and 16.5.2 and Fig. 16.5.1. The present compilation of 1D units is still growing rapidly as new structures are found in zeolites and related materials, and as new theoretical ones are invented. Indeed, a multiple infinity of units can be invented as new algorithms become worked out. One-dimensional units can be loosely classified into chains and columns. The simpler chains range from a single sequence of edges that may be nearly straight (e.g. z (2-repeat zig-zag), s (3-repeat ziz-zagstraight), c (4-repeat zig-straight-zag-straight), etc., forming an infinite series that has not yet been enumerated fully. Each chain has a sigma relative zz, ss, cc, etc. consisting of a single sequence of edgeshared 4-rings, each 4 being the sigma-product of an edge. In some 4-connected 3D nets, it is easy to select a chain: e. g. those 3D nets composed of the mathematical union of a chain perpendicular to a 3-connected 2D net. Thus the z chain is obtained by replacing alternate edges of the 2D net by a pair of zig-zag edges of a chain, and keeping the other edges horizontal (e.g., AEI, AEL, AFI, AFO, AHT, ATV, VFI], [99Han1]. The c chain can be similarly combined with a 3-connected 2D net by replacing pair of adjacent edges with a zig and a zag [99Han2]: to close each circuit geometrically in each 2D net, as many zigs as zags are needed thus requiring an even number of edges. Similarly, but with more complexity, enumeration was made for a s chain (ATT, JBW, LTL, SZF, ZST), [99Han3]. Chains may share vertices back-to-back (EDI, NAT, THO), [99Han3], and a systematic enumeration of 3D nets generated from the union of bs and 3-connected 2D nets is in progress. In many nets, especially those containing 5-rings, selection of chains is difficult, simply because the geometry between adjacent edges is not clearly distinguishable into straight and zigzag types. To what extent does one allow deviation from strict regularity? Again, it is easy to pick chains that have the zigs and zags in the same plane, but not easy to decide how much twisting is permissible. As a practical choice, distorted 1D units are listed if useful for mathematical characterization of the host 3D net. In some nets, it is more convenient to emphasize polyhedra sharing faces, and the selection of 1D chains is not compelling. Clearly, different choices can be made, and I will be delighted to consider new units upon notification by readers. A simple helix consists of a single spiral of edges obeying crystallographic symmetry, in this case one of the subset of 7 enantiomorphic pairs of 75 rod groups ([82Smi1]; Table 7.1 therein on p. 240). For every left-handed helix there must be a right-handed one: e.g. p61 and p65. An extraordinary helix is nzp with curved triplets of 4-rings marching around the axis one unit per turn. A third group of 1D units consists of tubes bounded by a net. Thus afv has a cross-section consisting of a crinkled 6-ring, and a wall composed of a hexagonal net arranged with one edge parallel to the axis of the tube. Related to it is atn with an octagonal cross-section and a wall of hexagons with one edge perpendicular to the axis. Systematic enumeration is in progress, and various other members of the family of tubes will become obvious on turning the pages of Fig. 16.5.1.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

181

A fourth group of 1D units consists of columns. Some are simply composed of in-line polyhedra sharing a face top and bottom. These columns were enumerated for the ABC-6 family of 3D nets, and aeb is a nice example. Many columns consist of polyhedra that project left and right: bea, beb, and bec are complex examples. Systematic enumeration is in hand, but slow progress is expected because of the many possible polyhedra, and the several ways of interconnecting the less-symmetric ones. Not all theoretical columns can be easily combined into 4-connected 3D nets, but the constraints have not yet been looked for. A fifth group consists of polyhedra connected by a single edge like a string of beads. Thus opposing vertices of a cube are linked by a single edge in afy. Such strings are very flexible. Somewhat more complex is ast with the opposite edges of a cube linked by two 2-connected vertices to generate a hexagon. A sixth group consists of polygons sharing edges, as in bre and bs. Left and right hexagons generate keb. To conclude this description, I emphasize that classification of 1D units is just beginning, and only the columns related to the ABC-6 nets have been fully enumerated [84Smi1]. I hope to begin submitting papers to the peer-reviewed literature in 2001 ready for a detailed classification of 1D units.

Landolt-Börnstein New Series IV/14

182

16.5 1D units

[Ref. p. 251

Table 16.5.1 1D units in inceasing order of R and N. R is the smallest number of edges along the chain repeat, N is the number of vertices in the chain repeat, and d is the repeat distance in Å for an edge of 3.1 Å (d varies considerably for flexible chains). The numbering of the symmetry groups for crystallographic rods is from ([82Smi1]; Table 7.1 therein on p. 240). A three-letter label in bold letters is used except for the simple units. A brief description is followed by a list of the 3D nets containing the building unit (Table 16.2.1). Updated to December 19, 1999. R

N

d

rod group

label

description

2

2

5

21(p c m m)

z

2

4

5

22 (p m m m)

kcb

2

4

5

21 (p c m m)

zz

2

6

5

47 (p 31 2)

zhp

2

6

5

22 (p m m m)

kcc

2

6

5

16 (p 2 m m)

hhz

2 2 2

6 8 8

5 5 5

2 (p 1 ) 41 (p 4/m m m) 40 (p 42/m m c)

kcd kay atn

2

8

5

21 (p c m m)

ton

2

8

5

16 (p 2 m m)

lao

2 2

8 8

5 5

16 (p 2 m m) 9 (p 1 1 2/m)

nao kce

2 2

10 10

5 5

22 (p m m m) 16 (p 2 m m)

kcf hst

simple zigzag, ABW, AEN, ATN, ATO, ATS, BIK, CAN, CAS, CFI, -CHI, DAC, EPI, JBW, MTT, MTW, OSI, TON, YUG, nets 1, 2, 3, 10, 90, 282, 400, 470, 597, 609, 749, 977, 978, 989, 1001, 1031, 1157, 1246 (part of zz) 6 sharing two opposite edges - when flattened, would equal two z joined by parallel edges, ABW, CGS, nets 2, 282, 751 (also any 3D net containing a hex 2D net) double zigzag, ABW, AEN, ATN, ATS, CAN, CFI, JBW, OSI, nets 470, 749, 989, 1246 zh'zh'zh'-tube, enantiomorphic with 48 (p322), ATO two z joined by parallel squares, = kds sharing 4, nets 282, 609 hhzhz-tube plus z, = pes sharing 2 edges, BIK, CAS, MTT, MTW, TON, VET zh'h'zh"h"-tube, nets 282, 609 lau sharing 4, ATO, OSI, -RON, net 3 (hz)4-tube, = kaa sharing 8, = kom sharing 8, ATN, nets 3, 402 lai sharing 6, (hhz)2-tube plus two z, JBW, MTT, MTW, TON, 1, net 977 hhhzhz-tube plus two z, = afi sharing three horizontal edges, nets 2, 956, 1001 hhhzhz-tube plus zz, ATS, OSI eun sharing three edges, = hhh with three z, CAS, MTW mtw sharing 4, MTW hsp sharing 5, CFI

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

Table 16.5.1 (continued). R

N

2

10

2 2 2 2 2

rod group

label

description

5

11 (p 1 1 21/m)

kcg

10 10 11 12 12

5 5 5 5 5

9 (p 1 1 2/m) 6 (p 1 1 m) 16 (p 2 m m) 75 (p 6/m m m) 74 (p 63/m m c)

kch kci kcj kck cnc

2 2 2 2

12 12 12 12

5 5 5 5

66 (p 6 2 m) 22 (p m m m) 21 (p c m m) 21 (p c m m)

voi kug mao sao

2

10

5

16 (p 2 m m)

hsu

2 2 2 2 2 2 2 2 2 3

12 16 16 16 16 16 16 16 20 3

5 5 5 5 5 5 5 5 5 7

16 (p 2 m m) 41 (p 4/m m m) 40 (p 42/m m c) 32 (p 4 2 m) 22 (p m m m) 21 (p c m m) 21 (p c m m) 16 (p 2 m m) 21 (p c m m) 47 (p 31 2)

kcl ati osi veu zzi hsr tno kcm kcn the

3

3

7.5

16 (p 2 m m)

s

3

4

7.5

22 (p m m m)

bs

3

4

7.5

22 (p m m m)

kua

3

5

7

66 (p 6 2 m)

kco

3

5

6.5

32 (p 4 2 m)

fib

two hhz sharing z plus extra vertex, CAS, MTT, MTW, TON lai sharing three edges, net 1 hhhhzhz-tube with one z & one zz, net 129 two hhz sharing z and extra vertex, MTT, TON ber sharing 6, nets 278, 978 (hz)6-tube, = kok sharing 12, ATO, CAN, net 470 can sharing 6, CAN kcf plus two outer z, MTW (hhhz)2-tube plus four z, BIK, 756 (hhhz)2-tube plus two zz, = vvs sharing 8, ABW, JBW hst plus two double-h linking 2-connected vertices, CFI hhhhhzhz-tube plus four z, CAS ocn sharing 8, ATN (hhz)4-tube plus four z, OSI (h2z)4 with sidepockets, VET (hhhzhz)2-tube plus four z , MTW (hhzhzhz)2-tube plus two z, CFI (hhhhz)2-tube plus six z, TON hhhhhhzhhz-tube plus six z, MTT (hhhhhz)2-tube plus four zz, ATS three-repeat helix, enantiomorphic with 48(p322), LAU, YUG, nets 90, 619, 1193 simple saw, BIK, DAC, EDI, EPI, FER, GOO, HEU, JBW, MFS, MON, MOR, RSN, SZF, VSV, ZON, nets 38, 100, 282, 575, 596, 601, 609, 979, 987, (also part of ss) back-to-back s, GOO, LAU, LOV, MON, RSN, VNI, VSV, YUG, ZON, nets 26, 43, 100, 598, 720, 751, 975 8 sharing opposite edges = teeth of two s joined by edge, JBW, RSN krs linked by edges along triad axis, EDI, THO; kfy is geometrically distinct vertex-shared des, EDI, NAT, THO

Landolt-Börnstein New Series IV/14

d

183

184

16.5 1D units

[Ref. p. 251

Table 16.5.1 (continued). R 3

N 6

d 7.5

rod group 22 (p m m m)

label bre

3

6

7.5

21 (p c m m)

scs

3

6

7.5

16 (p 2 m m)

ss

3

8

7.5

22 (p m m m)

kbb

3

8

7.5

22 (p m m m)

kbf

3

8

7.5

21 (p c m m)

kbv

3

8

7

16 (p 2 m m)

kcq

3 3 3

8 8 8

7 6.5 7

16 (p 2 m m) 10 (p 2/m 1 1) 7 (p m 1 1)

lvi brs kcp

3

9

7.5

16 (p 2 m m)

kcr

3

9

7.5

16 (p 2 m m)

kcs

3

9

7.5

16 (p 2 m m)

kpa

3

9

7.5

16 (p 2 m m)

kqy

3 3 3 3 3

9 10 10 10 10

7.5 7 7 7 7.5

1 (p 1) 22 (p m m m) 16 (p 2 m m) 10 (p 2/m 1 1) 10 (p 2/m 1 1)

goo krx vpe apt sfc

3 3 3 3 3 3

10 10 12 12 12 12

8 7 7 7.5 7.5 7

2 (p 1 ) 2 (p 1 ) 66 (p 6 2 m) 41 (p 4/m m m) 41 (p 4/m m m) 40 (p 42/m m c)

hel kyy tof kad sta alh

description shs, alternating 4 and 6 sharing edges, AFR, BRE, LAU, LOV, LTL, ZON, nets 26, 278, 282, 291, 609 alternating left and right 5 sharing edges, = sh's, BIK, BRE, EPI, FER, MAZ, MFS, MON, MOR, RSN, VSV, nets 100, 575, 596, 609, 987 double saw, ATT, LTL, MAZ, OFF, -WEN, nets 575, 601, 1177 bs joined by horizontal edges to give lov and 8, = lov/kaa sharing 6, LOV, RSN, net 26 bs joined by horizontal edges to give lov and 4, LOV, RSN left and right euo sharing edges to give c, RSN, VSV tes joined by two vertical edges, DAC, FER, MFS, MOR ohc sharing edges, AFR, ZON, net 720 Alberti f1, edge-shared bru, BRE triple-4 in rocker geometry linked by two edges from inner to outer vertices, relative of kge, PHI hhshs-tube plus one s, external single 4, = kdk cage sharing two edges, MFS, SZF ccshhs-tube, = pes joined by two vertical edges, MFS hhshs-tube plus one s, internal double-4, = koi & pes sharing five edges, MFS ccshs-tube plus one s, DAC, FER, MOR, nets 575, 601, 987 gos joined by 4, GOO tti joined by 4, AFR hhhss-tube plus two s, VSV lau sharing opposite edges, net 398 bre with 2-connected vertex of 6 becoming 4-connected vertex of sfi, WEI bru joined by 4, BRE, HEU kaa sharing one edge, net 398 (hss)3-tube, RSN ste sharing 4, DAC, LOV, net 100 alternating cub and lau sharing 4, net 291 Alberti h, ref. 7, bru sharing 4, net 100

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

185

Table 16.5.1 (continued). R

N

d

rod group

label

description

3 3 3

12 12 12

7.5 6.5 6.5

32 (p 4 2 m) 32 (p 4 2 m) 29 (p 41 2 2)

kcx ktf kct

3

12

7.5

22 (p m m m)

kbi

3

12

7.5

21 (p c m m)

csh

3

12

7.5

21 (p c m m)

kcy

3 3 3 3 3

12 12 12 12 12

7.5 7.5 7.5 6.5 7.5

18 (p m c 21) 16 (p 2 m m) 16 (p 2 m m) 15 (p m m 2) 11 (p 1 1 21/m)

zzo kox kps kcw mod

3 3 3 3

12 12 12 12

6.5 6.5 7.5 7.5

10 (p 2/m 1 1) 10 (p 2/m 1 1) 10 (p 2/m 1 1) 10 (p 2/m 1 1)

kcu kcv lum wwv

3 3 3 3 3 3 3 3 3

12 13 14 15 16 16 17 18 18

7.5 7.5 7 7.5 7.5 7.5 7.5 7.5 7.5

9 (p 1 1 2/m) 21 (p c m m m) 16 (p 2 m m) 16 (p 2 m m) 22 (p m m m) 16 (p 2 m m) 16 (p 2 m m) 75 (p 6/m m m) 66 (p 6 2 m)

kng kcz vpv kpp frt kda kdb kdc off

aww sharing 4, AWW krq sharing upper & lower 8, EDI four distorted s joined by horizontal edges, c-axis, NAT; enantiomorph 30(p4322) (hs)4-tube, ste sharing horizontal edges, kaa joined by four vertical edges forming two 4, LOV, LTL, nets 38, 575, 601, 1177 (cshs)2-tube, MAZ, MOR, RSN, VSV, nets 100, 575 kdw sharing inner edges of handles, = kbv plus horizontal handles, MON left and right mel sharing 6, net 1300 (hhs)2-tube plus two s, JBW (ccshhs)-tube plus two s, BIK kam sharing one edge, net 720 edge-shared tes plus extra edge, cshs.cshs.c(bridge), DAC, EPI, FER, MOR, nets 575, 601, 987 four distorted s joined by horizontal edges, THO ygw sharing 4, YUG alternating lau and bog sharing 6, LAU tte sharing 4, all parallel giving clino choice, RTE ygw sharing three edges, SZF two kcq sharing s, FER, MFS (hsthsshts)-tube, LOV, VSV two kcr sharing s, SZF (ccshs)2-tube, DAC, FER kdy sharing three edges, MON kdo cages sharing four edges, MFS ber/hpr sharing 6, = sigma of kck, net 279 (hs)6-tube, OFF, net 1177

3

18

7.5

66 (p 6 2 m)

ofr

can/hpr sharing 6, LTL, OFF, -WEN, net 1177

3

18

7.5

gme sharing 6, MAZ, OFF, net 575

18 18 18

7.5 7.5 7.5

66 (p 6 2 m) 52 (p 3 m) 22 (p m m m) 22 (p m m m)

tix

3 3 3

kdd kde kpk

3

18

7.5

16 (p 2 m m)

kdf

3 3 3

20 22 24

7.5 7.5 7.5

21 (p c m m) 22 (p m m m) 41 (p 4/m m m)

mdn szf too

toc sharing 6 with alternate 4, SOD dah sharing 4, DAC, MOR, net 575 (hhshs)2-tube plus two s, two internal single-4, SZF (hhhs)2-tube plus two ss, = gsm/kdq sharing 8, kfd = sigma kdf, ATT (cccshs)2-tube, MOR, net 987 hpr/son sharing 6, SZF pau sharing 8, net 1177

Landolt-Börnstein New Series IV/14

186

16.5 1D units

[Ref. p. 251

Table 16.5.1 (continued). R

N

3

24

3 3 3 3 3 3

d

rod group

label

description

7.5

36 (p 4 m m)

kdg

24 24 26 28 28 30

7.5 7.5 7.5 7.5 7.5 7.5

16 (p 2 m m) 10 (p 2/m 1 1) 22 (p m m m) 6 (p m 1 1) 10 (p 2/m 1 1) 16 (p 2 m m)

kdh rtf kdz wai wam kuw

3 3 4

36 36 4

7.5 7.5 9

75 (p 6/m m m) 66 ( 6 2 m) 29 (p 41 2 2)

lel mzz fhe

4

4

8.5

21 (p c m m)

c

4

6

8.5

40 (p 42/m m c)

ts

4

6

8.5

22 (p m m m)

kba

4

8

8.5

66 ( 6 2 m)

kbg

4

8

8.5

45 (p 3 )

afy

4

8

8.5

40 (p 42/m m c)

bhs

4

8

8.5

22 (p m m m)

kea

4 4

8 8

10.5 8.5

22 (p m m m) 21 (p c m m)

ked cc

rpa sharing 8 with alternate 4, AWW (actually p 8 2 m) (chhcshhhhs-tube with six s, MFS rte sharing 8, RTE frr sharing 6, FER wah sharing 10, SFF wan sharing 10, STF (chchhcshschh)-tube plus two s & three ss, net 575 lil sharing 12, LTL, net 1177 maz sharing 12, MAZ four-repeat helix, enantiomorphic with 30(p4322), GIS, JBW, nets 1, 400, 725 simple crankshaft, ABW, AEL, AET, AFI, AFO, AHT, APC, APD, ATT, ATV, AWO, BIK, -CHI, DAC, DFT, DON, EPI, FER, MAZ, MFS, MON, MOR, RSN, VFI, VSV, nets 2, 3, 5, 6, 9, 10, 38, 100, 470, 536, 539, 575, 596, 601, 609, 720, 725, 748, 956, 979, 980, 987, 1043, 1160, 1161, (also part of cc) twisted square, AFR, ANA, CZP, EAB, GIS, GME, GOO, LAU, MER, PAU, PHI, RHO, -RON, SAO, SOD, YUG, nets 278, 282, 984, 989 6 joined by edge; back-to-back c, nets 38, 956, 1001 kah joined by edge along trigonal axis, AEL, AET, AFI, AFO, AHT, APD, ATV, DFT, VFI, nets 2, 9, 10, 1001 cub joined by edges extending body diagonal, AFY bifurcated hexagon plus square, c4-tube, AEL, AET, AFI, AFO, AHT, APD, ATV, DFT, VFI, nets 38, 43, 1160, 1161 4 & 8 sharing opposite edges, ABW, AEN, STI, net 43 6/4/4 sharing opposite edges, net 538 double crankshaft, APC, ATT, ATV, AWO, DON, GIS, GME, MER, PHI, nets 6, 26, 598

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

187

Table 16.5.1 (continued). R

N

4 4 4 4

8 8 9 10

4

10

4 4 4 4 4

rod group

label

description

21 (p c m m) 16 (p 2 m m) 32 (p 4 2 m) 22 (p m m m)

keb kec kub ast

10

20 (p c c m)

sfa

10 10 10

10 8.5 8.5

16 (p 2 m m) 16 (p 2 m m) 16 (p 2 m m)

epi kee tts

10 12

10 8.5

5 (p 1 1 21) 74 (p 63/m m c)

kef afv

6 sharing 1- & 3-edges, AEN sti joined by vertical edge, net 720 hes sharing opposing vertices, JBW, net 1 cub with opposing edges linked by 6, = sigma of ts, AST, net 291 column of sfi joined by two edges to give 6, = ctt/ttc, WEI tes joined by handles, DAC, EPI cchh-tube plus c, = kdt & sti sharing 4, net 748 mel joined by two vertical edges, cccch, nets 1160, 1161 tfs sharing edge, net 751 c6-tube, AEL, AET, AFI, AFO, ATV, DON,

4

12

4

12

4

d 8.5 8.5 7 9.5

29 (p 41 2 2)

kek

8.5

22 (p m m m)

kaz

12

8.5

22 (p m m m)

kbx

4

12

8.5

22 (p m m m)

thr

4

12

8.5

21 (p c m m)

keg

4 4 4 4 4 4 4

12 12 12 12 12 12 12

8.5 8.5 8.5 10 8.5 8.5 9

21 (p c m m) 21 (p c m m) 19 (p 2 c m ) 18 (p m c 21) 15 (p m m 2) 11 (p 1 1 21/m) 10 (p 2/m 1 1)

kuv vfi kbp kel kei keh kej

4 4 4

12 12 12

8.5 9 8.5

10 (p 2/m 1 1) 8 (p c 1 1) 5 (p 1 1 21)

kyz kyg krc

4 4 4 4 4

12 14 16 16 16

9 8.5 8.5 8.5 8.5

4 (p 2 1 1) 16 (p 2 m m) 41 (p 4/m m m) 40(p 42/m m c) 40(p 42/m m c)

kyv kem ecc chf kbn

4

16

9

32(p 4 2 m)

ker

Landolt-Börnstein New Series IV/14

10

VFI, nets 2, 9, 10, 538, 956, 1001, 1160 4 sharing edge with fhe, enantiomorphic with 30(p4322), net 725 kaa joined by vertical 4; double kba related by sigma transformation, APD, nets 3, 6, 9, 10 bog and lov sharing 4; kbg sharing 2 edges, net 43 (cch)2-tube, lau joined by two edges, nets 3, 1161 kah sharing two edges to generate c, AEL, AET, AFI, AFO, AHT, APD, ATV, VFI, net 2 (chh)2-tube, knn sharing 6, net 538 double-c4, = two bhs sharing c, AET, AHT, VFI ccchch-tube; kaq sharing 6, nets 5, 9, 10, 536 kdn sharing 4, YUG kdq sharing edge, ABW, ATT, JBW sti sharing two edges to generate c, net 748 two opposite edges of hpr joined by two edges generating 4, AEI, CHA lau joined by two edges forming 4, net 398 kyf sharing outer 4, net 963 krb sharing side edge of handle with shared edge of 4's, net 725 kyu sharing 4, AEN kam/sti sharing 4, net 720 c8-tube, ATV, DFT; actually p84/mmc (ch)4-tube, MER, net 38 rotated bog sharing 4, AEL, AET, AFI, AFO, AHT, ATV, DFT gsm sharing 4, = sigma of kei, ATT, FER

188

16.5 1D units

[Ref. p. 251

Table 16.5.1 (continued). R

N

4 4 4

16 16 16

4 4 4 4 4

D

rod group

label

description

8.5 8.5 8.5

21 (p c m m) 21 (p c m m) 21 (p c m m)

for kbl ken

16 16 16 16 16

8.5 8.5 8.5 9 8.5

19 (p 2 c m ) 19 (p 2 c m ) 18 (p m c 21) 16 (p 2 m m) 16 (p 2 m m)

keo kep zzj kes kyc

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

16 16 16 16 16 16 16 16 16 16 18 20 20 20 20

10 10 8.5 9 9.5 10 10 10 10 10 8.5 8.5 8.5 8.5 8.5

15 (p m m2) 13 (p 2 2 2) 11 (p 1 1 21/m) 10 (p 2/m 1 1) 10 (p 2/m 1 1) 10 (p 2/m 1 1) 10 (p 2/m 1 1) 10 (p 2/m 1 1) 7 (p m 1 1) 2 ( 1) 49 (p 3 c) 41 (p 4/m m m) 22 (p m m m) 22 (p m m m) 21 (p c m m)

kev rti keq ket keu kew kex kys key kyt kez kfb kfa krg ape

4

20

8.5

21 (p c m m)

ktv

4 4 4 4

24 24 24 24

8.5 10 10 10

ave gmn lso ldo

(ccch)2-tube, net 720 left and right afs sharing 4, net 6 (hhc)2-tube with two c, = kdq sharing 6, ABW, nets 398, 536 kds sharing two vertices yielding two ts, net 282 ccccchch-tube, = kdl sharing 8, APD kqr/rotated kqr/kqr sharing 6, CGS afs & kal sharing 6, net 6 kem with extra handle which converts sti into ohc, net 720 vvn sharing 4, net 6 cle sharing 4, RTH two kee sharing c, net 748 fsi sharing 4, net 5 kau & lau sharing three edges, net 398 vvs sharing 4, ABW, net 398 ygw joined by two edges to yield 8, YUG krz sharing 4, net 400 kdm sharing 4, APC kjr sharing 4, net 400 three ts connected by horizontal edges, net 278 toc sharing 4, SOD (chchh)2-tube, net 748 kdu joined by four vertical edges, APD c10-tube, net 263, AEL, AFO, AHT; idealized geometry has 5-symmetry two ktw sharing pairs of 4 with handles related by c-glide, AET, AHT, VFI c12-tube, AFI; actually p126/mmc (ch)6-tube, GME can rotated by 180° and sharing 6, LOS los sharing 6, LOS

4

24

10

75 (p 6/m m m) 74 (p 63/m m c) 74 (p 63/m m c) 66 ( 6 2 m) 66 ( 6 2 m)

tsn

alternating gme and hpr sharing 6, GME

4 4 4

24 24 24

9 10 8.5

52 ( 3 m) 22 (p m m m) 21 (p c m m)

krv zzm kfc

4

24

8.5

21 (p c m m)

kfd

4 4

24 24

8.5 8.5

19 (p 2 c m) 16 (p 2 m m)

kij kfe

4 4

24 26

10 8.5

7 (p m 1 1) 66 ( 6 2 m)

mip kfg

afe sharing 12, AFY trc & ste sharing 8, net 1043 vvn sharing 8 to yield two cc, = (ccch)2-tube with two cc, net 6 left & right gsm sharing 8 to yield two cc, = (chhh)2-tube with two cc, ATT, GIS ccchhchh-tube with two parallel cc, APC chchhhhh-tube with two cc, = oto & phi sharing 8, PHI mil sharing 6, ESV kag connected by 4 edges which yield three 4 sharing edge on triad axis, net 538

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

189

Table 16.5.1 (continued). R 4 4 4

N 26 28 28

d 9.5 8.5 8.5

rod group 22 (p m m m) 22 (p m m m) 21 (p c m m)

label kfh ute vtn

4 4

28 28

10 9.5

20 (p c c m) 7 (p m 1 1)

sfd kfi

4 4 4 4 4 5 5

30 32 32 32 36 5 8

10 8.5 10 10 8.5 13 10

16 (p 2 m m) 41 (p 4/m m m) 10 (p 2/m 1 1) 10 (p 2/m 1 1) 74 (p 63/m m c) 16 (p 2 m m) 22 (p m m m)

kfj mer rtj rtk yee f kur

5 5

10 12

13 12.5

22 (p m m m) 22 (p m m m)

kfl kfm

5 5

14 14

10 11

66 (p 6 2 m) 22 (p m m m)

kfn kfo

5 5 5 5 5 5

17 20 20 20 20 20

13 11 12.5 12.5 10 11

7 (p m 1 1) 22 (p m m m) 22 (p m m m) 22 (p m m m) 21 (p c m m) 18 (p m c 21)

kfq kfs znh zzk kfr kft

5 5

24 24

12.5 12.5

41 (p 4/m m m) 10 (p 2/m 1 1)

kfu btg

5 5 5 5 5 5

26 26 26 26 28 28

12.5 12.5 12.5 12.5 11 12.5

16 (p 2 m m) 16 (p 2 m m) 7 (p m 1 1) 7 (p m 1 1) 32 (p 4 2 m) 18 (p m c 21)

kyj kyk kyi kyl zmb olm

5 5

30 40

11 12.5

75 (p 6/m m m) 41 (p 4/m m m)

kfv kfx

Landolt-Börnstein New Series IV/14

description trd sharing 6 with adjacent 4, AST (c6h)2, net 1161 c14-tube, AET, DON, net 1160; actually p147/mmc 10-ring channel: 48/84/6(3,3)6-tube, WEI per sharing 8 consisting of edges from three adjacent 4, AEI sgw sharing 6, SGT pau and opr sharing 8, MER rth sharing 8, adjacent 6 in equator, cf. rtk, RTH rth sharing 8, adjacent 6 along rod, cf. rtj, RTH c18-tube, VFI, actually p189/mmc AABAB-chain, ISV, MTW, nets 873-5, 1005 4 and 6 sharing opposing vertices with tetrahedral geometry, net 278 6 & 8 sharing opposite edges, AEN opposite edges of cub joined by two handles to give 8, LTA sgt joined by edge along triad axis, SGT hpr sharing opposite edges between 4 with 6, = sigma of kur, net 279 (fhh)2-tube plus one f pointing out, MTW red sharing opposite edges with 4, DOH cub/lau/lau sharing 4, ISV crossed hpr and opr sharing 4, TSC sgt sharing 4, SGT hhhhzhz-tube with one z & one zz, = kdp sharing four edges, net 749 cub/toc sharing 4, LTA bet and mtw sharing 4, tes sharing 5 with bet and mtw, nets 577-9, Beta*,-A,B,C 11-ring channel, net 875 cub/nug/tes/mel/nug sharing faces, net 875 11-ring channel, nets 873, 874 bet/tes/nug/mel/wwf sharing faces, nets 873, 874 zma sharing 4, net 1304 kab/wwf(left)/wwf(right) sharing 4 with external 4 generating mel (l & r), CON, net 1005 doh sharing 6, DOH grc sharing 8 with adjacent 4, LTA

190

16.5 1D units

[Ref. p. 251

Table 16.5.1 (continued). R 6

N 6

d 11

rod group 63 (p 62 2 2)

label qhe

6

10

14

52 (p 3 m)

kfy

6

10

14

40 (p 42/m m c)

kfz

6 6

10 12

14 13.5

21(p c m m) 40 (p 42/m m c)

sfb eue

6

14

13

66 (p 6 2 m)

bpi

6 6

14 15

12 14

52 (p 3 m ) 16 (p 2 m m)

kga kgb

6

16

14

22 (p m m m)

kgd

6 6 6 6 6

16 16 16 16 18

13 14 13 13 14

21(p c m m) 16 (p 2 m m) 16 (p 2 m m) 12 (p 2/c 1 1) 21(p c m m)

brt kge kut odc sss

6 6 6 6

18 20 20 20

14 11 14 16

10 (p 2/m 1 1) 66 (p 6 2 m) 16 (p 2 m m) 22 (p m m m)

kgf kgg kgh kuq

6 6 6

20 22 22

14 14 13.5

21(p c m m) 22 (p m m m) 8 (p c 1 1)

stl kgi pet

6 6 6 6 6 6 6 6

24 24 24 24 24 24 24 24

14 13.5 13 10 15 13.5 15 14

40 (p 42/m m c) 32 (p 4 2 m) 22 (p m m m) 22 (p m m m) 21(p c m m) 20 (p c c m) 20 (p c c m) 16 (p 2 m m)

wne kgk kgj kue kgn nas noa kgm

6

24

15

16 (p 2 m m)

kgo

description six-repeat helix, enantiomorphic with 64(p6422), net 90 43 edge-linked along triad,180° rotation between adjacent 43, NAT, THO; kco is geometrically distinct 6 rotated by 90° sharing tetrahedral vertex, = sigma of ts, MEP left-3/4/right-3/4 chain, WEI euo sharing 4 and top edge of handle, = kdw sharing edge, EUO, NES afo joined by edge, BPH lai joined by edge along triad axis, net 1 complex, pairs of tes sharing 5 connected by two handles sharing edge, FER hpr sharing opposite edges between 4 with 8, CHA, GME, LEV, OFF Alberti f2, bru sharing edges, BOG three adjacent 4 connected by 4, PHI krs joined by two edges perpendicular to m, AFN open-double-cube sharing edges, AFN eun (= 6-sharing pes) sharing two adjacent edges, EUO, NES eun sharing two horizontal edges, CAS doo connected by edge along hexad axis, DOH kam & ohc sharing edge, net 720 bog with horizontal handles sharing middle edges with handles of vertical bog, net 43 alternation of bru and sti sharing 4, STI red & 6 sharing opposite edges, MEP pen sharing two edges with each of two pen, mirror image pet', MEL, MFI rotated opr sharing 4, RHO kdr sharing 4, MEL toc & 4 sharing opposite edges, SOD opr & ste sharing 4, MER left & right oto sharing 4, PHI nna sharing three edges, EUO, NON nna sharing three edges, net 574, NON mla joined by edge perpendicular to its triad axis, MEP afi/kaq/rotated kaq sharing three edges, nets 9, 10

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

191

Table 16.5.1 (continued). R 6 6

N 24 24

d 13.5 15

rod group 8 (p c 1 1) 7 (p m 1 1)

label kgl kgq

6 6 6 6 6

24 28 28 28 30

14.5 13.5 11 12.5 14.5

2 (p 1 ) 41 (p 4/m m m) 32 (p 4 2 m) 10 (p 2/m 1 1) 67 (p 6 2 c)

beb kus zmc kux kgs

6 6 6 6 6 6 6 6 6

30 32 32 32 32 34 36 36 36

14 13.5 10 14 13.5 12.5 14 15 15

kgr kgt kuf kgu kul mio kgw eba aeb

6

36

15

22 (p m m m) 41 (p 4/m m m) 22 (p m m m) 21 (p c m m) 21 (p c m m) 21 (p c m m) 75 (p 6/m m m) 74 (p 63/m m c) 66 (p 6 2 m) 66 (p 6 2 m)

description pen & tes sharing 5, MEL, MFI kaq & kdj sharing alternately 4 and three edges, net 1001 alternating mtw and tes sharing faces, BEA*- B kry sharing 4, AFN zma sharing 10 with alternation, net 1304 12-ring channel, BEA*-A,B,C 4 linked by tetrahedral vertices of opposite polarity, net 279 red sharing 5, DDR, DOH, MTN cub & trd sharing 4, AST pau & ste sharing 8, MER phi sharing 4, PHI straight channel, NES mil sharing 8 in zigzag, ESV mla sharing 6, MEP; (actually pbar12.2m) rotated gme sharing 6, EAB, net 118 eab & hpr sharing 6, EAB

kgx

lio sharing 6, LIO

6

36

15

66 (p 6 2 m)

kgy

can & los sharing 6 with adjacent 4, LIO

6

36

15

66 (p 6 2 m)

kgz

can/can/rotated can sharing 6, LIO

6

36

15

eri sharing 6, ERI

36

15

66 (p 6 2 m) 66 (p 6 2 m)

kha

6

khb

can/hpr/rotated can/hpr sharing 6, ERI

6

36

15

52 (p 3 m)

khc

cha/hpr sharing 6, CHA

6 6

36 36

12 15

51 (p 3 c) 22 (p m m m)

kgv ktu

6

40

14.5

37 (p 4 c c)

khe kun

krf sharing 6 along body diagonal, ANA ktg sharing 4 and two vertical edges, each generating tes, EPI four kfz linked by horizontal edges around tetrad axis, net 291 grc sharing 6 without rotation, RHO

6

42

13

6 6 7 7

48 48 17 26

13 15 16 13.5

52 (p 3 m) 66 (p 6 2 m) 41 (p 4/m m m) 66 (p 6 2 m) 21 (p c m m)

krj roh mig ktx

7 7 8

48 48 8

16 16 20

69 (p 63/m) 52 (p 3 m) 21 (p c m m)

kuc khf p

8 8 8

16 16 16

16 20 17.5

66 (p 6 2 m) 21 (p c m m) 14 (p 2 2 21)

ost pp ktz

Landolt-Börnstein New Series IV/14

bpa/bph sharing 12, BPH grc & opr sharing 8 with adjacent 4, RHO mei sharing 3 and extra edge, MEI left & right non sharing an edge, EUO, NES, NON meg sharing 12, MEI grc/hpr sharing 6, KFI pentasil, AABABBAB-chain, BOG, MEL, MFI, MOR, TER, net 987; also occurs in pp cub joined by edge, LTA double-p-chain, BOG gos sharing outer vertices of 4, GOO

192

16.5 1D units

[Ref. p. 251

Table 16.5.1 (continued). R 8

N 18

d 17

rod group 40 (p 42/m m c)

label hen

8 8

20 24

17 18.5

21 (p c m m) 21 (p c m m)

heu khg

8 8

24 24

20 20

21 (p c m m) 21 (p c m m)

kri kuj

8 8 8 8

32 40 44 48

20 20 20 21

10 (p 2/m 1 1) 16 (p 2 m m) 21 (p c m m) 74 (p 63/m m c)

kza khi kui cna

8 8

48 48

20 21

74 (p 63/m m c) 66 (p 6 2 m)

szy afg

8 8 8 8 8 8 9 9

48 48 56 62 64 64 30 44

20 16.5 18.5 16 18.5 18.5 19 22

66 (p 6 2 m) 22 (p m m m) 21 (p c m m) 8 (p c 1 1) 41 (p 4/m m m) 40 (p 42/m m c) 52 (p 3 m) 22 (p m m m)

szz khj khk mrt khl khm khn kzh

9

54

22

66 (p 6 2 m)

kzg

description Alberti e, bru sharing opposing vertices, HEU, STI bru joined by edge, BRE, HEU, STI left/right tilted hpr sharing 1,4' opposite edges with those of 4, = sigma(v) of ost, AEI, KFI (h'h'p)2-tube, BOG, MFI, TER left & right kof sharing central edge of handle, MFI lau/kzb sharing 4, net 398 complex 10-ring channel parallel to a-axis, MEL straight channel, MFI can sharing 6 with alternation of adjacent & rotated 4, AFG gme/hpr/rotated gme/hpr sharing 6, AFX can & lio sharing 6 with adjacent 4, AFG aft-hpr sharing 6, AFX cub/grc sharing 4, LTA per/rotated per sharing 8, AEI mrr sharing 9, STT grc/pau sharing 8, KFI pau sharing nonplanar-8, KFI due/dum sharing 3, net 853 cub/lau/ftt/lau sharing 4 with four bog sharing 6 with lau & ftt, DFO double-stellated relative of nne, DFO

9

54

22.5

hpr/lev/inverted lev sharing 6, LEV

60 62 66 108 36

20 21 19 22 23.5

52 (p 3 m) 52 (p 3 m) 8 (p c 1 1) 75 (p 4/m m m) 75 (p 4/m m m) 10 (p 2/m 1 1)

kho

9 9 9 9 10

khp mru zni kzf kym

10

52

23.5

10 (p 2/m 1 1)

kyh

10

60

26.5

74 (p 63/m m c)

knc

10

60

26.5

66 (p 6 2 m)

knb

10 11 12 12

60 88 28 30

24.5 26 26 16

21 (p c m m) 41 (p 4/m m m) 74 (p 63/m m c) 61 (p 61 2 2)

khr clo esc nzp

12

32

28

22 (p m m m)

khs

12

40

22

52 (p 3 m)

kht

grc/toc sharing 6 with adjacent 4, LTA mrr sharing 7, STT hpr/ber/hpr/znl sharing 6, MSO eni/eni/kzd sharing 12 with adjacent 4, DFO mel/mel/nuh/bet/tes/mtw/tes/bet/nuh sharing faces, nets 873-5 complex 10,12-ring channel, NU-86A,B,C, nets 873, 874, 875 can/toc/rotated can/rotated toc, alternating 4, net 201 los/toc/rotated toc, alternating 4 at los/toc, net 201 oto/plg/rotated oto/rotated plg sharing 8, PAU grc/rpa/rotated rpa, -CLO afo joined by edge, AFS vertical-triple-4 sharing two edges with adjacent triple-4 of spiral, CZP lov/lov/rotated lov/rotated lov linked by two edges, APD cub/trd joined by edge along triad axis, AST

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

193

Table 16.5.1 (continued). R

N

d

rod group

label

description red/rotated red joined by edge along triad axis, MEP alternating mtw and tes sharing faces, enantiomorphic with 30(p4322), BEA*-A alternating mtw and tes sharing faces, BEA*- C cha/hpr/rotated cha/hpr sharing 6, AFT hpr/toc/wof/rotated toc sharing 6 with alternate 4 between toc & wof, EMT aft/hpr/gme/hpr sharing 6 with alternate 4 across hpr, AFT hpr/can/znf/can sharing 6 with inline 4 across can & hpr, SBS grc/pau sharing 4, KFI hpr/can/niw/rotated can sharing 6, inline double4’s between can & niw, SAT wou/rotated wou sharing 12, EMT afe/bph/rotated afe/rotated bph sharing 12, AFS grc/plg/plg sharing 6 with alternate 4, PAU grc/opr/pau/opr/pau/opr sharing 8 with adjacent 4, PAU grc/ltn/toc/ 4 -related toc sharing 4 with alternate 4, LTN sgw/6/sgt sharing 4 with 4 -related sgt/6, SGT ltn sharing 6 according to 41, enantiomorph has 30(p4322), LTN mtn/shared 5/inverted mtn/edge/red/edge, MTN

12

40

23

52(p 3 m)

khu

12

48

25

29(p 41 2 2)

bea

12 12 12

48 72 72

27 30 27

12(p 2/c 1 1) 74(p 63/m m c) 66(p 6 2 m)

bec khy khw

12

72

27

66(p 6 2 m)

khx

12

72

27

66(p 6 2 m)

zzn

12 12

72 72

26 31

22(p m m m) 66(p 6 2 m)

khv zzl

12 12 14 14

96 120 90 112

27 27 28.5 35

74(p 63/m m c) 74(p 63/m m c) 52(p 3 m) 41(p 4/m m m)

khz krw kia kib

15

104

37

32(p 4 2 m)

kic

16 16

64 88

34 37

32(p 4 2 m) 29(p 41 2 2)

kid kie

17

70

34

52(p 3 m)

kif

20

88

40

52(p 3 m)

kig

26

168

64

52(p 3 m)

kih

Landolt-Börnstein New Series IV/14

det/shared 6/inverted det/edge/dtr/shared 6/inverted dtr/edge, DDR can/grc/hpr/grc/ rotated can/toc/toc/toc sharing 6 with alternate 4, LTN

194

16.5 1D units

[Ref. p. 251

Table 16.5.2 1D units in alphabetical order. R

N

d

rod group

label

description

6

36

15

66 (p 6 2 m)

aeb

eab & hpr sharing 6, EAB

8 4

48 12

21 8.5

66 (p 6 2 m) 74 (p 63/m m c)

afg afv

4

8

8.5

45 (p 3 )

afy

3 4 3 4

12 20 10 10

7 8.5 7 9.5

40 (p 42/m m c) 21 (p c m m) 10 (p 2/m 1 1) 22 (p m m m)

alh ape apt ast

2 2

16 8

5 5

41 (p 4/m m m) 40 (p 42/m m c)

ati atn

4 12

24 48

8.5 25

75 (p 6/m m m) 29 (p 41 2 2)

ave bea

6 12 4

24 48 8

14.5 27 8.5

2 (p 1 ) 12 (p 2/c 1 1) 40 (p 42/m m c)

beb bec bhs

6 3

14 6

13 7.5

66 (p 6 2 m) 22 (p m m m)

bpi bre

3 6 3

8 16 4

6.5 13 7.5

10 (p 2/m 1 1) 21 (p c m m) 22 (p m m m)

brs brt bs

5

24

12.5

10 (p 2/m 1 1)

btg

4

4

8.5

21 (p c m m)

c

can & lio sharing 6 with adjacent 4, AFG c6-tube, AEL, AET, AFI, AFO, ATV, DON, VFI, cub joined by edges extending body diagonal, AFY Alberti h, ref. 7, bru sharing 4, net 100 c10-tube, net 263, AEL, AFO, AHT; idealized lau sharing opposite edges, net 398 cub with opposing edges linked by 6, = sigma of ts, AST, net 291 ocn sharing 8, ATN (hz)4-tube, = kaa sharing 8, = kom sharing 8, ATN, nets 3, 402 c12-tube, AFI; actually p126/mmc alternating mtw and tes sharing faces, enantiomorphic with 30(p4322), BEA*-A alternating mtw and tes sharing faces, BEA*- B alternating mtw and tes sharing faces, BEA*- C bifurcated hexagon plus square, c4-tube, AEL, AET, AFI, AFO, AHT, APD, ATV, DFT, VFI, nets 38, 43, 1160, 1161 afo joined by edge, BPH shs, alternating 4 and 6 sharing edges, AFR, BRE, LAU, LOV, LTL, ZON, nets 26, 278, 282, 291, 609 Alberti f1, edge-shared bru, BRE Alberti f2, bru sharing edges, BOG back-to-back s, GOO, LAU, LOV, MON, RSN, VNI, VSV, YUG, ZON, bet and mtw sharing 4, tes sharing 5 with bet and mtw, nets 577-9, Beta*,-A,B,C simple crankshaft, ABW, AEL, AET, AFI, AFO, AHT, APC, APD, ATT, ATV, AWO, BIK, -CHI, DAC, DFT, DON, EPI, FER, MAZ, MFS, MON, MOR, RSN, VFI, VSV, nets 2, 3, 5, 6, 9, 10, 38, 100, 470, 536, 539, 575, 596, 601, 609, 720, 725, 748, 956, 979, 980, 987, 1043, 1160, 1161, (also part of cc)

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

195

Table 16.5.2 (continued). R 4

N 8

D 8.5

rod group 21 (p c m m)

label cc

4 8

16 48

8.5 21

40 (p 42/m m c) 74 (p 63/m m c)

chf cna

2

12

5

74 (p 63/m m c)

cnc

3

12

7.5

21 (p c m m)

csh

6 4 4 12 6

36 16 10 28 12

15 8.5 10 26 13.5

74 (p 63/m m c) 41 (p 4/m m m) 16 (p 2 m m) 74 (p 63/m m c) 40 (p 42/m m c)

eba ecc epi esc eue

5 4

5 4

13 9

16 (p 2 m m) 29 (p 41 2 2)

f fhe

3 4 3 4 3 3 8

5 16 16 24 9 10 18

6.5 8.5 7.5 10 7.5 8 17

32 (p 4 2 m) 21 (p c m m) 22 (p m m m) 74 (p 63/m m c) 1 (p 1) 2 (p 1 ) 40 (p 42/m m c)

fib for frt gmn goo hel hen

8 2

20 6

17 5

21 (p c m m) 16 (p 2 m m)

heu hhz

2 2 2

16 10 10

5 5 5

21 (p c m m) 16 (p 2 m m) 16 (p 2 m m)

hsr hst hsu

3 2 4

12 8 12

7.5 5 8.5

41 (p 4/m m m) 41 (p 4/m m m) 22 (p m m m)

kad kay kaz

4

6

8.5

22 (p m m m)

kba

3

8

7.5

22 (p m m m)

kbb

Landolt-Börnstein New Series IV/14

description double crankshaft, APC, ATT, ATV, AWO, DON, GIS, GME, MER, PHI, nets 6, 26, 598 (ch)4-tube, MER, net 38 can sharing 6 with alternation of adjacent & rotated 4, AFG (hz)6-tube, = kok sharing 12, ATO, CAN, net 470 (cshs)2-tube, MAZ, MOR, RSN, VSV, nets 100, 575 rotated gme sharing 6, EAB, net 118 c8-tube, ATV, DFT; actually p84/mmc tes joined by handles, DAC, EPI afo joined by edge, AFS euo sharing 4 and top edge of handle, = kdw sharing edge, EUO, NES AABAB-chain, ISV, MTW, nets 873-5, 1005 four-repeat helix, enantiomorphic with 30(p4322), GIS, JBW, nets 1, 400, 725 vertex-shared des, EDI, NAT, THO (ccch)2-tube, net 720 (ccshs)2-tube, DAC, FER (ch)6-tube, GME gos joined by 4, GOO bru joined by 4, BRE, HEU Alberti e, bru sharing opposing vertices, HEU, STI bru joined by edge, BRE, HEU, STI hhzhz-tube plus z, = pes sharing 2 edges, BIK, CAS, MTT, MTW, TON, VET (hhzhzhz)2-tube plus two z, CFI hsp sharing 5, CFI hst plus two double-h linking 2-connected vertices, CFI ste sharing 4, DAC, LOV, net 100 lau sharing 4, ATO, OSI, -RON, net 3 kaa joined by vertical 4; double kba related by sigma transformation, APD, nets 3, 6, 9, 10 6 joined by edge; back-to-back c, nets 38, 956, 1001 bs joined by horizontal edges to give lov and 8, = lov/kaa sharing 6, LOV, RSN, net 26

196

16.5 1D units

[Ref. p. 251

Table 16.5.2 (continued). R 3

N 8

d 7.5

rod group 22 (p m m m)

label kbf

4

8

8.5

66 ( 6 2 m)

kbg

3

12

7.5

22 (p m m m)

kbi

4 4

16 16

8.5 8.5

21 (p c m m) 40 (p 42/m m c)

kbl kbn

4 3

12 8

8.5 7.5

19 (p 2 c m ) 21 (p c m m)

kbp kbv

4

12

8.5

22 (p m m m)

kbx

2

4

5

22 (p m m m)

kcb

2

6

5

22 (p m m m)

kcc

2 2

6 8

5 5

2 (p 1 ) 9 (p 1 1 2/m)

kcd kce

2 2

10 10

5 5

22 (p m m m) 11 (p 1 1 21/m)

kcf kcg

2 2 2 2 2 2 2 3

10 10 11 12 12 16 20 5

5 5 5 5 5 5 5 7

9 (p 1 1 2/m) 6 (p 1 1 m) 16 (p 2 m m) 75 (p 6/m m m) 16 (p 2 m m) 16 (p 2 m m) 21 (p c m m) 66 (p 6 2 m)

kch kci kcj kck kcl kcm kcn kco

description bs joined by horizontal edges to give lov and 4, LOV, RSN kah joined by edge along trigonal axis, AEL, AET, AFI, AFO, AHT, APD, ATV, DFT, VFI, nets 2, 9, 10, 1001 (hs)4-tube, ste sharing horizontal edges, kaa joined by four vertical edges forming two 4, LOV, LTL, nets 38, 575, 601, 1177 left and right afs sharing 4, net 6 rotated bog sharing 4, AEL, AET, AFI, AFO, AHT, ATV, DFT ccchch-tube; kaq sharing 6, nets 5, 9, 10, 536 left and right euo sharing edges to give c, RSN, VSV bog and lov sharing 4; kbg sharing 2 edges, net 43 6 sharing two opposite edges - when flattened, would equal two z joined by parallel edges, ABW, CGS, nets 2, 282, 751 (also any 3D net containing a hex 2D net) two z joined by parallel squares, = kds sharing 4, nets 282, 609 zh'h'zh"h"-tube, nets 282, 609 eun sharing three edges, = hhh with three z, CAS, MTW mtw sharing 4, MTW two hhz sharing z plus extra vertex, CAS, MTT, MTW, TON lai sharing three edges, net 1 hhhhzhz-tube with one z & one zz, net 129 two hhz sharing z and extra vertex, MTT, TON ber sharing 6, nets 278, 978 hhhhhzhz-tube plus four z, CAS hhhhhhzhhz-tube plus six z, MTT (hhhhhz)2-tube plus four zz, ATS krs linked by edges along triad axis, EDI, THO; kfy is geometrically distinct

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

197

Table 16.5.2 (continued). R 3

N 8

D 7

rod group 7 (p m 1 1)

label kcp

3

8

7

16 (p 2 m m)

kcq

3

9

7.5

16 (p 2 m m)

kcr

3

9

7.5

16 (p 2 m m)

kcs

3

12

6.5

29 (p 41 2 2)

kct

3 3 3 3 3

12 12 12 12 12

6.5 6.5 6.5 7.5 7.5

10 (p 2/m 1 1) 10 (p 2/m 1 1) 15 (p m m 2) 32 (p 4 2 m) 21 (p c m m)

kcu kcv kcw kcx kcy

3 3 3 3 3 3 3

13 16 17 18 18 18 18

7.5 7.5 7.5 7.5 7.5 7.5 7.5

21 (p c m m m) 16 (p 2 m m) 16 (p 2 m m) 75 (p 6/m m m) 52 (p 3 m) 22 (p m m m) 16 (p 2 m m)

kcz kda kdb kdc kdd kde kdf

3

24

7.5

36 (p 4 m m)

kdg

3 3 4

24 26 8

7.5 7.5 8.5

16 (p 2 m m) 22 (p m m m) 22(p m m m)

kdh kdz kea

4 4 4 4 4 4

8 8 8 10 10 12

8.5 8.5 10.5 8.5 10 8.5

21 (p c m m) 16 (p 2 m m) 22 (p m m m) 16 (p 2 m m) 5 (p 1 1 21) 21 (p c m m)

keb kec ked kee kef keg

4 4

12 12

8.5 8.5

11 (p 1 1 21/m) 15 (p m m 2)

keh kei

Landolt-Börnstein New Series IV/14

description triple-4 in rocker geometry linked by two edges from inner to outer vertices, relative of kge, PHI tes joined by two vertical edges, DAC, FER, MFS, MOR hhshs-tube plus one s, external single 4, = kdk cage sharing two edges, MFS, SZF ccshhs-tube, = pes joined by two vertical edges, MFS four distorted s joined by horizontal edges, c-axis, NAT; enantiomorph 30(p4322) four distorted s joined by horizontal edges, THO ygw sharing 4, YUG kam sharing one edge, net 720 aww sharing 4, AWW kdw sharing inner edges of handles, = kbv plus horizontal handles, MON two kcq sharing s, FER, MFS kdy sharing three edges, MON kdo cages sharing four edges, MFS ber/hpr sharing 6, = sigma of kck, net 279 toc sharing 6 with alternate 4, SOD dah sharing 4, DAC, MOR, net 575 (hhhs)2-tube plus two ss, = gsm/kdq sharing 8, kfd = sigma kdf, ATT rpa sharing 8 with alternate 4, AWW (actually p 8 2 m) (chhcshhhhs-tube with six s, MFS frr sharing 6, FER 4 & 8 sharing opposite edges, ABW, AEN, STI, net 43 6 sharing 1- & 3-edges, AEN sti joined by vertical edge, net 720 6/4/4 sharing opposite edges, net 538 cchh-tube plus c, = kdt & sti sharing 4, net 748 tfs sharing edge, net 751 kah sharing two edges to generate c, AEL, AET, AFI, AFO, AHT, APD, ATV, VFI, net 2 sti sharing two edges to generate c, net 748 kdq sharing edge, ABW, ATT, JBW

198

16.5 1D units

[Ref. p. 251

Table 16.5.2 (continued). R 4

N 12

d 9

rod group 10 (p 2/m 1 1)

label kej

4

12

10

29 (p 41 2 2)

kek

4 4 4

12 14 16

10 8.5 8.5

18 (p m c 21) 16 (p 2 m m) 21 (p c m m)

kel kem ken

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

16 16 16 16 16 16 16 16 16 16 16 18 20 20 24

8.5 8.5 8.5 9 9 9 9.5 10 10 10 10 8.5 8.5 8.5 8.5

19 (p 2 c m ) 19 (p 2 c m ) 11 (p 1 1 21/m) 32 (p 4 2 m) 16 (p 2 m m) 10 (p 2/m 1 1) 10 (p 2/m 1 1) 15 (p m m2) 10 (p 2/m 1 1) 10 (p 2/m 1 1) 7 (p m 1 1) 49 (p 3 c) 22 (p m m m) 41 (p 4/m m m) 21 (p c m m)

keo kep keq ker kes ket keu kev kew kex key kez kfa kfb kfc

4

24

8.5

21 (p c m m)

kfd

4

24

8.5

16 (p 2 m m)

kfe

4

26

8.5

66 ( 6 2 m)

kfg

4 4

26 28

9.5 9.5

22 (p m m m) 7 (p m 1 1)

kfh kfi

4 5 5

30 10 12

10 13 12.5

16 (p 2 m m) 22 (p m m m) 22 (p m m m)

kfj kfl kfm

description two opposite edges of hpr joined by two edges generating 4, AEI, CHA 4 sharing edge with fhe, enantiomorphic with 30(p4322), net 725 kdn sharing 4, YUG kam/sti sharing 4, net 720 (hhc)2-tube with two c, = kdq sharing 6, ABW, nets 398, 536 kds sharing two vertices yielding two ts, net 282 ccccchch-tube, = kdl sharing 8, APD two kee sharing c, net 748 gsm sharing 4, = sigma of kei, ATT, FER afs & kal sharing 6, net 6 fsi sharing 4, net 5 kau & lau sharing three edges, net 398 vvn sharing 4, net 6 vvs sharing 4, ABW, net 398 ygw joined by two edges to yield 8, YUG kdm sharing 4, APC three ts connected by horizontal edges, net 278 (chchh)2-tube, net 748 toc sharing 4, SOD vvn sharing 8 to yield two cc, = (ccch)2-tube with two cc, net 6 left & right gsm sharing 8 to yield two cc, = (chhh)2-tube with two cc, ATT, GIS chchhhhh-tube with two cc, = oto & phi sharing 8, PHI kag connected by 4 edges which yield three 4 sharing edge on triad axis, net 538 trd sharing 6 with adjacent 4, AST per sharing 8 consisting of edges from three adjacent 4, AEI sgw sharing 6, SGT 6 & 8 sharing opposite edges, AEN opposite edges of cub joined by two handles to give 8, LTA

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

199

Table 16.5.2 (continued). R

N

d

rod group

label

description

5 5

14 14

10 11

66 (p 6 2 m) 22 (p m m m)

kfn kfo

5 5 5 5

17 20 20 20

13 10 11 11

7 (p m 1 1) 21 (p c m m) 22 (p m m m) 18 (p m c 21)

kfq kfr kfs kft

5 5 5 6

24 30 40 10

12.5 11 12.5 14

41 (p 4/m m m) 75 (p 6/m m m) 41 (p 4/m m m) 52 (p 3 m)

kfu kfv kfx kfy

6

10

14

40 (p 42/m m c)

kfz

6 6

14 15

12 14

52 (p 3 m) 16 (p 2 m m)

kga kgb

6

16

14

22 (p m m m)

kgd

6 6 6 6 6 6 6 6 6 6 6

16 18 20 20 22 24 24 24 24 24 24

14 14 11 14 14 13 13.5 13.5 14 15 15

16 (p 2 m m) 10 (p 2/m 1 1) 66 (p 6 2 m) 16 (p 2 m m) 22 (p m m m) 22 (p m m m) 32 (p 4 2 m) 8 (p c 1 1) 16 (p 2 m m) 21(p c m m) 16 (p 2 m m)

kge kgf kgg kgh kgi kgj kgk kgl kgm kgn kgo

6

24

15

7 (p m 1 1)

kgq

6 6

30 30

14 14.5

22 (p m m m) 67 (p 6 2 c)

kgr kgs

6 6 6

32 32 36

13.5 14 12

41 (p 4/m m m) 21 (p c m m) 51 (p 3 c)

kgt kgu kgv

sgt joined by edge along triad axis, SGT hpr sharing opposite edges between 4 with 6, = sigma of kur, net 279 (fhh)2-tube plus one f pointing out, MTW sgt sharing 4, SGT red sharing opposite edges with 4, DOH hhhhzhz-tube with one z & one zz, = kdp sharing four edges, net 749 cub/toc sharing 4, LTA doh sharing 6, DOH grc sharing 8 with adjacent 4, LTA 43 edge-linked along triad,180° rotation between adjacent 43, NAT, THO; kco is geometrically distinct 6 rotated by 90° sharing tetrahedral vertex, = sigma of ts, MEP lai joined by edge along triad axis, net 1 complex, pairs of tes sharing 5 connected by two handles sharing edge, FER hpr sharing opposite edges between 4 with 8, CHA, GME, LEV, OFF three adjacent 4 connected by 4, PHI eun sharing two horizontal edges, CAS doo connected by edge along hexad axis, DOH kam & ohc sharing edge, net 720 red & 6 sharing opposite edges, MEP toc & 4 sharing opposite edges, SOD kdr sharing 4, MEL pen & tes sharing 5, MEL, MFI mla joined by edge perpendicular to its triad left & right oto sharing 4, PHI afi/kaq/rotated kaq sharing three edges, nets 9, 10 kaq & kdj sharing alternately 4 and three edges, net 1001 red sharing 5, DDR, DOH, MTN 4 linked by tetrahedral vertices of opposite polarity, net 279 cub & trd sharing 4, AST phi sharing 4, PHI krf sharing 6 along body diagonal, ANA

Landolt-Börnstein New Series IV/14

200

16.5 1D units

[Ref. p. 251

Table 16.5.2 (continued). R 6 6

N 36 36

D 14 15

rod group 75 (p 6/m m m) 66 (p 6 2 m)

label kgw kgx

description mla sharing 6, MEP; (actually pbar12.2m) lio sharing 6, LIO

6

36

15

66 (p 6 2 m)

kgy

can & los sharing 6 with adjacent 4, LIO

6

36

15

can/can/rotated can sharing 6, LIO

36

15

66 (p 6 2 m) 66 (p 6 2 m)

kgz

6

kha

eri sharing 6, ERI

6

36

15

66 (p 6 2 m)

khb

can/hpr/rotated can/hpr sharing 6, ERI

6 6

36 40

15 14.5

52 (p 3 m) 37 (p 4 c c)

khc khe

7 8

48 24

16 18.5

52 (p 3 m) 21 (p c m m)

khf khg

8 8 8 8 8 9

40 48 56 64 64 30

20 16.5 18.5 18.5 18.5 19

16 (p 2 m m) 22 (p m m m) 21 (p c m m) 41 (p 4/m m m) 40 (p 42/m m c) 52 (p 3 m)

khi khj khk khl khm khn

cha/hpr sharing 6, CHA four kfz linked by horizontal edges around tetrad axis, net 291 grc/hpr sharing 6, KFI left/right tilted hpr sharing 1,4' opposite edges with those of 4, = sigma(v) of ost, AEI, KFI complex 10-ring channel parallel to a-axis, MEL cub/grc sharing 4, LTA per/rotated per sharing 8, AEI grc/pau sharing 8, KFI pau sharing nonplanar-8, KFI due/dum sharing 3, net 853

9

54

22.5

52 (p 3 m)

kho

hpr/lev/inverted lev sharing 6, LEV

9 10 12

60 60 32

20 24.5 28

52 (p 3 m) 21 (p c m m) 22 (p m m m)

khp khr khs

12

40

22

52 (p 3 m)

kht

grc/toc sharing 6 with adjacent 4, LTA oto/plg/rotated oto/rotated plg sharing 8, PAU lov/lov/rotated lov/rotated lov linked by two edges, APD cub/trd joined by edge along triad axis, AST

12 12 12

40 72 72

23 26 27

52 (p 3 m) 22 (p m m m) 66 (p 6 2 m)

khu khv khw

12

72

27

66 (p 6 2 m)

khx

12 12 14

72 96 90

30 27 28.5

khy khz kia

14

112

74 (p 63/m m c) 74 (p 63/m m c) 52 (p 3 m) 41 (p 4/m m m)

35

kib

red/rotated red joined by edge along triad axis, grc/pau sharing 4, KFI hpr/toc/wof/rotated toc sharing 6 with alternate 4 between toc & wof, EMT aft/hpr/gme/hpr sharing 6 with alternate 4 across hpr, AFT cha/hpr/rotated cha/hpr sharing 6, AFT wou/rotated wou sharing 12, EMT grc/plg/plg sharing 6 with alternate 4, PAU grc/opr/pau/opr/pau/opr sharing 8 with adjacent 4, PAU

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

201

Table 16.5.2 (continued). R

N

D

rod group

label

description

15

104

37

32 (p 4 2 m)

kic

grc/ltn/toc/ 4 -related toc sharing 4 with alternate 4, LTN

16 16

64 88

34 37

32 (p 4 2 m) 29 (p 41 2 2)

kid kie

17

70

34

52 (p 3 m)

kif

sgw/6/sgt sharing 4 with 4 -related sgt/6, SGT ltn sharing 6 according to 41, enantiomorph has 30(p4322), LTN mtn/shared 5/inverted mtn/edge/red/edge, MTN

20

88

40

52 (p 3 m)

kig

26

168

64

52 (p 3 m)

kih

4 10

24 60

8.5 26.5

19 (p 2 c m) 66 (p 6 2 m)

kij knb

10

60

26.5

74 (p 63/m m c)

knc

3 3 3

12 12 9

7.5 7.5 7.5

9 (p 1 1 2/m) 16 (p 2 m m) 16 (p 2 m m)

kng kox kpa

3

18

7.5

22 (p m m m)

kpk

3 3 3

15 12 9

7.5 7.5 7.5

16 (p 2 m m) 16 (p 2 m m) 16 (p 2 m m)

kpp kps kqy

4

12

8.5

5 (p 1 1 21)

krc

4 8 6

20 24 48

8.5 20 13

22 (p m m m) 21 (p c m m) 66 (p 6 2 m)

krg kri krj

4 12 3 3 6

24 120 10 12 36

9 27 7 6.5 15

52 ( 3 m) 74 (p 63/m m c) 22 (p m m m) 32 (p 4 2 m) 22 (p m m m)

krv krw krx ktf ktu

4

20

8.5

21 (p c m m)

ktv

7

26

13.5

21 (p c m m)

ktx

8 3

16 4

17.5 7.5

14 (p 2 2 21) 22 (p m m m)

ktz kua

Landolt-Börnstein New Series IV/14

det/shared 6/inverted det/edge/dtr/shared 6/inverted dtr/edge, DDR can/grc/hpr/grc/ rotated can/toc/toc/toc sharing 6 with alternate 4, LTN ccchhchh-tube with two parallel cc, APC los/toc/rotated toc, alternating 4 at los/toc, net 201 can/toc/rotated can/rotated toc, alternating 4, net 201 ygw sharing three edges, SZF (hhs)2-tube plus two s, JBW hhshs-tube plus one s, internal double-4, = koi & pes sharing five edges, MFS (hhshs)2-tube plus two s, two internal single-4, SZF two kcr sharing s, SZF (ccshhs)-tube plus two s, BIK ccshs-tube plus one s, DAC, FER, MOR, nets 575, 601, 987 krb sharing side edge of handle with shared edge of 4's, net 725 kdu joined by four vertical edges, APD (h'h'p)2-tube, BOG, MFI, TER bpa/bph sharing 12, BPH afe sharing 12, AFY afe/bph/rotated afe/rotated bph sharing 12, AFS tti joined by 4, AFR krq sharing upper & lower 8, EDI ktg sharing 4 and two vertical edges, each generating tes, EPI two ktw sharing pairs of 4 with handles related by c-glide, AET, AHT, VFI left & right non sharing an edge, EUO, NES, NON gos sharing outer vertices of 4, GOO 8 sharing opposite edges = teeth of two s joined by edge, JBW, RSN

202

16.5 1D units

[Ref. p. 251

Table 16.5.2 (continued). R

N

d

rod group

label

description hes sharing opposing vertices, JBW, net 1 meg sharing 12, MEI opr & ste sharing 4, MER pau & ste sharing 8, MER kcf plus two outer z, MTW straight channel, MFI left & right kof sharing central edge of handle, MFI straight channel, NES grc sharing 6 without rotation, RHO bog with horizontal handles sharing middle edges with handles of vertical bog, net 43 4 and 6 sharing opposing vertices with tetrahedral geometry, net 278 kry sharing 4, AFN krs joined by two edges perpendicular to m, AFN (chh)2-tube, knn sharing 6, net 538 (chchhcshschh)-tube plus two s & three ss, net 575 12-ring channel, BEA*-A,B,C kem with extra handle which converts sti into ohc, net 720 kyf sharing outer 4, net 963 complex 10,12-ring channel, NU-86A,B,C, nets 873, 874, 875 11-ring channel, nets 873, 874 11-ring channel, net 875 cub/nug/tes/mel/nug sharing faces, net 875 bet/tes/nug/mel/wwf sharing faces, nets 873, 874 mel/mel/nuh/bet/tes/mtw/tes/bet/nuh sharing faces, nets 873-5 krz sharing 4, net 400 kjr sharing 4, net 400 kyu sharing 4, AEN kaa sharing one edge, net 398 lau joined by two edges forming 4, net 398 lau/kzb sharing 4, net 398 eni/eni/kzd sharing 12 with adjacent 4, DFO double-stellated relative of nne, DFO cub/lau/ftt/lau sharing 4 with four bog sharing 6 with lau & ftt, DFO

4 7 6 6 2 8 8

9 48 24 32 12 44 24

7 16 10 10 5 20 20

32 (p 4 2 m) 69 (p 63/m) 22 (p m m m) 22 (p m m m) 22 (p m m m) 21 (p c m m) 21 (p c m m)

kub kuc kue kuf kug kui kuj

6 6 6

32 42 20

13.5 13 16

21 (p c m m) 52 (p 3 m) 22 (p m m m)

kul kun kuq

5

8

10

22 (p m m m)

kur

6 6 4 3

28 16 12 30

13.5 13 8.5 7.5

41 (p 4/m m m) 16 (p 2 m m) 21 (p c m m) 16 (p 2 m m)

kus kut kuv kuw

6 4

28 16

12.5 8.5

10 (p 2/m 1 1) 16 (p 2 m m)

kux kyc

4 10

12 52

9 23.5

8 (p c 1 1) 10 (p 2/m 1 1)

kyg kyh

5 5 5 5 10

26 26 26 26 36

12.5 12.5 12.5 12.5 23.5

7 (p m 1 1) 16 (p 2 m m) 16 (p 2 m m) 7 (p m 1 1) 10 (p 2/m 1 1)

kyi kyj kyk kyl kym

4 4 4 3 4 8 9 9 9

16 16 12 10 12 32 108 54 44

10 10 9 7 8.5 20 22 22 22

10 (p 2/m 1 1) 2 (p 1 ) 4 (p 2 1 1) 2 (p 1 ) 10 (p 2/m 1 1) 10 (p 2/m 1 1) 75 (p 4/m m m) 66 (p 6 2 m) 22 (p m m m)

kys kyt kyv kyy kyz kza kzf kzg kzh

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

203

Table 16.5.2 (continued). R

N

rod group

label

description

2

8

5

16 (p 2 m m)

lao

10 7.5 10 7.5 7 5 7.5 8.5 16 12.5 10 7.5

66 ( 6 2 m) 75 (p 6/m m m) 74 (p 63/m m c) 10 (p 2/m 1 1) 16 (p 2 m m) 21 (p c m m) 21 (p c m m) 41 (p 4/m m m) 66 ( 6 2 m) 21 (p c m m) 7 (p m 1 1) 11 (p 1121/m)

ldo lel lso lum lvi mao mdn mer mig mio mip mod

62 62 36

16 21 7.5

mrt mru mzz

2 6 6 12

8 24 24 30

5 13.5 15 16

8 (p c 1 1) 8 (p c 1 1) 66 ( 6 2 m) 16 (p 2 m m) 20 (p c c m) 20 (p c c m) 61 (p 61 2 2)

hhhzhz-tube plus two z, = afi sharing three horizontal edges, nets 2, 956, 1001 los sharing 6, LOS lil sharing 12, LTL, net 1177 can rotated by 180° and sharing 6, LOS alternating lau and bog sharing 6, LAU ohc sharing edges, AFR, ZON, net 720 (hhhz)2-tube plus four z, BIK, 756 (cccshs)2-tube, MOR, net 987 pau and opr sharing 8, MER mei sharing 3 and extra edge, MEI mil sharing 8 in zigzag, ESV mil sharing 6, ESV edge-shared tes plus extra edge, cshs.cshs.c(bridge), DAC, EPI, FER, MOR, nets 575, 601, 987 mrr sharing 9, STT mrr sharing 7, STT maz sharing 12, MAZ

4 3 4 3 3 2 3 4 7 6 4 3

24 36 24 12 8 12 20 32 17 34 24 12

8 9 3

nao nas noa nzp

6 3

16 18

13 7.5

12 (p 2/c 1 1) 66 ( 6 2 m)

odc off

3 5

18 28

7.5 12.5

66 ( 6 2 m) 18 (p m c 21)

ofr olm

2 8 8

16 16 8

5 16 20

40 (p 42/m m c) 66 ( 6 2 m) 21 (p c m m)

osi ost p

6

22

13.5

8 6

16 6

6 3 4 4

48 24 16 32

Landolt-Börnstein New Series IV/14

d

8 (p c 1 1)

pet

20 11

21 (p c m m) 63 (p 62 2 2)

pp qhe

15 7.5 10 10

41 (p 4/m m m) 10 (p 2/m 1 1) 13 (p 2 2 2) 10 (p 2/m 1 1)

roh rtf rti rtk

hhhzhz-tube plus zz, ATS, OSI nna sharing three edges, EUO, NON nna sharing three edges, net 574, NON vertical-triple-4 sharing two edges with adjacent triple-4 of spiral, CZP open-double-cube sharing edges, AFN (hs)6-tube, OFF, net 1177 can/hpr sharing 6, LTL, OFF, -WEN, net 1177 kab/wwf(left)/wwf(right) sharing 4 with external 4 generating mel (l & r), CON, net 1005 (hhz)4-tube plus four z, OSI cub joined by edge, LTA pentasil, AABABBAB-chain, BOG, MEL, MFI, MOR, TER, net 987; also occurs in pp pen sharing two edges with each of two pen, mirror image pet', MEL, MFI double-p-chain, BOG six-repeat helix, enantiomorphic with 64(p6422), net 90 grc & opr sharing 8 with adjacent 4, RHO rte sharing 8, RTE cle sharing 4, RTH rth sharing 8, adjacent 6 along rod, cf. rtj, RTH

204

16.5 1D units

[Ref. p. 251

Table 16.5.2 (continued). R

N

3

3

2

12

3

6

4

10

6 3

d

rod group

label

description

7.5

16 (p 2 m m)

s

5

21 (p c m m)

sao

7.5

21 (p c m m)

scs

10

20 (p c c m)

sfa

10 10

14 7.5

21 (p c m m) 10 (p 2/m 1 1)

sfb sfc

4 3

28 6

10 7.5

20 (p c c m) 16 (p 2 m m)

sfd ss

6

18

14

21 (p c m m)

sss

3 6 3 8 8 3

12 20 22 48 48 3

7.5 14 7.5 20 20 7

41 (p 4/m m m) 21 (p c m m) 22 (p m m m) 74 (p 63/m m c) 66 ( 6 2 m) 47 (p 31 2)

sta stl szf szy szz the

4

12

8.5

22 (p m m m)

thr

3 2 3 2

18 16 12 8

7.5 5 7 5

66 ( 6 2 m) 21 (p c m m) 66 ( 6 2 m) 21 (p c m m)

tix tno tof ton

3 4

24 6

7.5 8.5

41 (p 4/m m m) 40 (p 42/m m c)

too ts

4 4

24 10

10 8.5

66 ( 6 2 m) 16 (p 2 m m)

tsn tts

4 2

28 16

8.5 5

22 (p m m m) 32 (p 4 2 m)

ute veu

simple saw, BIK, DAC, EDI, EPI, FER, GOO, HEU, JBW, MFS, MON, MOR, RSN, SZF, VSV, ZON, nets 38, 100, 282, 575, 596, 601, 609, 979, 987, (also part of ss) (hhhz)2-tube plus two zz, = vvs sharing 8, ABW, JBW alternating left and right 5 sharing edges, = sh's, BIK, BRE, EPI, FER, MAZ, MFS, MON, MOR, RSN, VSV, nets 100, 575, 596, 609, 987 column of sfi joined by two edges to give 6, = ctt/ttc, WEI left-3/4/right-3/4 chain, WEI bre with 2-connected vertex of 6 becoming 4-connected vertex of sfi, WEI 10-ring channel: 48/84/6(3,3)6-tube, WEI double saw, ATT, LTL, MAZ, OFF, -WEN, nets 575, 601, 1177 eun (= 6-sharing pes) sharing two adjacent edges, EUO, NES alternating cub and lau sharing 4, net 291 alternation of bru and sti sharing 4, STI hpr/son sharing 6, SZF gme/hpr/rotated gme/hpr sharing 6, AFX aft-hpr sharing 6, AFX three-repeat helix, enantiomorphic with 48(p322), LAU, YUG, nets 90, 619, 1193 (cch)2-tube, lau joined by two edges, nets 3, 1161 gme sharing 6, MAZ, OFF, net 575 (hhhhz)2-tube plus six z, TON (hss)3-tube, RSN lai sharing 6, (hhz)2-tube plus two z, JBW, MTT, MTW, TON, 1, net 977 pau sharing 8, net 1177 twisted square, AFR, ANA, CZP, EAB, GIS, GME, GOO, LAU, MER, PAU, PHI, RHO, -RON, SAO, SOD, YUG, nets 278, 282, 984, 989 alternating gme and hpr sharing 6, GME mel joined by two vertical edges, cccch, nets 1160, 1161 (c6h)2, net 1161 (h2z)4 with sidepockets, VET

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

205

Table 16.5.2 (continued). R 4 2 3 3 4

N 12 12 10 14 28

d 8.5 5 7 7 8.5

rod group 21 (p c m m) 66 (p 6 2 m) 16 (p 2 m m) 16 (p 2 m m) 21 (p c m m)

label vfi voi vpe vpv vtn

3 3 6 3

28 28 24 12

7.5 7.5 14 7.5

6 (p m 1 1) 10 (p 2/m 1 1) 40 (p 42/m m c) 10 (p 2/m 1 1)

wai wam wne wwv

4 2

36 2

8.5 5

74 (p 63/m m c) 21(p c m m)

yee z

2

6

5

47 (p 31 2)

zhp

5 6 5 9 2

28 28 20 66 4

11 11 12.5 19 5

32 (p 4 2 m) 32 (p 4 2 m) 22 (p m m m) 75 (p 4/m m m) 21 (p c m m)

zmb zmc znh zni zz

2 4 5 12

16 16 20 72

5 8.5 12.5 31

22 (p m m m) 18 (p m c 21) 22 (p m m m) 66 (p 6 2 m)

zzi zzj zzk zzl

4 12

24 72

10 27

22 (p m m m) 66 (p 6 2 m)

zzm zzn

3

12

18 (p m c 21)

zzo

Landolt-Börnstein New Series IV/14

7.5

description double-c4, = two bhs sharing c, AET, AHT, VFI can sharing 6, CAN hhhss-tube plus two s, VSV (hsthsshts)-tube, LOV, VSV c14-tube, AET, DON, net 1160; actually p147/mmc wah sharing 10, SFF wan sharing 10, STF rotated opr sharing 4, RHO tte sharing 4, all parallel giving clino choice, RTE c18-tube, VFI, actually p189/mmc simple zigzag, ABW, AEN, ATN, ATO, ATS, BIK, CAN, CAS, CFI, -CHI, DAC, EPI, JBW, MTT, MTW, OSI, TON, YUG, nets 1, 2, 3, 10, 90, 282, 400, 470, 597, 609, 749, 977, 978, 989, 1001, 1031, 1157, 1246 (part of zz) zh'zh'zh'-tube, enantiomorphic with 48 (p322), ATO zma sharing 4, net 1304 zma sharing 10 with alternation, net 1304 cub/lau/lau sharing 4, ISV hpr/ber/hpr/znl sharing 6, MSO double zigzag, ABW, AEN, ATN, ATS, CAN, CFI, JBW, OSI, nets 470, 749, 989, 1246 (hhhzhz)2-tube plus four z , MTW kqr/rotated kqr/kqr sharing 6, CGS crossed hpr and opr sharing 4, TSC hpr/can/niw/rotated can sharing 6, inline double4’s between can & niw, SAT trc & ste sharing 8, net 1043 hpr/can/znf/can sharing 6 with inline 4 across can & hpr, SBS left and right mel sharing 6, net 1300

206

16.5 1D units

[Ref. p. 251

aeb

afg

afv

afy

alh

ape

apt

ast

Fig. 16.5.1 Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

207

ati

atn

ave

bea

beb

bec

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

208

16.5 1D units

[Ref. p. 251

bhs

bpi

bre

brs

brt

btg

bs

c

cc

chf

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

209

clo

cna

cnc

csh

eba

ecc

esc

epi

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

210

eue

frt

16.5 1D units

fhe

f

gmn

[Ref. p. 251

fib

goo

for

hel

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

hen

hst

16.5 1D units

heu

hsu

hhz

kad

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

211

hsr

kay

212

16.5 1D units

kaz

kba

kbg

kbi

kbb

kbl

[Ref. p. 251

kbf

kbn

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

kbp

kbv

kcc

kcd

kbx

kce

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

213

kcb

kcf

214

16.5 1D units

[Ref. p. 251

kcg

kch

kci

kcj

kck

kcl

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

kcm

kcp

kcn

kcq

kco

kcr

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

215

kcs

216

kct

kcx

16.5 1D units

kcu

[Ref. p. 251

kcv

kcy

kcw

kcz

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

217

kda

kdb

kdc

kdd

kde

kdf

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

218

16.5 1D units

kdg

kea

kdh

keb

[Ref. p. 251

kdz

kec

ked

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

kee

kef

kei

kej

keg

kek

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

219

keh

kel

220

kem

keq

16.5 1D units

ken

ker

[Ref. p. 251

keo

kes

kep

ket

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

keu

kev

key

kez

kew

kfa

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

221

kex

kfb

222

16.5 1D units

kfc

kfd

kfg

kfh

[Ref. p. 251

kfe

kfi

Fig. 16.5.1 (continued) Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref.p.251]

kfj

kfo

16.5 1D units

kfl

kfq

kfm

kfr

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

223

kfn

kfs

224

16.5 1D units

kft

kfy

kfu

kfz

[Ref. p. 251

kfv

kga

kfx

kgb

kgd

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref.p.251]

16.5 1D units

225

kge

kgf

kgg

kgh

kgi

kgj

kgk

kgl

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

226

16.5 1D units

[Ref. p. 251

kgl

kgm

kgn

kgo

kgq

kgr

kgs

kgt

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref.p.251]

kgu

kgy

16.5 1D units

kgv

kgz

227

kgw

kha

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

kgx

khb

228

16.5 1D units

khc

khe

khi

khj

khf

khk

[Ref. p. 251

khg

khl

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref.p.251]

16.5 1D units

khm

khn

khr

khs

kho

kht

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

229

khp

khu

230

16.5 1D units

khv

khw

khz

kia

khx

kib

[Ref. p. 251

khy

kic

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref.p.251]

16.5 1D units

kid

kih

kie

kij

kif

knb

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

231

kig

kng

232

16.5 1D units

kox

kpp

[Ref. p. 251

kpa

kps

kpk

kqy

krc

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref.p.251]

16.5 1D units

krg

krv

kri

krw

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

233

krj

krx

234

16.5 1D units

ktf

ktu

ktz

kua

ktv

kub

[Ref. p. 251

ktx

kuc

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref.p.251]

16.5 1D units

235

kue

kuf

kug

kui

kuj

kul

kun

kuq

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

236

kur

kuw

16.5 1D units

kus

[Ref. p. 251

kut

kux

kuv

kyc

kyg

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref.p.251]

kyh

kyl

16.5 1D units

kyi

kym

kyj

kyk

kys

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

237

kyt

238

kyv

kzf

16.5 1D units

kyy

[Ref. p. 251

kyz

kzg

kza

kzh

lao

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

ldo

lum

lel

lvi

lso

mao

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

239

mdn

240

mer

mip

16.5 1D units

mig

[Ref. p. 251

mio

mod

mrt & mru not drawn

mzz

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

nao

odc

16.5 1D units

nas

noa

off

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

241

nzp

ofr

242

16.5 1D units

olm

osi

pet

pp

[Ref. p. 251

ost

qhe

p

roh

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

rtf

rtk

16.5 1D units

rti

243

rtj

sao

s

scs

Fig. 16.5.1 (continued). Drawings of 1D units in near alphabetical order.

Landolt-Börnstein New Series IV/14

sfa

244

sfb

sss

16.5 1D units

sfc

sta

[Ref. p. 251

sfd

stl

ss

szf

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

szy

tix

16.5 1D units

szz

tno

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

245

the

thr

tof

246

ton

tts

16.5 1D units

too

[Ref. p. 251

ts

ute

tsn

veu

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

vfi

voi

vpv

vtn

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

247

vpe

wai

248

wam

yee

16.5 1D units

wne

[Ref. p. 251

wwv

zhp

z

zmb

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

Ref. p. 251]

16.5 1D units

zmc

zz

znh

zzi

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

249

zni

zzj

250

zzk

zzn

16.5 1D units

[Ref. p. 251

zzl

zzm

zzo

Fig. 16.5.1 (continued). Drawings of 1D units in alphabetical order.

Landolt-Börnstein New Series IV/14

17 References

17

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71Bru1 71Bru2 71Fis1 71Mar1

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71Mit1

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77Ave1 77Bae1

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82Gar1 82Gra1 82Hel1 82Lag1 82Mac1 82Sat1 82Smi1 82Sol1 82Tag1 82Wil1

83And1 83Lie1 83Mer1 83Nym1 83Sch1 83Sen1 83Sla1 83Smi1 83Wel1 83Wel2

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84Koc1 84Nym1 84Par1 84Par2 84Smi1 84Smi2 85Ben1 85Chi1 85DeW1 85Fer1

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86Wel1 86Wel2 87Bar1 87Fis1 87Giu1 87Goel

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Andersson S., Hyde, B.G., Larsson, K., Lidin, S.: Chem. Rev. 88 (1988) 221.

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17 References 90Bru1 90Gab1 90Gut1 90Han1 90Han2 90Han3 90Hos1 90Kni1 90Koh1 90Lid1 90Lim1 90Mar1 90Mar2 90Mar3 90Plu1 90Sti1 91Atw1

91Der1 91Eng1 91Eng2 91Goe1 91Hyd1 91Kap1 91Kot1 91Kra1 91Li1 91Lid1 91Mac1 91Mül1 91Nic1 91Nis1 91O’Ke1 91O’Ke2 91O’Ke3 91Ros1 91van1 91Veg1 92Akp1 92Bau1 92Bec1

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260 92Bei1 92Beu1 92Bru1 92Cat1 92Cra1 92Dee1 92Dee2 92DeM1 92Ing1 92Jac1 92Jef1 92Kim1 92Lar1 92Mie1 92Mül1 92New1 92O’Ke1 92O’Ke2 92O’Ke3 92Pri1 92Rob1 92Stu1 92Szo1 92Tho1 92Tsi1 92Van1 92Woo1 93And1 93And2 93Bau1 93Bru1 93Bru2 93Fei1 93Fér1 93Fow1 93Gha1 93Har1 93Hei1

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97And1 97Bad1

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96Alf1 96Boy1 96Cav1 96Gar1 96Gro1 96Hig1 96Kho1 96Kli1 96Kos1 96Lew1 96Lid1 96Loi1 96Mei1 96O’Ke1 96O’Ke2 96Sat1 96Tre1

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17 References 97Cam1 97Hyd1 97Lil 97Lob1 97Lob2 97McC1 97O’Ke1 97Sap1 97Sch1 97Stu1 97Thi1 97Tre1

98Abr1 98Akp1 98Bu1 98Bu2 98Bu3 98Cat1 98Coo1 98Fra1 98Gie1 98Li1 98O’Ke1 98O’Ke2 98O’Ke3 98Oli1 98Rei1 98Ren1 98Tan1 98Vil1 98War1 98Zon1

99Bar1 99Bar2

99Bau2 99Bau3

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18 Acknowledgements

Acknowledgements Joseph J. Pluth gave invaluable help with computer programs and graphical display of the 2D nets. Koen Andries did an independent analysis of 3D nets in 1990 to check for sub-units against those determined by me. We agreed very closely. In general, he obtained several more subunits than I did. I accepted with pleasure the extra ones that seemed to be of topologic and crystal-chemical significance, but omitted some that seemed too bizarre to have importance to synthesis chemists. He drew most of the polyhedral and 1D units, most of which I have redrawn to heighten the perspective. I take full responsibility for any errors, and want to emphasize how much intellectual credit goes to Koen’s original work. David Brown read thoroughly an earlier trial review of net topology, and I have tried to follow most of his extremely detailed advice on how to rewrite it into the present introduction. The present review of net topology is very condensed indeed, and I hope to expand it into a more accessible monograph and textbook to meet David’s justified suggestion that it is too brief. Werner H. Baur and Reinhard X. Fischer provided invaluable editorial guidance. I thank all the zeolite crystallographers and mathematicians, too numerous to list, who have provided information and critical advice: many of them have been or are members of the Structure Commission of the International Zeolite Association. Douglas Coombs kindly communicated the papers that lead to the Report on Nomenclature of Zeolite Minerals to the International Mineralogical Association. Direct financial support to the Consortium for Theoretical Frameworks from NSF, ACS-PRF, UOP, Exxon Educational Foundation, Mobil Corporation and Chevron Corporation is highly appreciated. I apologize for the five-year delay in production of this manuscript that arose mainly from the administrative load while I was Executive Director of the Center for Advanced Radiation Sources responsible for the initial design and fund raising for three sectors at the Advanced Photon Source. Some lack of consistency in the style of drawing results from incompatibility between the first primitive computer programs and the latest sophisticated ones. Most of the final drawings were made with MacPro, many after translation from MacDraw and its updates. I thank Fred Arnold for assistance with conversion from Mac to PC format. Finally, I thank Hannes Krüger (Bremen) for changing the hard copy figures into machine readable form and Elke Eggers (Bremen) for the formatting and preparation of the camera-ready copy of the manuscript.

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