2-isohedral triangulations

KENNETH

2-ISOHEDRAL

HOLLADAY

TRIANGULATIONS

ABSTRACT. A triangulations is 2-isohedral iff there are exactly two orbits of triangles under the triangulation's symmetry group. 2-isohedral triangulations are classified using incidence symbols. This also determines the homeomeric types of 2-isohedral triangulation. There are 38 types. The proof is by aggregation into isohedral tilings and by reflection axis splitting.

A tiling is a locally finite cover of the Euclidean plane by closed topological disks called tiles, such that the intersection of any two different tiles is either empty or a connected subset of the boundary of each. A non-empty intersection of three or more tiles must be a point, called a vertex of the tiling; each tile has a finite number of vertices. The vertices of a tile divide its boundary into arcs (called edges) that are the intersection of pairs of tiles (which are said to be adjacent). Although we will need to use more general tilings; this paper is about triangulations, where each tile has exactly three vertices and the edges are line segments. We call the tiles triangles but note that a tiling whose tiles are geometric triangles need not be a triangulation. Figure 1 gives an example of this; the equilateral triangles must be considered hexagons since they are adjacent to six other tiles. The number of different tiles containing a vertex is called the valence of the vertex. Any tiling has an associated face graph whose nodes are the tiles and with an arc joining two nodes iff their cells are adjacent. Clearly the face graph of a triangulation is an infinite planar cubic graph. Two tilings are the same topological type iff they have isomorphic face graphs. In this paper we will classify triangulations with only two 'kinds' of triangle. To make this notion precise requires the use of symmetry groups. A symmetry of a triangulation is a Euclidean motion that sends triangles (of the triangulation)into triangles.We will usually identify a symmetry with the permutation of the triangles that it induces. The symmetry group of a triangulation T~

Fig. 1. A tiling by triangles that is not a triangulation.

Geometriae Dedicata 15 (1983) 155-170. 0046-5755/83/0152-0155502.40. © 1983 by D. Reidel Publishing Company.

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written S(T), is the group of all symmetries of T. We require our triangulations to be periodic; that is, the symmetry group must contain two independent translations. It is well known that this condition limits the symmetry group to one of 17 types; we will use the international symbols for these groups. Our final condition on triangulations is that they be 2-isohedral. This means that there are exactly two orbits of triangles under the action of the symmetry group. A tiling with one orbit of tiles is said to be isohedral; isohedral tilings are classified in [ 1]. We use the notation of the lists in [ 1] and [2] throughout this paper. In [2] it is shown that there are four topological types of triangulation with isohedral representations. We will obtain 2-isohedral representatives of all four of these types and of 17 new topological types. Classifying triangulations by topological type is too coarse a division to be considered complete. The topological type does not determine the symmetry group or its precise action on the tiles. Following in the path of Griinbaum and Shephard, we define a finer classification scheme based on incidence symbols. As further evidence of the value of this scheme, we will show that it also solves the problem of homeomeric classification of 2isohedral triangulations. The 21 topological types with 2-isohedral representatives give rise to 38 homeomeric types. The incidence symbol of a triangulation describes the symmetries of the triangles and how the triangles are fitted together. Incidence symbols for isohedral tilings (which we will use extensively here) are discussed in [1] and [2]. The incidence symbol is composed of two parts: the tile symbol and the adjacency symbol. They are both obtained from a labelling of the edges. Every edge receives a letter and possibly a direction from each of the two triangles containing it. First we must arbitrarily designate the triangles of one orbit under the symmetry group as being orbit I and the triangles of the other orbit as being orbit 2. In this paper we will take orbit 1 to be the more asymmetric triangles if possible. We will use lower case letters a, b, c for edges of orbit-1 triangles and capital A, B, C for edges of orbit-2 triangles. The letters are initially all regarded as giving a direction to the edge they label. We fix one orbit-1 triangle, label one of its edges a and direct it counterclockwise. Applying the operations of the symmetry group carries the label to all of the other triangles of the orbit. If there is a reflection of the group whose axis is the perpendicular bisector of the chosen ct edge, so that the edge is carried to itself with direction reversed, we consider the edge to be undirected. If the chosen triangle still has unlabelled edges, take the first one proceeding counterclockwise, label it b, direct it counterclockwise and proceed as above. If there still is an unlabelled edge, label it c and proceed as before. Having labelled the orbit-1 triangles, we label the orbit-2 triangles in the same fashion. The initiaI orbit-2 triangle is chosen adjacent to the initial orbit-1 triangle and if the orbit-1 triangle is adjacent to more than one orbit-2 triangle, we pick the one sharing the alphabetically least edge.

2-ISOHEDRAL T R I A N G U L A T I O N S

157

To reduce the arbitrariness somewhat, if there are undirected edges, we will arrange to give them the labels a or A. Since there is at most one orbit of undirected edges per orbit of triangles, we can do this. The tile symbol records the ways that the symmetry group can map a triangle to itself. The subgroup of symmetries that send a triangle to itself is called the stabilizer. The symbol for a triangle depends only on the orbit and not the particular choice of triangle. The tile symbol for a triangle is obtained by listing the labels of the edges in counterclockwise order starting from a or A. If an edge is undirected, its letter has no superscript in the symbol. If the edge is directed consistently with counterclockwise orientation, the letter has a ' + ' superscript. If it is directed inconsistently, the letter has a ' - ' superscript. There are four possibilities for the stabilizer of a triangle. Their names and tile symbols are: scalene, a +b +c +; isosceles, ab +b - ; cyclic, a +a +a + ; regular, aaa.

Just as the tile symbol describes how the symmetries relate a triangle to itself, the adjacency symbol describes how the symmetries relate a triangle to its neighbors. The procedure is the same for both orbits so we will just construct the adjacency symbol for orbit 1. All edges given the label a by one of their triangles will be given a fixed label (including direction) by their other triangle. This label gives the first letter of the adjacency symbol for orbit-1 triangles. Similarly the next letter comes from the label that always occurs with b (if there is any b) and the last letter comes from the label with c. If the edge is undirected, it will be undirected on both sides and its letter in the adjacency symbol has no superscript. If the edge is directed, it will be directed on both sides and its letter gets a superscript ' + ' if the two directions are opposite and a superscript ' - ' if they are the same. The incidence symbol of an orbit's triangles is the tile symbol followed by a semicolon and then the adjacency symbol. If this incidence symbol is not part of a larger incidence symbol, it would be enclosed in brackets. The incidence symbol of a 2-isohedral triangulation is the incidence symbol of the orbit-1 triangles followed by a slash and then the incidence symbol of the orbit-2 triangles. The symbol is enclosed in brackets. The incidence symbol of a triangulation is not unique since some arbitrary choices are made in the construction. Two incidence symbols, possibly from different triangulations, are said to differ trivially iff they could be made the same by changing the orbit numbers or choices of edge labels. Triangulations of different topological types can have the same incidence symbol, so we make the following definition. Two 2-isohedral triangulations are the same 2-isohedral type iff they are the same topological type and they can be given the same incidence symbols. THEOREM 1. There are exactly 38 2-isohedraI types o f triangulation. The proof is broken up into several propositions. The various types are

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TABLE

1

Orbit 1 2-1HT Topological number type (1) (2)

Symmetry S(T) (3)

Orbit 2

Incidence symbol (4)

Vertex (5)

Edge (6)

Aspects (7)

ctfl7 ~[# ct,87 ctfly ctfl7 otflfl ctfl)' ~tfl)'

6D, 6R 2D, 2R 3D, 3R 3D, 3R 4D, 4R 8 4D, 4R 2D, 2R

Incidence symbol (4)

l 2 3 4 5 6 7 8

3.122;3.122 3122;3.122 4.6.12;4.6.12 4.6.12;4.6.12 4.82;4.82 4.8~;4.8 ~ 4.82;4 8 ~ 4.8~;4.8 ~

p6m cmm p3ml p31m p4rn p4g p4g pm#

a+b÷c+;B÷b c a+b+c+;B+b*c a+b+c÷;A+b c a + b + e + ; A + b B* a + b ~ c + ; A ~ b ca+b+c+;A+c~b+ a + b + c + ; A +b-B+ a+b+c+;A+b+c

or.fie ~t.Bfl aft)' c~fl'e ~tfl)' afly ~f17 ctflfl

AB+B ;Aa + AB+B ;Aa + A+B+C+;a+B CA~B+C+;a+c+c A+B~C÷;a*B C A+B+C+;a÷B-C A+B+C+;a+c+C A÷B+C+;a+B-C +

9

4.8-~;4.82

pmg

a+b+c+;A+C÷c +

otfl~t

cq6)'

2D, 2R

A+B~C+;a+B

10

4.82;4.82

prara

ab*b

;aB

~o~fl

~.B[3

2

AB+B-;Ab

b+

-

Vertex (5)

Edge (6)

Aspects (7)

flflot ~8,6ct flot6 flo:y flo:,5 fl76 [3cty [3~fl

6aot 6otot ct& o:?~ ~,& a)'6 ~6 otBe

6 2 3D, 3R 3D, 3R 4D, 4R 4D, 4R 4D, 4R 2D, 2R

,6otot

o:6fl

2D, 2R

e,totfl

)'.Off

2

11

4.8z;4.82

cm

a+b+c+;A+B-c -

~.otfl

ctfl?

ID, IR

A+B+C+;a+b-C

-

~tfl

~t66

ID, IR

12

4.8~;4.82

p2

a+b+c*;A+b+B +

cq3fl

~7

2D

A+B+C+;a+c+C ÷

fl~.fl

ct76

2D

13

63;6 J

p3lm

a+b+c+;A+b-c

"

otct,6

~tfl'f

3D, 3R

A*A+A~;a

~tctez

cto~

2

14

63;63

cmm

a+b+c+;a

b+B +

~tflfl

aft7

2D, 2R

AB+B;Ac

flflct

37~/

2

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

63;6 ~ 6~;6 s 6~;6 ~ 6~;63 6~;63 63;63 3.6 24;3.242 4 6 . 2 4 ; 4 ~ 24 4.6.18;4~. 18 4.6.8;4812 4.122;4~ 1 2 ~ 1 4.8. t2;4~.9 3.8~; 6.8 ~ 3.82;6.82 3.8.16;3.162 4.8.16;4e.16 4.8.12;4~.12 4.72;7 ~ 3.9~;9 ~ 4.68;6.8 ~ 4.68;6z.8 5~.8; 5.8 ~ 4.12';4z.12#2 4 10~;4z 10

PmO pmo pm 0 PgO p~ p2 p6m p6m p6m p6m p6m p6m p6m p31m p4m p4m p4m p4g p3ml prom cmm cram cmm cram

a+b+ c + ; A +B÷ c " a+b+c+;A+b+c a+b+b ; A b + a*b+c+;A+C c + a+b+c+;A+b*c + a~b+c+;A+b~c + a+b÷c+;B+b-c a+b* c+ ; A + b - c a+b+c+;B~b c a+b+c~;A÷b ca+b+c+;A+b ca+b+c+;B+b-c ab*b ; A b a+b+c+;A+c+b ÷ a*b+c+;B+b-c a * b ÷ c + ; A + b ca*b÷c+;B+b c a+b+c+;B+c+b + ab+b ; A b a+b+c+;B+b c a+b+ c+ ; A + b c a÷b+c÷;B+b c ÷ a÷b+c÷;A+b c ~ a÷b÷c~;B+b-c ÷

~flfl ctflfl ct~tet ct~tct ct~ct ~a~ ct.8~ afl~, otfl'f aft)' ~fl)' ~tfl'~ ctotfl ctotfl ~fl)' otfl7 ~tfl)' ~tctfl ~tafl ~tfl? ct[3"~ ctfla o:[3~t o:flct

~f17 ct~)' ~t,~,8 ot,~7 c~fl7 o:t~ ~fl)' ¢tfl7 ~IJ7 aft)' aft), ~fl)' o:.B~ ~.flfl ~,~ ~tfl~, c~)' ct.~ ctfl~8 aft'./ otfl)' ~y o.~? ~t~7

2D, 2R 2D, 2R 2 4 21),2R 2D 6D, 6R 6D, 6R 6D, 6R 6D, 6R 6D, 6R 6D, 6R 6 3D, 3R 4D, 4R 4D, 4R 4D, 4R 8 3 2D, 2R 2D, 2R 2D, 2R 2D, 2R 2D, 2R

A+B+C+ ; a ~ b + C + flctfl A+B+C+;a+B+C fl~t~t AB÷B-;aB + ~o:o~ A + B + C + ; a * B + b - ~.~tct A + B + C + ; a + C B - ototr~ A ~ B + C * ; a + B + C * ctct~t AB+B;Aa + flfl~t A+B+C+ ; a + B C - 13ot6 AB+B-;Aa + flfl~ A * B + C + ; a + B - C - 13ot6 A ÷ B ~ C * ; a + B - C - fl~t& AB~B-;Aa + flfl~t AB+B-;a B~7 A+B+C+;a+BC - c~ 7 AB+B ;Aa + flflct A+B+C+;a+B C ,8o:6 AB+B ;Aa* flfla AB+B-;Aa + ~t~tct AAA;a ctot~ AB*B ;Aa + flfl~ A+B+C+ ; a + B + C fl~tct AB+B-;Aa + fl~ct A+B+C~;a+B C fl~7 AB+B-;Aa * ~fl~.

~tfl6 ct6e ct?y o:6,6 a66 ct~s 6~ a6.e 6a~t or& ct&. b~ta ~)')' ~t)'6 6a~t ~6e 6ctct )'~to~ a~tot 6~tct ot6e 6o~ct st& 6o:ct

2D, 2R 2D, 2R 2 4 2 2D 6 6D, 6R 6 6D, 6R 6D, 6R 6 2 3D, 3R 4 4D, 4R 4 4 1 2 2D, 2R 2 2D, 2R 2

+

NOTES TO TABLE 1. Column (1) gives list number. Column (2) gives topological type as explained before Theorem 2. Column (3) gives the symmetry group of the filings of the type. Column (4) gives the incidence symbol of the appropriate orbit triangle as explained before Theorem 1. Column (5) gives the vertex transitivities with ~t being the start vertex of side a and continuing arou nd orbit l and then orbit 2 in a counterclockwise order. Column (6) gives the edge transitivities with ~t being side a and continuing around as in column (5). Two triangles are of the same aspect iff one is a translate of the other. Column (7)gives the number of aspects, where D and R mean direct and reflected aspects.

TABLE 2-IHT number

2. Degrees of freedom Degeneracy conditions

Angle relation

1 2 3 4 5

~ = zt/6, A + 2. fl = 2re

A +2.fl=2rc o:=A=7+B=~r/3 c~=A=rc/6, y+B=~z ~=A=rr/4, y+B=zc

1 2 1 1 1

6 7

fl = ~ = zr/4, A = ~z/2 o:=A=rc/4,7+B=rc

1 1

8 9

fl+F=~ 7+B=n

3 2

10 11 12

a+A=rc ct + A = T t , f l = B fl+F=n

1 2 2

B o t h gives P3 - 10 fl = ~z/4 gives P a - 10 = n/2 gives P3 - 9 + B = ~/2 gives 2 - I H T 1 0

13 14 15 16 17 18 19 20

~ = A = n/3 fl=B fl=B,y=F fl=B None None B = F None

1 2 2 3 2 4 3 4

fl = ~/2 gives P 3 - - 9 fl = ~/3 gives P3 - 14 fl = 7 gives P3 - 13 M o n o h e d r a l , ct = fl gives P3 - 13 = F gives P3 - 12 fl = B gives P3 - 13 7 = B gives P3 - 11 fl = y gives 2 - I H T 1 7 fl = ~ a n d B = F gives 2 - I H T 1 7

21 22 23 24 25 26 27

ct = c~ = c~ = c~ = c~ = c~ = c~ =

z/3, 7 + B = n / 6 z~/3, A = ~/2, y + B = rc ~/3, 7 + B = ~z re/3, A = re/6, 3' + B = ~t re/6, A = r~/2, fl + F = rc rc/6, 7 + B = r~ 2n/3, A = r~/3

28 29 30 31

fl = y = re/6, A = e = zt/4, y + B = c¢=rc/4, A = r c / 2 , ~ = r~/4, ~ + B =

32 33 34 35 36 37 38

fl = 7 = ~z/4 c~ = 2rc/3 c~ = zt/2, fl = B ct = r~/2, fl = B y + B = ~/2 A = ~z/2, ? + B = z 7+B=~z

~z/3 Tr/4 y+B=Tz zc

1 1 1 1 l 1 0 l l l 1 1 0 1 2 2 2 2

fl = 27z/3 gives P3 - 1 ct = ~z/6, fl = 2zt/3 gives P~ - 1 fl = fl = fl = B = fl = fl = fl + =

~/2 gives P3 - 8 n/3 gives P3 - 8 ~/4 gives P3 - 10 n / 4 gives P3 - 10 n/4 gives P3 - 10 F a n d y = B gives P3 - 9 F = ~/2 gives 2 - I H T 1 0 A makes monohedral

fl = B a n d 7 = F g i v e s P3 - 11 fl = F a n d y = B gives P3 - 11 Not monohedral Not monohedral Not monohedral Not monohedral Not monohedral Not monohedral Not monohedral B = ~z/3 gives 2 - I H T 2 7 Not monohedral Not monohedral Not monohedral Not monohedral Not monohedral Not monohedral A = Ir/2 m a k e s m o n o h e d r a l Not monohedral Not monohedral Not monohedral

NOTES TO TABLE 2 : T h e a n g l e s o f a t r i a n g l e a r e l a b e l l e d as follows. F o r a scalene t r i a n g l e , a n g l e ~ is o p p o s i t e t h e side l a b e l l e d a, etc. F o r a n isosceles t r i a n g l e , ct is the a p e x a n g l e a n d fl the b a s e angle. F o r cyclic a n d r e g u l a r t r i a n g l e s , the a n g l e is c~. C a p i t a l letters d e n o t e o r b i t - 2 triangle. T h e a n g l e r e l a t i o n s a r e in a d d i t i o n to t h o s e i m p o s e d b y the tile s y m b o l a n d s u m o f a n g l e s e q u a l s n. C o l u m n 3 gives the n u m b e r o f real p a r a m e t e r s n e e d e d to d e t e r m i n e a tiling o f t h e t y p e u p to similarity.

160

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described in Tables 1 and 2 and illustrated in Figure 3 at the end of the paper. The following proofs will use two basic techniques: aggregation and reflection axis splitting. Aggregation is used to construct an isohedral tiling from a 2-isohedral tiling by combining tiles. Typically a triangle from one orbit is combined with a triangle from the other orbit that is adjacent to it. In effect we create an aggregate tile by disregarding an orbit of edges. We cannot completely ignore these edges, however, because that could easily alter the symmetry group. The aggregate tile must be considered a marked tile, with the disregarded edge being the mark. Note that an aggregate tile composed of two triangles need not be a quadrilateral. If an edge of one triangle is collinear with an edge of the other, the aggregate tile could combine the two edges into one and be triangular (i.e. be adjacent to only three aggregate tiles). For an aggregation to be valid, we must show that every triangle is in a unique aggregate tile and that the symmetry group is transitive on the aggregate tiles. If, for example, both orbits are scalene triangles, this is automatically the case. Each triangle has a unique edge from the disregarded orbit and hence is part of a unique aggregate tile. Since the symmetry group is transitive on the orbit of disregarded edges, it is transitive on the aggregate tiles. We will use the technique of reflection axis splitting to analyse triangulations containing isosceles triangles. By introducing new edges that split isosceles triangles down the middle (along the reflection axis), we replace the isosceles triangles by scalene triangles while retaining a 2-isohedral (or possibly isohedral) triangulation. The effect is to make an orbit 'larger' by reducing the size of the stabilizer. Whereas before there were two symmetries sending a given isosceles triangle to any triangle of its orbit; after the split there is only one symmetry sending a given scalene triangle to any triangle of its orbit. Note that a split could create the possibility of new symmetries that would make the new triangulation isohedral. This happens with 2IHT34 for example. P R O P O S I T I O N 1. A 2-isohedral triangulation cannot have both orbits of triangles regular. The only 2-isohedral type with one orbit of regular triangles is 2-IHT33. Proof. Regular triangles are equilateral. If both orbits were regular, the triangulation would not be 2-isohedral; it would be the isohedral triangulation P3 - 14 (in the notation of [2] ). If the orbit-2 triangles of a triangulation are regular; the orbit-1 triangles must be isosceles if they are not regular, since they have an undirected edge. We can apply aggregation to create an isohedral tiling. The aggregate tiles are composed of a regular triangle and its three isosceles neighbors. If the isosceles triangle has tile symbol ab+b -, the aggregate tile will be a hexagon with tile symbol b+b-b+b-b+b -.

2-ISOHEDRAL T R I A N G U L A T I O N S

161

IH19 is the only tiling with this tile symbol, and it gives the incidence symbol [ab +b- ; Ab-/AAA;a] for the original triangulation. This is the 2-isohedral type 2-IHT33. Note that the aggregate tile is really a hexagon and not a triangle because consecutive edges of any aggregate tile are oppositely directed and cannot be combined to form a single edge. P R O P O S I T I O N 2. A 2-isohedral triangulation cannot have both orbits of triangles cyclic. 7he only 2-isohedral type with one orbit of cyclic triangles is 2-IHTI3. Proof. Since a cyclic triangle is equilateral, both orbits cannot be cyclic for the same reason as in Proposition 1. For triangulations with one orbit cyclic, we first assume the other orbit is scalene. We will apply aggregation with the aggregate tiles composed of a cyclic triangle and its three scalene neighbors. The aggregate tile is either a hexagon or a triangle. If the tile is a hexagon and we say that the scalene triangle's a edge is adjacent to the cyclic triangle, then the hexagon's tile symbol will be b+c+b+c+b+c +. From [1], we see that there is only one such tiling, IH10. But to get IH10 with straight edges, we introduce reflection symmetry that the marking of the tile does not destroy. If the tile is a triangle, the resulting aggregate tiling is made of cyclic triangles and is either IH89 or IH90. In IH90, adjacent tiles are interchanged by a half-turn and we would need the lengths of the b and c edges to be equal so as to send triangles into triangles in the original triangulation. This makes the original triangulation into the isohedral P 3 - 14. In IH89, on the other hand, the reflection interchanging adjacent aggregate tiles is compatible with b and c different lengths. The resulting original triangulation (which is 2-IHT13) will be 2-isohedral as long as the lengths are different. Now we investigate the possibility of one cyclic orbit and one isosceles orbit. Since there is but one kind of undirected edge to use, the isosceles triangles must be joined in pairs base to base. If we made a reflection axis split, we would obtain a 2-isohedral triangulation with cyclic and scalene triangles. The scalene triangles would be fitted together into rhombi composed of four triangles. Since the only 2-isohedral triangulation with cyclic and scalene triangles is 2-IHT13, and it has no such rhombi, we see that the cyclic and isosceles case does not occur. P R O P O S I T I O N 3. A 2-isohedral triangulation with both orbits scalene and only one common edge between the two orbits must be one of the types numbered: 3, 5, 6, 8, 16, 19, 20, 22, 24, 25, 28, 30, 35, 37. Proof. We proceed by cases, considering the various patterns of adjacency symbol. First we consider incidence symbols of the form [a+b+c+; A+bWe~/A+B+C+;a+BYCZ] where the edges of both orbits have the same label on both sides. An aggregate tile formed by disregarding the a-A edges

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can be either a quadrilateral or a triangle. If it is a quadrilateral, the aggregate tiling's incidence symbol will be [b+c+B+C + ;bWcXBrCZ]. This pattern occurs four times but IH48 is not realized by the marking consisting of a diagonal. IH46 gives 2-IHT20, IH49 gives 2-IHT16 and IH54 gives 2-IHT35. To get a triangular aggregate tile we need a pair of consecutive edges, one from each orbit of triangles. Both edges must have a ' - ' superscript in the adjacency symbol. If we say that c and B combine to form an edge labelled d, the aggregate tiling's incidence symbol will be [b+d+C+; bWd-CZ]. This pattern occurs five times but some cases have more than one edge that could be d, so eight tiling arise this way. IH77 gives 2-IHT22, 2-IHT24 and 2-IHT25; IH78 gives 2-IHT37; IH80 gives 2-IHT5 and 2-IHT30; IH85 gives 2-IHT8; and IH87 gives 2-IHT3. We now consider combinations of adjacency symbols where an orbit of edges has both labels of one orbit of triangles but the other orbit of triangles has the same label on both sides of an edge. The aggregate tile cannot be triangular. Except for the superscripts on the adjacency symbol, a quadrilateral aggregate tile's incidence symbol is [b +c +B +C +; cbBC]. There are three tilings of this type; IH30, IH53 and IH56. There will be only one suitable diagonal to insert; for IH30 this yields 2-IHT28, for IH53 this yields 2-IHT19, for IH56 this yields 2-IHT6. The last case consists of combinations where both orbits of triangle have both letters on single edges. Again a triangular tile is impossible. The aggregate tile's adjacency symbol will have letters cbCB (with some superscripts). The only tilings matching this pattern are IH31, IH33, IH44 and IH55. None of these can be realized with straight edges; we always get 'extra' reflection symmetries that the diagonal marking cannot destroy. Thus this case does not occur. PROPOSITION 4. A 2-isohedral triangulation with both orbits scalene and two orbits of common edges must be one of the types numbered : 4, 7, 9, 11, 12, 15, 18. Proof. By our convention that the original orbit-2 triangle be opposite the alphabetically least possible edge of the original orbit-1 triangle, the first capital letter of the orbit-i adjacency symbol will have a ' + ' superscript. We will take this alphabetically least edge to be the a-A edge and we disregard it to create the aggregate tile. We do the quadrilateral case first and then the triangle case. The quadrilateral's incidence symbol could be of the form [b+c+B+C + ; BXcYb~C~]. This pattern occurs in IH42, IH45, IH47, IH50 and IH51. Straightening the edges of IH42 and IH47 introduces new symmetries that the diagonal marking does not destroy. New symmetries are also introduced by straightening the edges of IH45 but the diagonal destroys them and we get 2-IHT11. Inserting either diagonal in IH50 yields 2-IHT15 and in IH51 yields 2-IHT18.

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163

The other possible pattern for the incidence symbol is [b+c+B+C+; bXBrcYCZ]. This occurs in IH30, IH53 and IH56. We looked at adding one diagonal to these tilings back during the proof of Proposition 3. Adding the other diagonal this time, we get 2-IHT4 from IH30, 2-IHT18 from IH53 and 2-IHT7 from IH56. If the aggregate tile is triangular, the broken edge could have either ' + ' o r ' - ' for its superscript. If it is' - ' and we say c and B are combined to form the new (broken) edge d, the incidence symbol will be [b+d+C+; CXd-bX]. Three tilings in [1] have this pattern: IH38 gives 2-IHT4, IH81 gives 2-IHT7 and IH83 gives 2-IHT11. If the superscript is' + ', we will need the edges to be combined (c and B again) to be the same length so that the half-turn symmetry of the new edge, d, will interchange tiles in the original triangulation; that is, the disregarded edge must go to the midpoint of the broken edge. The incidence symbol of the aggregate tile will have the pattern [b+d+C+; bXd+Cy] and this occurs in IH78, IH84 and IH85. Straightening the edges of IH78 introduces new symmetries that the marking (a disregarded edge) does not destroy. IH84 gives 2-IHT12 and IH85 gives 2-IHT9. PROPOSITION 5. There are no 2-isohedral triangulations with both orbits of trianyles scalene and all three orbits of edges common to both orbits of triangles. Proof For this (vacuous) case, we alter our labelling convention. After selecting one orbit of triangles as orbit 1 and labelling the edges a, b, c in counterclockwise order, we label the edges of the orbit-2 triangles by simply capitalizing the letters on the other side of each edge. The edges of the orbit-2 triangles are then directed by assuming an alphabetical orientation. Thus it is possible that none of the orbit-2 triangles receive a counterclockwise orientation. The two orbits have congruent triangles and there are two relative positions for adjacent congruent triangles. Ignoring the rest of the tiling, the two triangles are interchanged by either a reflection or a half-turn. In these circumstances, the reflection or half-turn must extend to a symmetry of the whole tiling. The extension interchanges the two orbits of triangles but all that changes is capital versus lowercase letters. The actual letter and the relative position of the triangle on the other side of an edge is always the same. This new symmetry shows that the allegedly 2-isohedral tiling is in fact isohedral. P R O P O S I T I O N 6. A 2-isohedral triangulation with one orbit of triangles scalene and one orbit isosceles must be one of the types numbered: 1, 2, 14, 21, 23, 26, 29, 31, 32, 34, 36, 38. Proof Since there is but one orbit of undirected edges, the isosceles triangles must be adjacent at the base in pairs. Applying a reflection axis split produces a 2-isohedral or isohedral result. We treat the 2-isohedral case first. Both orbits of triangles are scalene after the split and we take the new triangles to be orbit 2. If the new edge on the reflection axis is labelled C

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and the new edge formed from half of the old undirected edge A is labelled D, the tile symbol of the new triangles is D + B + C +. We suppose that the edge of the orbit-1 triangle on the other side of B is labelled a. The adjacency symbol of orbit 2 is D - a + C -. We now consider the various possibilities for the orbit-1 adjacency symbol. First we suppose the original triangulation's orbit-1 adjacency symbol has the form B +bXe y. The split triangulation will have incidence symbol [a + b +c + ; B+bXcY/D+B+C+; D - a + C - ] . This pattern occurs in 2-IHT numbers 3, 5, 22, 24, 25, 30, 35 and 37. The orbit-2 triangles that fit together to make the isosceles triangles must occur in rhombic-shaped groups of four. 2-IHT3, 2-IHT5 and 2-IHT24 have no such rhombi. The other five do have the rhombic groups and removing one or the other of their diagonals gives ten results. 2-IHT22 yields 2-IHT21 and 2-IHT23; 2-IHT25 yields 2-IHT1 and 2-IHT26; 2-IHT30 yields 2-IHT29 and 2-IHT31; 2-IHT35 yields 2-IHT14 and 2-IHT36; and 2-IHT37 yields 2-IHT2 and 2-IHT38. Next we suppose that the original triangulation's orbit-1 adjacency symbol has form B+eXb ~. The split triangulation will have the incidence symbol [a +b +c + ; B+ cXbX/D +B+ C + ; D - a +C - ]. This pattern occurs in 2-IHT6 and 2-IHT28. There are no rhombic groups in 2-IHT28 but in 2-IHT6 they do occur. Removing either diagonal of 2-IHT6 yields 2-IHT32. Finally, we consider the possibility that reflection axis splitting converts a 2-isohedral triangulation into an isohedral triangulation. The rhombic groups created by the split would have to be matched by rhombic groups of four of the original scalene triangles (which would have a [a + b + c + ; B~b- c - ] pattern adjacency symbol). Since they have no such rhombic groups, the split tiling cannot be of topological types [3.122 ] or [63]. But [4.6.12] (represented by P3 - 8 ) and [4.82] (represented by P 3 - 9) do have such rhombic groups. Considering the rhombi as aggregate tiles, we get P~ - 42 from P3 - 8 and P4 - 55 from P3 - 9. P4 - 42 cannot be properly 2-colored and so it is impossible to produce a 2-isohedral result by reversing a split. P 4 - 55 can be properly 2-colored and we can reverse a split leading to P 3 - 9 and get 2-IHT34. Note that P 3 - 1 0 (also representing [4.82]) can be reverse split (in a fashion that does not work on P3 - 9) to yield some but not all instances of 2-IHT32. The construction of 2-IHT32 from 2-IHT6 given earlier in this proof yields all instance of 2-IHT32. P R O P O S I T I O N 7. A 2-isohedral triangulation with both orbits of triangle isosceles must be one of the types 2-IHTIO, 2 - I H T 1 7 or 2 - I H T 2 7 . Proof We have two basic cases: the two orbits of triangle are adjacent at their bases either to triangles of the other orbit or else both to triangles of their own orbit. In the first case we have kite-shaped groups of two triangles that can split into a group of four. Checking vertices of valence 4 in the lists of Propositions 3 and 4, we find possible reverse splits in some instances

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of 2-IHT8 and of 2-IHT24. 2-IHT8 yields 2-IHT17 and 2-IHT24 yields 2-IHT27. In the second basic case, we can split one orbit of isosceles triangles and obtain a triangulation from the list of Proposition 6. In one of those, 2-IHT34, a reverse split is possible and it yields 2-IHT10. Throughout the proofs of the seven propositions, we have assumed the triangles are not marked and that their edges are straight. Relaxing either of these conditions allows new types not on the above list. If any kind of mark is allowed, its presence versus absence can be used to 2-color any isohedral triangulation of the topological types [4.6.12], [4.82] and [63]. This gives a '2-isohedral' triangulation of the type covered by Proposition 5. If curved edges are allowed, we can get the 2-isohedral 'triangulation' illustrated in Figure 2. This tiling has two orbits of tiles and each tile is adjacent to exactly three other tiles. But the tiling cannot be realized by unmarked tiles with straight edges.

Fig. 2. 2-isohedral tiling where each tile has three neighbors. There is no unmarked tiling with straight edges homeomeric to this tiling.

In Table 1 we use a simple system for designating topological types. The lexicographically least order of the list of valences of the vertices of a triangle describes the triangle. We separate the numbers by periods and use exponents to indicate repetition. The s y m b o l for the topological type of a given 2isohedral triangulation gives the valence list for the orbit-1 triangles followed by a semicolon and the valence list for orbit 2. Theorem 1 shows that the symbol uniquely determines the topological type except for the case 4.122 ; 42.12. Here there are two types. We distinguish these two by adding # 1 to the type represented by 2-IHT25 and # 2 to the type represented by 2-IHT37. Theorem 1 also shows that the homogeneous topological types (both valence lists the same) are exactly the four types with isohedral representations. The classification of 2-isohedral triangulations by 2-isohedral type also solves the problem of classifying 2-isohedral triangulations by homeomeric type. We say that a homeomorphism, h, of the plane maps a tilin 9 T 1 onto a tiling T 2 iff h maps tiles of T 1 onto tiles of T 2 and h- 1 maps tiles of T 2 onto tiles of T~. T 1 and T 2 are homeomeric iff there is, in addition to h, an isomor-

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phism a:S(T1)~S(T2) such that a(s) (h(p))= h(s(p)) for all seS(T1) and all points p. Being homeomeric is an equivalence relation and the equivalence classes are called homeomeric types. T H E O R E M 2. 2-isohedral triangulations are homeomeric iff they are the same 2-isohedral type. Thus there are exactly 38 homeomeric types of 2-isohedral

triangulation. Proof. A homeomorphism like h induces an isomorphism between the face graphs of T 1 and T2, so they are the same topological type. We can use the isomorphism cr to carry a labelling of T1 over to T2. The action of a symmetry s in T 1 is copied by the action of cr(s)in T 2. The incidence symbol for T z obtained from this carried-over labelling will be the same as that of T v Thus T~ and T 2 have the same 2-isohedral type. Now let us assume that T I and T2 are the same topological type and they have been labelled so as to give the same incidence symbol. We want an isomorphism between the face graphs that sends the orbit-1 triangles of T 1 onto the orbit-1 triangles of T 2 and similarly for orbit 2. The orbit numbers of one of the tilings may have to be reversed to get the same sort of triangle to be orbit 1 in both tilings. This happens because in some tilings, 2-IHT22 for example, the orbits are distinguishable but they can be interchanged without changing the incidence symbol. Edges of the face graph correspond to edges of the tiling. Hence the face graph isomorphism gives a bijection between the edge sets of T 1 and T 2 . By considering vertices to be intersections of edges, we also get a bijection between the vertex sets of T~ and T 2 . Once we have a face graph isomorphism that preserves symmetry group orbits, we would like it to preserve edge labels. That is not necessarily the case. In some tilings, 2-IHT3 for example, replacing the original orbit-1 triangle by a reflection of it does not change the incidence symbol even though it does change the labels of some orbit-2 edges. So we may have to switch the original orbit-1 triangle in one of the tilings before we get the face graph isomorphism to preserve edge labels. We can now use the edge label preserving face graph isomorphism to construct the homeomorphism and symmetry group isomorphism that shows Tt and T 2 are homeomeric. If we consider T 1 and T 2 to be locally finite simplicial complexes, the face graph isomorphism gives us a map from vertices of T1 to vertices of T 2 that is a simplicial map. The homeomorphism h is the linear (in terms of barycentric coordinates - see section 3.2 of [3]) map uniquely determined by this vertex map. To construct the symmetry group isomorphism, we need an edge whose label is directed. By Proposition 10 there will be such an edge. Since having a directed labelled edge of a triangle is equivalent to having a flag, there will be a unique motion sending a fixed directed labelled edge to any other edge with the same label. Given a symmetry s of T~, we define e(s) to be the symmetry of T 2 that sends

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2-1HT 1

2-IHT 2

2-IHT3

2-1HT4

2-IHT 5

2-1HT6

2-IHT7

2-1HT8

2-IHTg

2-IHT 10

2-1HT11

2-I HT 12

2-IHT13

2-1HT 14

2-IHT 15

Fig. 3

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2-IHT16

2-1HT 17

2-IHT18

2-1HT19

2-1HT20

2-IHT21

2-IHT 22

2-1HT 23

2-IHT24

2-1HT 25

2-IHT26

2-IHT27

2-IHT28

2-IHT29

2-1HT30

2-ISOHEDRAL TRIANGULATIONS

2-1HT3*

2-IHT32

169

2-IHT33

/¢/ 2-IHT35

2-IHT34

2-IHT36

/¢/

2-IHT37

2-IHT38

Fig. 3. The original orbit-1 ceUismarkedwithadot and its side a is marked byanarrowifdirected and a line segment if undirected. The A side of the original orbit-2 triangle is similarly marked.

the edge c o r r e s p o n d i n g to o u r fixed edge to the edge c o r r e s p o n d i n g to the image u n d e r s of the fixed edge. It is easy to show that a is an i s o m o r p h i s m a n d that h a n d G m a k e T 1 h o m e o m e r i c to T 2 . ACKNOWLEDGEMENT I w o u l d like to t h a n k B. G r f i n b a u m a n d G. C. Shephard for their m a n y helpful suggestions, Marie a n d Olga for their typing, a n d my wife, Wendy, for her support. BIBLIOGRAPHY 1. Gffinbaum, B. and Shephard, G. C. : 'The Eighty-One Types of Isohedral Tilingsin the Plane'. Math. Proc. Camb. Phil. Soc. 82 (1977), 177-196. 2. Grfinbaum, B. and Shephard, G. C. : 'Isohedral Tilings of the Plane by Polygons'. Comment. Math. Helvetiei 53 (1978), 542-571. 3. Spanier, E. H.: Algebraic Topology. McGraw-Hill, New York, 1966, pp. 108-116.

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Author's address :

K. W. Holladay, University of New Orleans, Department of Mathematics, Lake Front, N e w Orleans, (Received February 14, 1983)

La. 70148 U.S.A.