Treasures Inside the Bell: Hidden Order in Chance

TREASURES INSIDE THE BELL HIDDEN

ORDER

IN

CHANCE

C A R L O S E. P U E N T E

TREASURES HIDE Till: BELL HIDDEN ORDER IN CHANCE

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TREASURES IISIDE THE RELL H I D D E N ORDER I N C H A N C E

Carlos E. Puente Institute of Theoretical Dynamics, The University of California Davis, USA

V f e World Scientific w l

• Hong Kong New Jersey • London • Singapore Si

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Scripture texts in this work are taken from the New American Bible with Revised New Testament and Revised Psalms © 1991,1986,1970 Confraternity of Christian Doctrine, Washington, D.C. and are used by permission of the copyright owner. All Rights Reserved.

TREASURES INSIDE THE BELL Hidden Order in Chance Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-140-6 ISBN 981-238-141-4 (pbk)

Printed in Singapore.

How precious to me are your designs, 0 God; how vast the sum of them! Were I to count, they would outnumber the sands; to finish, I would need eternity. Psalm 139:17-18

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To my Father, companion and friend

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Preface The bell curve, also known as the normal or Gaussian distribution, is one of the most ubiquitous mathematical objects in science. Introduced in the early nineteenth century by mathematical prodigy Karl Friederich Gauss, the curve's fundamental nature stems from the renowned central limit theorem, first proven by Pierre-Simon Laplace, that states that a bell is the universal curve found when a multitude of unrelated (independent) quantities are added together. Recently, generalized versions of the central limit theorem leading to Gaussian distri­ butions over one and higher dimensions, via arbitrary iterations of simple mappings, have been discovered by the author and his collaborators. The purpose of this book is to reveal how these new constructions lead to infinite exotic kaleidoscopic decompositions of twodimensional circular bells in terms of beautiful deterministic patterns possessing arbitrary n-fold symmetries, patterns that, while reminding us of the infinite structure previously found in the celebrated Mandelbrot set, turn out to contain natural shapes such as snow crystals and biochemical rosettes, including life's own DNA. This book is divided into three main sections. A general introduction to the ideas is given first, followed by a gallery of patterns found inside the bell when the iterations are guided by the binary expansion of the omnipresent number ir, including a potpourri of images showing curious pattern evolutions and collages of bell patterns. Hoping to capture the most general readership, the introduction relies on several diagrams and uses as few equations as possible, leaving the technical details to a set of notes at the end of the book. This set of notes also points out relevant references, contains a sample program the reader may implement in his/her own computer in order to explore the many treasures found inside the bell, and includes the specific information for all the images in the book. To allow the reader to further appreciate the work, the book includes a CD containing selected bell patterns whose interesting evolutions may be readily visualized on a personal computer. This book could not have been accomplished without the invaluable assistance of many people. First, my warmest thanks go to Michael F. Barnsley, whose lovely ideas shaped IX

X

Preface

the direction of my research, giving birth to this work. Second, I would like to thank my collaborators throughout the years, whose dedication is reflected in the pages that follow, in particular: Enrique Angel, German Poveda, Marc Bierkens, Jorge Pinzon, Miguel Lopez, Jose Angulo, Aaron Klebanoff, John Wagner, Nelson Obregon, Demiray Simsek, Akin Orhun, Bellie Sivakumar, Stephen Bennett and Marta G. Puente. Third, I would like to acknowl­ edge the institutions that have supported my research, especially and rather appropriately Pacific Bell, whose timely donation a few years ago allowed study of the bell in peace, and the University of California, Davis where I have been given the freedom to roam beyond my natural confines as a hydrologist. I owe much to Dick Odgers, Daniel Fessler and my UCD colleagues for their trust. Fourth, my sincere thanks go to World Scientific for their professionalism in producing such a beautiful book. I am indebted to Stanley Liu and Ian Seldrup. Many thanks go also to my dear friend Fernando Duarte for his artistic cover. Finally, I would like to acknowledge the loving support of my beloved family and friends. By your constancy, you nurture and inspire my yielding to a majestic bell, one ever conducting. Carlos E. Puente Davis, California March 2002

Contents

Preface

ix

Part 1 I n t r o d u c i n g t h e Bell 1 The Notion of Iterations 2 Interpolating Wires 3 Textures on Interpolating Wires 4 The Bell as a Shadow 5 Wires and Bells in Higher Dimensions 6 Exotic Patterns Inside the Bell 7 Natural Sets Inside the Bell 8 More Treasures Inside the Bell

1 1 3 6 8 11 15 23 23

Part 2 Gallery D e s i g n s Radial Patterns Rotational Patterns Evolutions & Quilts

33 35 49 63

Part 3

73

Technical N o t e s

Index

96

xi

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Part 1

Introducing the Bell

This section of the book attempts to explain, step by step, what is needed in order to understand and recreate the lovely patterns that are found inside the bell. The material is based on a comprehensive set of diagrams and graphs, relegating the more formal details to a set of notes at the end of the book. 1

The Notion of Iterations

We begin by introducing a rather simple process named the chaos game. This is a game of chance and to play it one needs a die. To start, choose three non-aligned points and label them according to the sides of the die, 1-2, 3-4, and 5-6, as in Figure 1 (left). Then, select a point, say the one marked as 1-2, and roll the die. Suppose a 3 comes up, then move

Figure 1. First two stages of the chaos game. 1

2

Introducing the Bell

Figure 2. The chaos game after 500 (left) and 8,000 (right) tries.

from the current position (1-2) to the middle of the vertex marked with 3, namely the point shown in Figure 1 (left). Roll the die again and suppose a 5 appears. Then, move from the last point to the mid-point of the vertex given by the outcome of the die, as shown in Figure 1 (right). Repeat the process many times and see what happens. As may be hinted, the game needs to be played for a while in order to elucidate its outcome. What appears after 500 tries, and as implemented on a computer playing with fair pseudo-random numbers, may be seen in Figure 2 (left). The game ultimately generates, the celebrated Sierpinski triangle, as depicted after 8,000 rolls in Figure 2 (right). This set, known as the game's attractor, turns out to be a rather peculiar object that has a deterministic and infinite structure of "middle triangles taken away from triangles," and appears independently of the throws of the die and of the point selected to start the game. Although the formation of the triangle point by point may be surprising, its structure may be elucidated once one considers the precise rules that are being repeated, that is, iterated, again and again. If (x, y) represents a generic position, such rules are (x + x

y+ y\

Wn{x,y) = y—-—,n —^—nJ , where (xn,yn),n — 1,2,3, are the coordinates of the three vertices. Clearly, applying the mappings to the three vertices themselves gives the precedence diagram shown in Figure 3, one that depicts (using alternative line types for the three rules) the boundary conditions for the chaos game. With this pictorial aid, the deterministic structure of the Sierpinski triangle readily appears through successive iterations of the rules starting at the newly defined points and following infinite ternary trees depicting all possibilities, as in Figure 4. Notice how the triangular gaps of decreasingly small sizes in Figure 2 represent regions in space that are never visited by any of the rules. The chaos game turns out to recreate the Sierpinski triangle because following any single non-trivial branch of a ternary tree (that is, one that jumps to all the rules as guided by a fair die) rooted anywhere ultimately covers the attractor densely everywhere. This happens because the rules, by always traveling inwards, pull all successive points towards a unique and stable state. 1

3

Interpolating Wires

Figure 3. Boundary conditions for construction of the Sierpinski triangle.

Figure 4. Ternary tree depicting recursive construction of the Sierpinski triangle.

2

Interpolating Wires

As iterating a set of simple rules gives rise to interesting sets, it is relevant to study the attr actors generated by other sets of mappings. For this purpose, exchange "move to the mid-point" by the following set of two rules: wi(x,y) = ( - -x, x + di-y w2(x,y) = (- -x + - , l-x

(1) + d2-y) ,

where d\ and d2 are (free) parameters whose magnitude is less than one.

(2)

4

Introducing the Bell

Figure 5. Alternative wires interpolating {(0,0), (1/2,1), (1,0)} for sign combinations on the parameters d\ and d-2 in Equations (1) and (2). \d\\ = je^l = z. Left, z= 0.5. Right, z = 0.8.

Figure 6. Boundary conditions for construction of interpolating wires.

It happens that these simple mappings generate alternative attractors shaped as convo­ luted wires, that interpolate the points (0,0), (1/2,1) and (1,0). As portrayed in Figure 5, for alternative magnitudes and sign combinations on the parameters, such sets are functions from x to y that are either "cloud" profiles (Case + + : d\ = d2 = z) or "mountain" bound­ aries (Case H—: d\ — —d% = z; Case : —d\ = —d2 = z; and Case —\-: the reciprocal of Case -I—, not shown) that have, as the Sierpinski triangle, a rather striking repetitive nature. Notice how wires corresponding to a magnitude of z equal to 0.8 are increasingly rough and considerably longer than those for z at 0.5. As may be easily verified, Wi(0,0) = (0,0), tui(l,0) = (1/2,1), w2(0,0) = (1/2,1), and 102(1,0) = (1,0). Hence, mappings (1) and (2) yield, in parallel to Figure 3, the precedence diagram of Figure 6. Because the more general mappings may also be shown to travel inwards, they define a unique attractor which emanates from point (1/2,1) following an infinite binary tree, as depicted in Figure 7. For instance, at the first level of the construction,

Interpolating Wires

5

Figure 7. Binary tree illustrating construction of interpolating wire. Locations of nodes in the horizontal are plotted at the proper scale starting at (1/2,1), indicating that w\ travels to the left and w-i does so to the right of the mid-point.

Figure 8. Recursive construction of an interpolating wire (Case + + ) . First points are found adding z to mid-points in lines joining interpolating points. Other points are obtained adding successive powers of z from successive mid-points, as shown.

the obtained values give wi(l/2,1) = (1/4,1/2 + di), w2(l/2,1) = (3/4,1/2 + d2), that may be readily noticed in Figure 5. In the end, a chaos game dictated by fair (or biased) coin tosses may be employed to obtain the unique attractor induced by the simple mappings (1) and (2). As before, the resulting attractor turns out to be fully determined as it may be generated recursively via a geometric procedure that parallels "taking middle triangles away from triangles," as illustrated in Figure 8.2 The Sierpinski triangle and the interpolating wires (for z > 1/2) are examples of de­ terministic fractal sets defined over the plane. 3 These are objects that fill two-dimensional space to varying degrees and have fractal dimensions between 1 and 2, D — In 3/ In 2 ~ 1.585

6

Introducing the Bell

for the triangle and D = 1 + ln(2 • z)/ln(2) for the interpolating wires (that is, more as z increases, as seen in Figure 5). Notice that as z tends to its maximal value of 1, the inter­ polating wires, which are topologically one-dimensional strings, tend to fill the plane (that is, D tends to 2.)4 3

Textures on Interpolating Wires

While constructing a unique interpolating wire via the chaos game, one could also compute the relative frequencies that points make as they densely sample the attractor. Although the same set is ultimately painted via alternative iteration rules, usage of fair or biased coins yields alternative stable textures within the wire.5 For initial points equally spaced in x, as used thus far, the overall trends are as follows. When the coin is fair, a branch that goes left or right 50% of the time fills up all values within the wire evenly. This may be seen in Figure 7, by noticing the precise equally-spaced locations for branches in the x component. What is found in terms of texture is thus shown on the left in Figure 9, a uniform measure over the wire set. As displayed on the right in Figure 9, when the coin used is biased, for example, w\ used 70% of the time and w2 the remaining 30%, no longer is the uniform texture found but instead one obtains a spiky multi-layered texture, a singular multifractal measure over the wire, one that decomposes the wire into infinitely many disjoint and intertwined sets corresponding to equal (multiplicative) textures. Such a detailed and thorny texture appears because of the independence of coin tosses and may be visualized overlaying the recursive construction of such a measure in Figure 10 with the binary tree in Figure 7.6'7 Although coin tosses are employed to find a wire and its texture, the results on Figure 9 stress that chance has no influence over the objects the chaos game generates. Once the

Figure 9. Alternative textures generated on a wire via the chaos game. Fair coin (left), biased coin traveling 70% to the left and 30% to the right (right). Interpolating points: {(0,0), (1/2,1), (1,0)}, dx = 0.5, d2 = - 0 . 5 .

7

Textures on Interpolating Wires

p = 70%

q = 30%

0

(a)

1

(b)

(c)

(d)

L.LL, L L L,.. L,. ,

U„L, L,...

(e)

Figure 10. Multiplicative process defining a multifractal texture. From a uniform measure (a) to 2 1 2 = 4,096 spikes in (e). Every rectangle splits into two rectangles that have half the parent's size (horizontal length). The smaller rectangles have areas that are 70% (left) and 30% (right) of the parent's area. The first four stages are drawn to scale, but the last diagram uses a vertically-reduced scale of the true figure, which would otherwise not fit on the page.

bias or fairness of the coin is set, such a choice defines a multifractal or uniform measure that surely expresses the ultimate texture found on the same attracting wire. The peculiar nature of the measures on wires may be further explored by defining projec­ tions of wire textures along arbitrary lines. This entails moving along such a line, over the x-y plane, identifying all points within the wire that are perpendicular to such a location, and adding the textures of all those points. It turns out that such an operation is easily performed while playing the chaos game, as the sought textures are found by computing a histogram of obtained points projected over the desired line. Figure 11 illustrates these ideas when the textures in Figure 9 are projected over the x and y axes. As is seen, the obtained "shadows" along the x axis, dx, corresponding to places where the wire is intersected only once, simply give either a uniform distribution or a multifractal measure defined over the wire's domain. Along the y axis, the shadows of the wire texture, dy, become more difficult to describe as they typically add the textures found on several points, that is, two or more for the examples shown.

Introducing the Bell

8

(a)

(b)

Figure 11. Projection of a wire over x and y as given by the chaos game. Fair coin (left), biased coin traveling 70% to the left (right). Interpolating points: {(0,0), (1/2,1), (1, 0)}, dx = 0.5, d2 = - 0 . 5 . The one-dimensional histograms and wire shown approximate the true objects via 1,024 equally-spaced bins containing 50 million pseudo-random coin tosses.

As seen on the left in Figure 11, a uniform texture on such a wire results in a symmetric dy whose shape is proportional to the number of times the wire y = f(x) is crossed by horizontal lines. Observe how the wrinkles of the wire indeed reflect the shadow dy, with larger spikes corresponding to the confluence of wire crossings. As observed on the right in Figure 11, when the texture over the same wire becomes multifractal, the corresponding projection dy is no longer symmetric. This happens due to the inherent biases on the chaos game, which prefers (by layers) some points within the wire more than others. As may be appreciated, dy gives rise to an irregular and rather complex (but deterministic) object whose shape reminds us of patterns commonly observed while displaying variations of natural phenomena, either in time or in space along a given line.8 As the interpolating wires define continuous functions from x to y, the measures dx and dy shown in Figure 11 are functionally related to each other, dy is the derived measure of dx via the function / that a wire represents, or more simply, dy is found transforming dx via such a function / . 4

The Bell as a Shadow

Figure 12 shows the wires and respective projections off them over both axes (dx and dy) when the three points {(0,0), (1/2,1), (1,0)} are interpolated employing the H— rule, that is, d\ = — d2 = z in Equations (1) and (2), via a fair coin implementation of the chaos game, where the parameter z is varied from 0 up to a value of 0.999.

The Bell as a Shadow

(c)

9

(d)

Figure 12. From a uniform measure to the bell curve. Interpolating points {(0,0), (1/2,1), (1, 0)}, d\ = —d-2. = z, (a) z = 0, (b) z = 0.5, (c) z = 0.8, (d) z = 0.999. The one-dimensional histograms and wires shown approximate the true objects in 1,024 equally spaced bins containing 50 million points gathered via the chaos game. The averaged wires for high fractal dimensions have been stretched vertically for aesthetic purposes.

As may be seen, the uniform projection, always found over x, yields a variety of symmetric, derived measures over y. When z — 0, the wire simply gives linear interpolation between the three original points and hence defines a uniform measure over y. This is easily seen noticing that there are exactly two points that are crossed by an arbitrary horizontal line (except at the upper tip). When z = 0.5, the mountain-like profile gives the distribution of crossings dy already shown in Figure 11(a). As z is increased beyond 0.5, the wire becomes infinite, progressively stretches up and down, and gives shadows over y that tend to group around a central tendency. When z = 0.8, the wire attains a fractal dimension D ~ 1.678

Introducing the Bell

10

and dy exhibits two main mounds, one on each side (up-down) of the wire. When z — 0.999, the wire almost fills up two-dimensional space as D « 1.999, and dy closely approximates a bell curve.9 Even though the shown measure dy for z — 0.999 is not a true bell curve, it hints at the result that is obtained as z tends to its maximal value of 1. As such a limit is approached, the resulting wire grows to fill two-dimensional space, that is, D tends to 2, and the crossings of such a deterministic object yield a distribution that indeed tends to the bell curve.10'11

(c)

(d)

Figure 13. From a multifractal measure (70%-30% biased coin) to the bell curve. Interpolating points {(0, 0), (1/2,1), (1,0)}, di = -d2 = z, (a) z = 0, (b) z = 0.5, (c) z = 0.8, (d) z = 0.999. The one-dimensional histograms and wires shown approximate the true objects in 1,024 equally spaced bins containing 50 million points gathered via the chaos game. The averaged wires for high fractal dimensions have been stretched vertically for aesthetic purposes.

Wires and Bells in Higher Dimensions

11

Figure 13 shows the counterpart of Figure 12 when a biased coin, with a 70%-30% split, is used to define wires and textures. As seen for z = 0, a multifractal measure over x gives, as expected, another multifractal texture over y, one that is found adding the respective textures of the two horizontal crossings that the two line segments have. When z = 0.5, and as already portrayed in Figure 11(b), an asymmetric texture dy is defined which shares commonly observed erratic behavior as found on several applications. As z is increased beyond 0.5, the wires become fractal, and give derived measures dy that are progressively smooth. Surprisingly, and as found for a fair coin, the limiting derived measure dy, as z tends to 1, is also a symmetric bell curve. It happens that a bell is universally found from the same limiting wire, irrespective of the bias of the coin, and also for a vast family of textures that only excludes discrete jumps. 12 Thus, such a limiting wire perfectly filters all kinds of thorny measures and transforms them into smooth bells, whose means and standard deviations depend explicitly on the texture dx.13 Remarkably, a change of perspective by 90 degrees provides a smooth bell for a wide variety of both smooth or rough (that is, singular) measures over x and hence yields an unforeseen connection between plane-filling fractal functions and the central limit theorem.14 As turbulence phenomena are associated with multifractal behavior, plane-filling fractal interpolating functions also provide an unexpected bridge from disorder to order, one in which the opposite behaviors given by turbulence and the bell (that is, dissipation and conduction of heat) appear as two sides of the same infinite attractor. 15 5

Wires and Bells in Higher Dimensions

The simple mappings given in Equations (1) and (2) may be extended to accommodate a larger number of coordinates. For example, the two mappings: Wi(x,y,z)

= (- ■ x,H ■ x + di ■ y + hi ■ z,x + li ■ y + m,i ■ z) ,

w2(x, y,z) — ( - • x + - , —H ■ x + d2 ■ y + h2 ■ z + H, -x + l2 ■ y + m2 ■ z + 1 j , may be employed to produce a fractal interpolating function that lies in three-dimensional space, a wire, from x to the plane y-z, whose fractal dimension D could be any number in the interval [1, 3), and that interpolates the set of three points {(0, 0, 0), (1/2, H, 1), (1, 0, 0)}. 16 As illustrated in Figure 14, for a uniform texture over the shown wire, the concept of derived distributions is readily generalized in order to define projections not only over y or over z, but also over the y-z plane. These joint measures, over two dimensions, yield a wide variety of deterministic patterns whose shapes are uniquely parameterized in terms of the coefficients H,d1,d2,hi,h2,l\,l2, mi and m2, as defined in Equations (3) and (4), and the bias of the coin that allows a construction via the chaos game. As found in the one-dimensional case, it turns out that some of the shadows obtained over two dimensions closely resemble some of nature's irregular patterns, for example, weather radar images and maps of pollution concentrations. 17

(3) (4)

12

Introducing the Bell

Figure 14. A wire over three dimensions and its shadow over the plane y-z. Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, dx = d2 = 0.7, hx = h2 = -0.606, h = l2 = 0, m1 = m 2 = 0.35, fair coin.

Figure 15 shows the typical behavior obtained when the fractal dimension of the resulting wire is increased from 1 to 3. Analogous to the graphs previously shown for two-dimensional wires, each block on Figure 15 contains: the xy and xz projections of the wire itself; the implied projections over the x, y and z axes, dx, dy and dz; and a contour plot of the joint measure obtained over the plane y-z, dyz, that is, as seen from above. As is observed, an increase in fractal dimension results (as before) in progressive filtering of a multi-layered texture over x and yields joint measures that converge to a smooth bell, but now over two dimensions.18 It turns out that all interpolating wires whose fractal dimensions approach the maximal value of 3 yield joint derived measures (over y-z) that closely approximate bells over two dimensions. These space-filling objects naturally extend what was previously reported over

Wires and Bells in Higher Dimensions

13

(c)

(d)

Figure 15. From a multifractal measure (70%-30% coin bias) to a bell over two dimensions. Interpolating points {(0, 0,0), (1/2,1,1), (1, 0,0)}, dx = d2 = 0, hi = -h2 = r, h = l2 = r, m1 = m2 = 0. (a) r = 0.25, (b) r = 0.5, (c) r = 0.75 and (d) r = 0.999. All graphs summarize what is found via 15 million chaos game calculations. The averaged wires and one-dimensional histograms are based on 512 bins and the twodimensional histograms are based on 64 x 64 bins. For aesthetic purposes the averaged wires are shown vertically stretched.

two dimensions and correspond, for the two mappings in Equations (3) and (4), to 16 cases of sign combinations as follows. If the parameters for the mappings are given in polar coordinates, that is, representing a location in the plane not by their Cartesian coordinates, but using a distance from the origin and an angle from the x axis, Vf'cos^

-ri2)sin#i2)\

r[ sin d[

r\' cos d\

?2 cos#2

—r2 sin# 2

r 2 sin 02

r2 cos 92

and

I

Introducing the Bell

14

(c)

(d)

Figure 16. The geometry of correlation. Interpolating points {(0,0, 0), (1/2, H, 1), (1,0,0)}, d\ = di = 0, hl = -h2 = 0.999, h = -l2 = 0.999, mi = m 2 = 0. (a) i J = 1, (b) H = 0.41, (c) H = 0 and (d) i? = —0.41. All graphs summarize what is found via 15 million chaos game calculations. The averaged wires and one-dimensional histograms are based on 512 bins and the two-dimensional histograms are based on 64 x 64 bins. The coefficients of correlation for these bells are: (a) p = 1, (b) p = 0.7, (c) p = 0 and (d) p = —0.7. For aesthetic purposes the averaged wires are shown vertically stretched.

then joint Gaussians are found over the y-z plane whenever the magnitudes of the distances r i > r i J r2 a n d r2 (which could be either positive or negative) all tend to one, and when 0\ — d[ + &i7r and #2 — #2 + ^2^, for any k\ and k2 that are integers.19 It happens that alternative cases result in bells whose circular or elliptical nature, that is, their correlation, depends on: (a) the sign combinations on the parameters r^\ (b) the angles 6^\ (c) the coordinates of the points being interpolated, and (d) the bias of a coin used in chaos game calculations.20 To illustrate such behavior, Figure 16 shows that there are indeed space-filling wires, all based on a fair coin, that give bells with arbitrary correlations just by varying the coordinates of an interpolating point, that is, the height in y on the mid-point denoted by H. Notice how the correlation depends on the "likeness" of the xy and xz projections of the corresponding wire, that leads to the unforeseen conclusion that the correlation on two-dimensional bells is dictated by the precise geometry of space-filling wires.

Exotic Patterns Inside the Bell

6

15

Exotic Patterns Inside the Bell

Even though the extension from two-dimensional wires, that is, Equations (1) and (2), to three-dimensional wires, that is, Equations (3) and (4), is natural, the added structure on the higher dimensional mappings hampers our ability to produce an elegant general proof of the validity of the Gaussian limit.21 To this effect, we decided to study the nature of the convergence towards a two-dimensional bell to elucidate if simple trends emerged that would suggest a pathway for an alternative proof. As the two-dimensional "bells" in Figures 15 and 16 are histograms over the y and z axes and over the y-z plane of 15 million chaos game points on the limiting wire, the idea was to study how circles and ellipses would be formed when the number of points was reduced to groups of a few thousand at a time. Intuitively, we expected circles to be made of circles and disks over the y-z plane, patterns that would dance "independently" as they made-up the bell. However, we were in for a surprise, for we found that, for many combinations on the parameters r^ and 9^, such sets of a few thousand points yielded exquisite decompositions of bells in terms of crystal-like patterns that had arbitrary symmetries. It turns out that such specific patterns occur whenever all r ^ ' s tend to 1 in magnitude and when the angles on both mappings, that is, d[ , 6\' and #2 , #2 , are properly "syn­ chronized." For example, they happen for 9\ — 9X ' — 9[ and #2 = #2 = #2 when both 9\ and #2 divide 2w (360 degrees) and when such angles are multiples of one another. To illustrate the nature of these findings, Figures 17 to 20 show twenty successive sets of 10,000 points each on the y-z plane (left to right and bottom to top) that are found playing the chaos game to build a wire that interpolates the generic set of points {(0,0,0), (1/2,1,1), (1,0,0)}, such that all coefficients r^ are negative and equal, that is, (1)

(2)

(1)

(2)

n

n n r i n

r = r\' = r\' = r2 = r\ = —0.9999. Figure 17 shows what is found when the two angles 9\ and 6>2, defined above, take on the values of 120 and 60 degrees, respectively, and when a fair coin is employed in order to play the chaos game. As may be seen, with the outcomes of mappings w\ and W2 colored red and blue, the patterns exhibit a clear six-fold symmetry (360/60) and include, surprisingly and quite noticeably, shapes that resemble those present in marine microorganisms. Figure 18 is simply the counterpart of Figure 17 and shows what happens when the coin bias is changed to 30%-70%, giving a multifractal texture over the three-dimensional wire. Notice how these six-fold patterns, now more blue than red, also define beautiful shapes that include many interesting and non-trivial rosettes. Figure 19 shows what is found when the two angles 9\ and #2 take on values of 90 degrees each and when the two mappings (3) and (4) are iterated according to a fair coin. Observe now that varied beautiful patterns with four-fold symmetry (360/90) arise, shapes that ultimately coalesce to define a circular bell. Figure 20 further illustrates the trends encountered by having angles 9^ = 72 and 92 = —72 degrees. Notice how, in this case, exotic patterns having ten-fold symmetry (2 • 360/72) decompose the bell. Observe how the precise shapes from frame to frame are not easily anticipated, as they depend on the peculiar choice of heads and tails employed in selecting the two mappings Wi and W2.22

16

Introducing

the Bell

Figure 17. Sequential p a t t e r n s decomposing the two-dimensional circular bell (left to right and b o t t o m to top). Interpolating points: {(0,0, 0), ( 1 / 2 , 1 , 1 ) , (1, 0 , 0 ) } , Parameters: r[ = r[ = r2 = r2 = r = - 0 . 9 9 9 9 , #i = 120, 92 = 60, fair coin. Dots per pattern: 10,000, Scale of each box: - 9 2 . 92. Seed for pseudo random tosses: —153.

Exotic Patterns Inside the Bell

17

Figure 18. Sequential patterns decomposing the two-dimensional circular bell (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r[ ' = r[' = r 2 = r2 = r = -0.9999, 0i = 120, 02 = 60, biased coin 30-70% split, Dots per pattern: 10,000, Scale of each box: - 7 1 , 71, Seed for pseudo random tosses: —153.

18

Introducing the Bell

Figure 19. Sequential patterns decomposing the two-dimensional circular bell (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r[' = r\ ' = r2 = r2 = r = —0.9999, 61 = 90, 82 = 90, fair coin, Dots per pattern: 10,000, Scale of each box: —68, 68, Seed for pseudo random tosses: —153.

Exotic Patterns Inside the Bell

19

Figure 20. Sequential patterns decomposing the two-dimensional circular bell (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r[' = r[' = r% = »2 = r = -0.9999, 0i = 72, 02 = - 7 2 , fair coin, Dots per pattern: 10,000, Scale of each box: -169, 169, Seed for pseudo random tosses: —153.

20

Introducing the Bell

The figures just explained illustrate that the usage of chance gives, rather surprisingly, very precise shapes that define treasures inside the bell: exotic kaleidoscopes of crystal­ like symmetric patterns made explicit via the chaos game, that, when added, amazingly result in the harmonious circular (or elliptic) bell over two dimensions.23 Given the fact that a close approximation of a bell arises from a large number of mapping iterations, for example, 15 million as in Figures 15 and 16, the patterns depicted in Figures 17 to 20 represent only a tiny sample of the behavior (forever) concealed inside bivariate Gaussian distributions. 24 As may be hinted from the sequences shown, following a path of iterations, that is, a branch on the tree of Figure 7, results in an ever-changing dance of patterns, that is nicely captured by the metaphor of an aleph as described in the celebrated story by Argentine

Figure 21. Sample patterns inside the circular bell having radial symmetry. The sets have been rotated for aesthetic purposes.

Exotic Patterns Inside the Bell

21

philosopher and writer Jorge Luis Borges.25 Quite literally, choosing a set of parameters r^ whose magnitude is close to the limiting value of 1 and synchronized angles 9^ yields a "point of light" from which to see a vast number of evolving symmetric shapes. As alternative coin tosses elicit different sequences of patterns and as alternative behavior is obtained depending on the signs of the parameters r^\ the chaos game generates a vast number of "kaleidoscopic Gaussian alephs."26 In the end, and as illustrated in Figures 21 and 22 and in the accompanying CD, the circular bell contains patterns having arbitrary n-fold symmetries, sets that may be classified as having radial or rotational traits. 27 Even though pseudo-random numbers are naturally used to specify suitable iteration paths, it is relevant to emphasize that the geometric sets decomposing the bell are, in the end, deterministic designs that lie hidden inside the bell. These designs represent "alter­ native universes" that strongly depend on the actual path of iterations traveled. Invariably,

Figure 22. Sample patterns inside the circular bell possessing rotational traits.

22

Introducing the Bell

however, they provide striking crystalline kaleidoscopic mandalas having unpredictable dy­ namics, that, by varying data points, free parameters and iteration paths, define a great many nets of gems.28 Altogether, they expound gigantic jigsaw puzzles of infinite varieties, whose pieces remarkably interlock to yield bells, suggesting that inside the bell there is hidden order in chance. It is worth remarking, as already explained regarding the filtering of singular measures into bells, that it is near the Gaussian limit, when all parameters r\[> have magnitudes tending to 1, and only near such a limit, where exotic behavior is found. In fact, if these parameters are all sufficiently small (say less than 0.99 in magnitude) evolutions that parallel the ones given become rather predictable, as all sequential patterns, with say 10,000 dots, give basically the same approximation of a unique and non-crystalline attractor.

Figure 23. Sample ice crystals inside the circular bell.

Natural Sets Inside the Bell

7

23

Natural Sets Inside the Bell

Besides the mathematical relevance of the curious exotic decompositions of the bell over two dimensions, it is particularly striking to realize that some of those patterns closely resemble a host of relevant natural shapes.29 As an example, Figure 23 shows that the circular bell contains beautiful designs that correspond to the magnificent structure of the ever-changing ice crystals.30 Certainly, and more impressively, the bell also includes, as members of gigantic jigsaw puzzles, some key geometric structures of particular relevance to life itself. This is illustrated in Figure 24 that portrays: (from top to bottom) the foot-and-mouth disease virus, the E. coli chaperonin GroEl protein, the B-DNA rosette (found projecting the double helix over a plane perpendicular to its main axis), and the Salmonella typhimuriumbacteria; as found in nature (left) and as approximated inside the bell (right). 31 The patterns inside a circular bell that evoke the structure of the virus, the protein and the bacteria were chosen from pages similar to those displayed in Figures 17 to 20, but yielding the appropriate number of, in order, 5, 7 and 11 tips. Even though improved fittings may surely be found for these biochemical units, the displayed approximations should help the reader appreciate that iteration paths exist that generate via the simple mappings w± and w2 in Equations (3) and (4) (or others similar to them) objects close to those natural sets.32 As may be noticed, patterns inside the bell may be found that closely capture the very detailed structure of the ten-fold B-DNA rosette, of universal importance in the transmission of hereditary genetic information for all known life forms.33 Even though alternative iteration paths may be selected to give other close renderings of the beautiful rosette and despite the fact that a connection between the biochemistry of the DNA molecule and a geometric rosette inside the bell is yet to be established, the appearance of such a pattern indeed suggests that the Gaussian distribution may play a central role beyond the familiar fields of physics, probability and statistics. As hinted by the patterns shown thus far,34 the bell represents an unexpected repository of shapes that, by providing suitable blueprints of key building blocks in physics and biology,35 establishes a new paradigm for addressing the very relevant questions pertaining to the origins of order, one containing a rather minimal computational complexity that may be coined "order at the plenitude of dimension." m 8

More Treasures Inside the Bell

While one ponders the equation that defines the beautiful standard bell curve

fix) = - L e~x2/2> one readily recognizes three key irrational numbers that help us understand generic shapes in our daily life: squares in y/2, spirals in e, and circles in w. As these numbers generate, in their binary expansions, sequences of digits that appear to be guided by "chance," (that is, \[2 = 1.0110100..., e = 10.101101... ,TT = 11.001001...) it becomes natural to ask

24

Introducing the Bell

Figure 24. Biochemical patterns inside the circular bell.

More Treasures Inside the Bell

25

Figure 25. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of \/2 (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r\ ' = rf] = r^] = r f ' = r = -0.99999999, 0i = 0, 92 = 30, Dots per pattern: 25,000, Scale of each box: -173, 173.

if they encode, via iterations of the simple mappings (3) and (4), relevant patterns inside two-dimensional bells. In this spirit, Figures 25 to 28 present some examples of what the bell contains when such numbers set up a dialogue between mappings w\ and u>2,foralternative space-filling wires passing by the generic interpolating points, that is, {(0, 0,0), (1/2,1,1), (1, 0,0)}, that result in, 12-, 8-, 6- and 10-fold symmetry, in that order. As seen in these figures, the digits of these irrational numbers (and surely others) do provide beautiful decompositions of the circular bell. Notice by comparing Figure 17 with Figure 27 that usage of the "random" digits of n yields other exotic patterns that, although resembling the ones reported via pseudo-random numbers, are (as expected) quite distinct. As seen in Figure 28, and via enlargements in Figures 29 and 30, the digits of % sur­ prisingly abridge relevant natural patterns, as the second set, made up of 20,000 dots, gives

26

Introducing the Bell

Figure 26. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of e (left to right and bottom to top). Interpolating points: {(0,0, 0), (1/2,1,1), (1,0,0)}, Parameters: —r[ — r\ ' = { ] rW = r 2 =r = 0.99999999, 6>i = 0, 92 = 45, Dots per pattern: 25,000, Scale of each box: -479, 479.

a topologically correct approximation to the projection of our B-DNA, as already seen in Figure 24. Notice the appearance of rings and spokes in the real and generated patterns that leads us to an unforeseen connection between the geometric structure of life (albeit frozen over two dimensions) and the binary digits of n through the bell.37 This intriguing linkage insinuates an unexpected avenue for finding "meaning" in the intrinsic "randomness" in 7r, and leads us to wonder what else could its binary expansion (or others) encode inside the bell (or others) in two and in higher dimensions. A preliminary analysis of the first 100,000 binary digits of y/2 and e reveals that they do not encode via the generic interpolating points, either every 10,000 or 20,000 dots, the B-DNA rosette. However, these results do not preclude the existence of such a shape later on or via alternative wires passing by other points and rather invite us to further study the mysteries of arbitrary symmetries that are encoded in the bell via such key numbers (and

More Treasures Inside the Bell

27

Figure 27. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of i: (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r\ = r[ = 2) =r = -0.9999, 6»i = 120, 62 = 60, Dots per pattern: 10,000, Scale of each box: - 9 5 , 95. r W = r2

28

Introducing the Bell

Figure 28. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of 7r (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: —r[ = r\ = r(21] = rf] =r = 0.99999999, 61 = 36, 02 = 36, Dots per pattern: 20,000, Scale of each box: -556, 556.

More Treasures Inside the Bell

29

Figure 29. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of n (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: —r\ — r[ — r{21] = r{22) =r = 0.99999999, 9X = 36, <92 = 36, Dots per pattern: 20,000, Scale of each box: -420, 420.

others). For in a manner that boggles the mind, a simple wire may code, via one (or many) irrational number(s), a multitude of pertinent patterns! In what follows, a rather incomplete gallery of the lovely patterns that lie inside the circular bell via the binary digits of n (also found in the accompanying CD) is given. This includes patterns obtained when the mappings in (3) and (4) are replaced by others of similar structure that do not result in functions but on more general attractors of maximal dimen­ sions that also result in bells over two dimensions.38'39 Such results, that include progressive buildups and collages, further invite the reader to explore the remarkable silent bell.40

30

Introducing the Bell

Figure 30. The planar structure of B-DNA (left) and a topologically correct rendering as found inside the bell via the binary digits of IT (right).

More Treasures Inside the Bell

The Silent Bell The bell peals intensely revealing mighty deed, in shadows, as by magic, designs of life crossbreed. Its central theme exalts independence as a feast, reparation of the broken in enduring code within. O aum of pure beauty, giving harmony to chance, 0 quintessential simplicity ascribing a meaning to IT. In fullness of dimension, while defeating all strife, dwells by ardent iteration a reflection of God's art.

31

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Part 2

Gallery Designs

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Radial Patterns

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Radial Patterns

37

38

Gallery Designs

Radial Patterns

39

40

Gallery

Designs

Radial Patterns

41

42

Gallery Designs

Radial Patterns

43

44

Gallery Designs

Radial Patterns

45

46

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Radial Patterns

47

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Rotational Patterns

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Rotational Patterns

51

52

Gallery Designs

Rotational Patterns

53

54

Gallery Designs

Rotational Patterns

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56

Gallery Designs

Rotational Patterns

57

58

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Rotational Patterns

59

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Rotational Patterns

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Evolutions & Quilts

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Evolutions & Quilts

65

G6

Gallery Designs

Evolutions & Quilts

67

68

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Evolutions & Quilts

69

70

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Evolutions & Quilts

71

72

Gallery Designs

Part 3

Technical N o t e s

1. The chaos game was introduced by Michael F. Barnsley. For a complete mathematical treatment of iterations of contractile mappings and the related iterated function system, IFS, theory, the reader is referred to Fractals Everywhere, by M. F. Barnsley, Academic Press, 1988. Such a reference also contains examples of a host of attractors (other than the Sierpinski triangle) that include the celebrated Barnsley's fern. 2. The notion of interpolating wires was also introduced by Michael F. Barnsley. These functions may pass by an arbitrary number of points and are constructed as follows. Given a set of N + 1 data points {(xn, yn) : n = 0 , 1 , . . . , N} with x0 < • ■ • < x^, define N affine mappings wn, in matrix notation:

wn (X] = (an \y)

\cn

° ] (^

dn) \y,

where |
to yield an = {xn ~ xn_i)/(xN

-

x0),

e n = {XN ■ %n-i ~ xQ ■ xn)/{xN cn = {Vn - yn-i)/(xN

- xQ),

- x0) - dn{yN - y0)/{xN

In = (XN ■ yn-i - x0 • yn)/(xN

-

x0),

~ x0) - dn(xN ■ y0 - XQ ■ yN)/(xN

-

x0).

Then, as explained by Barnsley in Fractals Everywhere, the mappings generate the graph G of a function passing by the data points, G = U^Li wn{G). G exists and is deterministic as it is the unique fixed point of a contractile set of mappings. 73

Technical Notes

74

3. The fractal (box-counting) dimension of a set G on the plane is defined as follows. Consider boxes of a given size 5 and count how many of them, that is, N(S), are needed in order to cover the set G. Study the power-law relationship that exists between 5 and N(8), that is, N(5) ~ 5~D, when 5 tends to zero. Then, the negative of the exponent, that is, D, is the fractal dimension of G. 4. As proven in Fractals Everywhere, note 2 is given by

the fractal dimension D for the most general case in N

E Klan_1 = 1 71=1

whenever J2n=i \dn\ > 1 and when all points are not aligned. Otherwise, D ~ 1. Even though not all attractors are fractal, the interpolating wires are known as fractal interpolating functions. 5. This result is known as Elton's ergodic theorem. For details the reader is referred to Fractals Everywhere, by M. F. Barnsley. 6. lip and q denote a general left-right coin bias (that is, p = 0.7, q = 0.3, as in the text), the process in Fig. 10 defines a binomial multiplicative cascade. This happens because, in the n t h generation, the distribution of areas yields pk ■ q(n~k\ k = 0 , 1 , . . . , n, whose occurrence is given by a binomial coefficient: /n\ \k)

=

n\ k\-{n-k)\

'

where k\ = k ■ (k — 1) • (k — 2) 2 - 1 denotes the factorial operation, and 0! = 1. As n tends to infinity, the domain [0,1] may be decomposed in terms of infinitely many layers pk-q(-n~k\ that define infinitely many intertwined Cantor sets, having fractal dimensions between 0 and 1. For further details on multifractal measures, the reader is referred to Fractals, by J. Feder, Plenum Press, New York, 1988. 7. When points are not equally spaced in x, for example, {0,rci,l}, the resulting wire would be sampled uniformly if the function operating to the left is used precisely X\% of the time. Any other choice for a coin would imply a texture on the wire that is singular and multifractal. For details, the reader is referred to "Deterministic fractal geometry and probability," by C. E. Puente, International Journal of Bifurcations and Chaos 4(6), 1613 (1994). 8. By varying the parameters that define a wire and its texture, one may obtain a wide variety of interesting shadows that resemble natural sets and include (deterministic) series that may be labeled "chaotic" and "random." These unexpected results suggest a deterministic Platonic framework for studying natural patterns, that is, one referring to the notion of "shadows" as given in Plato's Republic via his famous allegory of the caveman. For details on these ideas the interested reader is referred to "Multinomial multifractals, fractal interpolators, and the Gaussian distribution," by C. E. Puente, Physics Letters A 1 6 1 , 441 (1992); "A deterministic geometric representation of temporal rainfall: Results for a storm in Boston," by C. E. Puente and N. Obregon, Water Resources Research 32(9),

Technical Notes

75

Case + Derived Measure Moments Order

z = 0.9999

z = 0.999999

z = 0.99999999

3 4 5 6 7 8 9 10 11 12

0.000000000 2.999365046 0.000000000 14.990478199 0.000000000 104.866744091 0.000000000 943.002111830 0.000000000 10362.054001516

0.000000000 2.999993650 0.000000000 14.999904750 . 0.000000000 104.998666509 0.000000000 944.979997736 0.000000000 10394.669964570

0.000000000 2.999999937 0.000000000 14.999999048 0.000000000 104.999986665 0.000000000 944.999799975 0.000000000 10394.996699588

AT(0,1

0.0 3.0 0.0 15.0

0.0 105.0

0.0 945.0

0.0 10395.

2825 (1996); "A geometric Platonic approach to multifractality and turbulence," by C. E. Puente and N. Obregon, Fractals 7(4), 403 (1999); "Projections of fractal functions: A new vision to nature's complexity," by C. E. Puente, N. Obregon, O. Robayo, M. G. Puente and D. Simsek, Fractals 7(4), 387 (1999); and "Chaos and stochasticity in deterministically generated multifractal measures," by N. Obregon, C. E. Puente and B. Sivakumar, Fractals 10(1), 91 (2002). 9. The Gaussian distribution for a variate Y is given by My)

=

~JWo e x p { _ ( y " m )V2^ 2 } ,

where m and a2 represent the mean and variance of Y, and exp is the exponential function. 10. A proof of this fact is advanced in "The Gaussian distribution revisited," by C. E. Puente, M. M. Lopez, J. E. Pinzon and J. M. Angulo, Advances in Applied Probability 28, 500 (1996). The proof consists of showing (by induction) that the moments of the measure dy, which may be computed analytically via recursive expressions to yield ratios of polynomials on the parameter z, converge (after proper standardization and when z tends to one) to the moments of the standard Gaussian distribution 7V(0,1), that is, zero for all odd moments; and (2m — 1) • (2m — 3) • ... • 3 • 1 = (2m — 1)!! for the even moments of order 2m. For the set up of points used in the text, the mean and the variance yield fj, = 1/2 and 2 a = 1/(12 — 12z2), and hence a2 tends to infinity as z tends to unity. The convergence to the standard bell is illustrated in the table above, computed using the Maple V symbolic language package with a precision of 250 digits. Notice how the convergence of moments happens in an orderly fashion, with lower order (even) moments tending to the Gaussian values faster than higher order (even) moments.

Technical Notes

76

11. The other wires in Fig. 5, corresponding to alternative combinations on the scaling parameters d\ and d2, yield the following results for the set up of three points given in the text. When both scalings are positive, the "angel wings" shown in Fig. 8 "fly" to infinity, yielding a mean it = 1/(2 — 2z) and a variance a2 = 1/(12 — 12z2) that both tend to infinity as z tends to one. As the coefficient of variation, a//j,, contains the factor (1 — z)1/2, the + + wire gives in the limit a bell concentrated at infinity! In consonance with the previous note, the convergence to the standard Gaussian distribution is illustrated below for the first few moments. Notice how the long tail of the derived measure (as it is anchored by the initial points) results in slow convergence to zero on the odd order moments. When both scaling parameters di and d2 are negative, the symmetric "mountain profiles" shown in Fig. 5 appear to give in the limit yet another bell with mean fj, = 1/4 and variance a2 = 1/(12 — 12z2), as hinted at by the standardized table below. However, this happens not to be the case, for if the mean is calculated by levels fol­ lowing the binary tree portrayed in Fig. 7, one finds that the mean for all values up to a level n is f^-

2n + z(-2z)n (2»+i + l)(* + l ) '

and hence in the limit, when z tends to one, there are not one but two bells whose means oscillate between 0 and 1/2. 12. A proof of this remarkable fact is given in "The Gaussian distribution revisited," by C. E. Puente, M. M. Lopez, J. E. Pinzon and J. M. Angulo, Advances in Applied Probability 28,

Case + + Derived Measure Moments Order

z = 0.9999

z = 0.999999

z = 0.99999999

3 4 5 6 7 8 9 10 11 12

-0.032656325 3.001639399 -0.326658273 15.035259628 -3.432783098 105.643263393 -41.260116637 956.901068056 -568.823119975 10628.732737598

-0.003265983 3.000016400 -0.032659923 15.000352666 -0.342932064 105.006430686 -4.115251502 945.118861509 -56.586203271 10397.330855430

-0.000326599 3.000000164 -0.003265986 15.000003527 -0.034292860 105.000064307 -0.411514386 945.001188600 -5.658324296 10395.023307907

Af(0,1) 0.0 3.0 0.0 15.0 0.0 105.0 0.0 945.0 0.0 10395.0

77

Technical Notes

Case - Derived Measure Moments Order

z = 0.9999

z = 0.999999

z = 0.99999999

w, i)

3 4 5 6 7 8 9 10 11 12

0.000002939 2.999080244 0.000029380 14.986209090 0.000308244 104.807032780 0.003694899 942.107516384 0.050733888 10347.314696658

0.000000003 2.999990800 0.000000029 14.999862001 0.000000309 104.998068023 0.000003704 944.971020552 0.000050923 10394.521843197

0.000000000 2.999999908 0.000000000 14.999998620 0.000000000 104.999980680 0.000000004 944.999710200 0.000000051 10394.995218301

0.0 3.0 0.0 15.0

0.0 105.0

0.0 945.0

0.0 10395.0

500 (1996). The proof relies on showing that arbitrary portions of the wire, with end points given by images of the initial points via arbitrary usage of mappings w± and W2, may be built as fractal interpolating functions, in terms of suitable functions w\ and w^, that themselves give limiting bells from uniform textures over their domain. All these bells have finite means and all share variances that tend to infinity in the same general fashion, that is, as 1/(1 — z2) as mentioned in the previous notes. This fact is then used to show that a diffuse measure, that is, one with a continuous cumulative distribution that may be approximated piecewise as a sum of uniform measures over disjoint subintervals, is transformed by the limiting wire into the weighing of a great many bell curves (that is, not Gaussian distributions), that surprisingly coalesce to produce yet another bell curve. 13. For the set up of points in Figs. 12 and 13, the mean of the limiting bell satisfies /x = p, where p is the redistribution parameter of the (binomial) multifractal measure. The Gaussian results remain valid for all sign combinations on the free parameters d\ and d2 (see note 11) and are naturally extended to plane-filling fractal interpolating functions passing by an arbitrary number of points, that is, for those defined in note 2. 14. The construction of a bell as a projection off a plane-filling fractal interpolating function results in a central limit theorem as follows. If X denotes an arbitrary diffuse measure over x, and Y(z) = fz{X) represents the derived measure found via a fractal interpolating function with parameter z, then the random variables W\ = Y(Q), W2 = y(3/4) — Y(0), Wz = y(8/9) - y(3/4), ...,Wn = Y(l- 1/n2) - Y(l - l / ( n - l) 2 ) satisfy

£ w i = y(i--i/n 2 ),

Technical Notes

78

and such a sum converges, after proper normalization, to the standard Gaussian distribution. It turns out that such a central limit theorem is not trivial because the variates Wi are (a) dependent (although weakly), (b) non-stationary, and (c) unbounded. For further details the interested reader is referred to "The Gaussian distribution revis­ ited," by C. E. Puente, M. M. Lopez, J. E. Pinzon and J. M. Angulo, Advances in Applied Probability 28, 500 (1996). 15. For interesting treatments of turbulence and its relation to multifractals, the reader is referred to Turbulence, by U. Frisch, Cambridge University Press, 1995; "Fractals and multifractals in fluid turbulence," by K. R. Sreenivasan, Annual Reviews of Fluid Mechanics 23, 539 (1991); and references therein. 16. Fractal interpolating functions in three dimensions may be constructed following a procedure analogous to the one given in note 2 for two dimensions. Given a set of N + 1 points {(xn, yn, zn) : n — 0 , 1 , . . . , N} with XQ < ■ • ■ < XN, one uses N affine mappings: (X\

wn

(an 0 0 \ (x\

y +

Cn CLn tln

y

In

\9n)

w

whose coefficients an, cn, kn and en, fn, gn are defined via the conditions: fx0^

wr,

yo

W

(Xn^

yn

2/n-l

V Zn-1J

\Zn)

to yield an = (xn - xn-i)/(xN

-

x0),

cn = {yn - j/n-i - dn ■ (yN - y0) - hn ■ (zN -

z0))/(xN

K = {Zn - Zn-1 ~ In ■ (VN ~ Vo) ~ ™n • (zN ~ Z0))/(xN en = (XN ■ Xn-1 - X0 ■ Xn)/{XN fn = Vn-1 - dn-y0-hn-

Z0-

Xo)

- XQ) ,

- XQ) , Cn- XQ,

9n = zn~i - ln ■ 2/o - rnn ■ z0 - kn ■ x0 . A unique attractor G = \J^=1wn(G) exists when the mappings are contractile, that is, when the maximal eigenvalue of A^An is less than unity, and An is the parameter sub-matrix: An —

Un •'■n

in rnn

For details the reader is referred to Fractals Everywhere, by M. Barnsley, Academic Press, 1988 and to "Gaussians everywhere," by C. E. Puente and A. D. Klebanoff, Fractals 2(1), 65 (1994).

Technical Notes

79

17. For further developments on the projection ideas to model spatial patterns and their dynamics, the interested reader is referred to "A fractal-multifractal approach to geostatis­ tics," by C. E. Puente, in Geostatistics for the Next Century, ed. R. Dimitrakopoulos, Kluwer Academic Publishers, Dordrecht, 1994, pp. 476-487; "A new approach to hydrologic model­ ing: derived distributions revisited," by C. E. Puente, Journal of Hydrology 187, 65 (1996); "A fractal-multifractal approach to groundwater contamination. 1. Modeling conservative tracers at the Borden site," by C. E. Puente, O. Robayo, M. C. Diaz, and B. Sivakumar, Journal of Stochastic Environmental Research and Risk Assessment 15(5), 357 (2001); and "A fractal-multifractal approach to groundwater contamination. 2. Predicting conservative tracers at the Borden site," by C. E. Puente, 0 . Robayo, and B. Sivakumar, Journal of Stochastic Environmental Research and Risk Assessment 15(5), 372 (2001). 18. The joint Gaussian distribution for a vector (Y, Z) is given by

fy,z{y,z)

= ——

j== -exp{-l/2[(y,z)

- (mY,mz)]B

1

[(y,z) -

(mY,mz)}T},

where (mY, mz) are the means of individual bells (a vector), B is the covariance matrix for variates Y and Z, B~x is its inverse, | • | denotes a matrix determinant, T represents the transpose of a vector, and exp is the exponential function. The covariance matrix is

B =

(

°Y

\ payerz

P°YO-Z\

a\

) '

where aY and az are the variances of the individual bell curves Y and Z, and p represents their correlation, — 1 < p < 1. Once Y and Z are standardized, subtracting their mean and dividing by their standard deviation (that is, the square root of the variance), p dictates the shape of the distribution as follows. When p = ± 1 , fY,z(y,z) looks from above like a straight line, namely Y = Z or Y = —Z, respectively. When 0 < p < 1, the joint distribution yields elliptical contours from above, ellipses whose major axis is Y = Z. When — 1 < p < 0, the ellipses have as major axis Y = — Z. When p = 0, the contours are circles and the variates Y and Z become independent. 19. These conditions on the parameters r^ and 6^ yield the maximal norm of unity on the eigenvalues of A^An as reported in note 16. For further details the reader is referred to "Gaussians everywhere," by C. E. Puente and A. D. Klebanoff, Fractals 2(1), 65 (1994). 20. For the three interpolating points {(0, 0, 0), (1/2, Hu H2), {1,H3, HA)}, and for dx = 0[ = 0[ and 62 = #2 = #2 > the 16 sign combination cases in the following table may be divided into four major subgroups.

80

Technical Notes

Case

rj[/

7*1

7*2

^2

Case

rj/

r['

r^

+ + + +

+ +

+ + +

+

9 10 11 12 13 14 15 16

— — —

+

+ +

+

—

— —

—

—

+

—

+

—

— —

— —

— —

—

+ + +

Group Group Group Group

A B C D

= = = =

{Case {Case {Case {Case

1, 2, 3, 6,

Case Case Case Case

7, 4, 5, 8,

—

+ + —

+ —

Case Case Case Case

—

+ + — —

—

+ + + — —

+

—

+

?2 —

+ + — — —

+ —

10, Case 16}, 13, Case 15}, 12, Case 14}, 9, Case 11},

whose correlations are as follows: Group A: p = 0, except when $i = kix, 62 = for, which give p = 1 or p — —1, k, I integers. Group B: p = 0, except when 9\ — kit, which gives arbitrary correlation p, which depends on 62 and the interpolating coordinates, k an integer. Group C: p = 0, except when 92 = for, which gives arbitrary correlation p, which depends on #! and the interpolating coordinates, I an integer. Group D: p — 0, except when 6 = 9i — 02 + kn, which gives arbitrary correlation p, which depends on 9 and the interpolating coordinates, k an integer. For additional details the interested reader is referred to "Gaussians everywhere," by C. E. Puente and A. D. Klebanoff, Fractals 2(1), 65 (1994). 21. Although the joint moments of the derived measure dyz may indeed be calculated analytically and recursively based on a uniform measure over the wire, as was done for the bell over one dimension (see note 10), the final expressions are cumbersome and do not allow writing a general proof by induction. Those formulas have been used, however, to show that the low order joint moments of dyz, in addition to those of dy and dz, indeed converge (when properly normalized) to the moments of the Gaussian distribution over two dimensions. 22. The patterns in Figs. 17-20 were obtained performing the coin tosses according to the pseudo-random number generator routine rani as defined in Numerical Recipes: The Art of Scientific Computing, by W. H. Press, S. A. Teukolsky, W. T. Vettterling and B. P. Flannery, Cambridge University Press, 1989. 23. Decompositions of elliptical bells are found to be not just elliptical renditions of patterns found inside circular bells. Instead, these typically are stretched broken patterns aligned with the major axes of the limiting ellipse.

Technical Notes

81

24. As noted by Michael F. Barnsley, close inspection of the patterns in Fig. 19 re­ veals that they arise not from one but from four bells over two dimensions, centered at (0,0), (1/2,1/2), (1,0) and (1/2, - 1 / 2 ) . It turns out that the results reported earlier for a bell over one dimension, note 11, may be generalized for the generic interpolating points {(0,0,0), (1/2,1,1), (1,0,0)} and for 0X = 9{1] = 9{? and 92 = 9{21] = 9{22) according to the sixteen cases defined in note 20. Numeric calculation of means over both y and z by levels, that is, following a binary tree as the one in Fig. 7, reveals that the most commonly derived measures generated by the space-filling three-dimensional wires are indeed single bells with finite means and with variances that tend to infinity. But there are exceptions along the lines #i = kn, 92 = kir and 9\ = 92 + kir, for an integer k, and for some sign combination cases on the wire's scalings, as follows. When 9 — 9\ — kir, all cases in groups A and C and cases 8 and 9 give a single bell with finite means whose correlation is zero, cases 2 and 6 yield a single bell whose mean converges to (oo, oo), cases 4 and 11 result in another bell whose mean converges to (oo, — oo), and cases 13 and 15, which generate non-circular bells as a function of 9, are actually two oscillating bells, as in the one-dimensional case explained in note 11. When 9 — 92 — kir, the results are similar to the ones just explained. All cases in groups A and B and cases 8 and 9 give a single circular bell with finite means, cases 3 and 6 result in a single bell centered at (oo, oo), cases 5 and 11 yield bells with means (—00,00), and cases 12 and 14 result in elliptical patterns arising from two oscillating bells. When 9 — 9i — 92+kn, all cases in groups B and C and cases 7 and 10 give a single circular bell with finite mean, cases 6 and 11 yield single bells whose mean tends respectively to (00, 00) and (00, —00), and cases 8 and 9, yielding non-zero correlation, correspond (contrary to the results just reported for the other lines) to single bells with finite means. Cases 1 and 16 appear to converge to a single bell with a finite center, but calculations reveal that they contain oscillatory behavior that encompasses 2n/9 bells, when such a ratio is integer. For case 1, the bell's center travels following a circle, centered at the apparent single mean, according to the simple dynamics 4>n+i = 4>n + 9. The radius of such a circle increases as 9 decreases, for it is found that the line joining the points (0,0) and (1/2,1/2) is always inscribed within such a circle. When 2ir/9 = n, the circle map generates a regular polygon having n sides and hence such a case corresponds to n bells that cycle in such a circle. When 9 does not divide 2n, case 1 generates infinitely many bells whose means travel a circle in a non-periodic fashion. For case 16, as happens in Fig. 19, there are also bells galore. This case also yields simple dynamics on a circle, 4>n+\ = 4>n + IT — 9, which result in stars inscribed on a finite circle that contains the line joining the points (0,0) and (1/2,1/2). When 2n/9 — n, the simple map travels within the circle and yields sharp oscillatory behavior on the means of the implied n bells. As before, this case yields infinitely many bells when 9 does not divide 2ir. It should be stressed that cases 1 and 16 only give many bells along the line 9 = 9\ — 92 + kir, for outside such a line the dynamics no longer happen along a circle but rather are attracted towards the center yielding, at the end, a single bell with finite mean, as reported earlier.

Technical Notes

82

9

Seed

Initial point

Scale of box

2TT/3

-111 -153 -111 -111 -111 -111 -153 -111 -153

60,001 1 1 140,001 20,001 1 20,001 160,001 180,001

-103,103 -163,163 -272,272 -388,388 -432,432 -468,468 - 5 6 4 , 564 -784,784 -753,753

TT/2 2TT/5 TT/3 2TT/7 TT/4 2TT/9 TT/5 TT/6

Figure 21.

9

Seed

Initial point

Scale of box

TT/2

-111 -153 -111 -111 -111 -153 -153 -153 -153

1 100,001 1 20,001 100,001 20,001 1 40,001 20,001

-78,78 -65,65 -87,87 -93,93 -97,97 -119,119 -150,150 -115,115 -151,151

2TT/5 TT/4 2TT/9 TT/5 TT/6 TT/9 TT/10 TT/12

Figure 22.

25. Borges' famous tale may be found in The Aleph and Other Stories, Edited and translated by N. T. di Giovanni, E. P. Dutton & Co., 1970. 26. As may be expected, usage of nearly space-filling fractal interpolating functions passing by more than three points (that is, the construction in note 16) yields many more beautiful alephs. 27. The nine radial patterns in Fig. 21 are based on sign combinations that belong to case 2 (that is, r[ = r[ = r^ = —r|> — r, in Group B) as defined in note 20. Those with rotational characteristics in Fig. 22 are found via combinations in case 1 (that is,

Technical Notes

83

r

r r i i 2 — r 2 — r ) m Group A). All patterns are found for wires that interpolate the set {(0, 0,0), (1/2,1,1), (1,0, 0)}, with 91 = 92 = 9 and r = 0.99999999. All sets include 20,000 dots found via fair coin tosses performed using routine rani as defined in Numerical Recipes. The Art of Scientific Computing, by W. H. Press, S. A. Teukolsky, W. T. Vettterling and B. P. Flannery, Cambridge University Press, 1989. The specific parameters per frame are given in the previous page (left to right and bottom to top). Angles are in radians.

28. Even though all patterns inside the bell are found to be rather beautiful, their degree of sharpness depends, in a non-trivial way, on the iteration path used. Also, varying the parameters according to the four groups in note 20 results in individual sets that have varying degrees of aesthetic appeal. The obtained behavior depends on the choices of signs on the coefficients r^ as follows. Patterns on group A that correspond to sign combinations that agree by map (that is, either both positive or both negative) and that yield sets with rotational traits, and patterns on groups B and C, whose signs agree only on one of their maps and that result in radial symmetric sets, are rather sharp and well defined. In contrast, patterns on group D, that correspond to coefficients r^ whose signs do not agree on either of their maps, may often appear blurred or fuzzy (depending on the choice of angles 9^). These radial symmetric sets do require a larger number of points per frame in order to have comparable appeal. 29. Alternative algorithms, other than iterations of simple mappings, may be employed in order to generate (other) beautiful n-fold symmetric patterns. Interesting examples may be encountered in Symmetry in Chaos, by M. Field and M. Golubitsky, Oxford University Press, 1992 and Physics of Chaos in Hamiltonian Systems, by G. M. Zaslavsky, Imperial College Press, 1998. For further examples and for a lucid treatment of symmetry in nature the reader is referred to What Shape is a Snowflake?, by I. Stewart, W. H. Freeman and Company, 2001. 30. The "matches" shown in Fig. 23 were found by filling a template of an observed ice crystal using a suitable sequence of iterations, pasting appropriate sequences of heads and tails as generated by routine rani from Numerical Recipes: The Art of Scientific Com­ puting, by W. H. Press, S. A. Teukolsky, W. T. Vettterling and B. P. Flannery, Cam­ bridge University Press, 1989. All patterns shown are given by wires that pass by the points {(0,0,0), (1/2,1,1), (1,0,0)} and that have as parameters 9X — 92 = 7r/3 and —r[ — r[' — ?2 = r 2 = 0.99999. The number of points making up the figures depends on the pattern considered and are as follows (left to right and bottom to top): 109,000, 121,000, 164,000, 136,000, 103,000, 126,000, 83,000, 144,000 and 151,000. The scales of their boxes are: -435, 435; -400, 400; - 4 8 1 , 481; -437, 437; - 3 9 1 , 391; -440, 440; -417, 417; -410, 410 and —414, 414. The templates were made from the catalog Snow Crystals, by W. A. Bentley and W. J. Humphrey, Dover Publications Inc., 1966. For other examples of "ice crystals" generated via the binary digits of IT, the reader is referred to "Snow Crystals inside the Bell," by C. E. Puente and M. G. Puente, submitted to Symmetry, 2003. 31. The key references for the real patterns in Fig. 24 are: "The three-dimensional structure of foot-and-mouth disease virus at 2.9 A resolution," by R. Acharya, E. Fry, D. Stuart,

84

Technical Notes

Set

\r\3,\

91

02

Virus Protein Bacteria

0.999 0.99992 0.99997

2TT/5

-2TT/5

2vr/7

8TT/7

2TT/11

4TT/11

#3

Seed

Initial point

pi

p2

Ps

2TT/11

-22,571 -8,103 -92,767

34,001 20,001 360,001

2/5 1/2 1/3

3/5 1/2 1/3

1/3

Figure 24.

G. Fox, D. Rowlands and F. Brown, Nature 337, 709 (1989); "The crystal structure of the bacterial chaperonin GroEl at 2.8 A," by K. Braig, Z. Otwinoski, R. Hedge, D. C. Boisvert, A. Joachimiak, A. L. Horwich and P. B. Sigler, Nature 371, 578 (1994); and Biochemistry, by D. Voet and J. G. Voet, John Wiley & Sons, 1995. 32. The patterns "inside" the bell corresponding to the virus and the protein in Fig. 24 come from wires that interpolate {(0, 0, 0), (1/2,1,1), (1, 0, 0)} and are made of 2,000 and 10,000 points, respectively. The bacteria set is made of 40,000 dots arising from three mappings that interpolate {(0, 0, 0), (1/2,1,1), (1, 0, 0), (3/2,1,1)}. These sets have all their scalings r\ positive and were found implementing the chaos game via routine rani as defined in Numerical Recipes: The Art of Scientific Computing, by W. H. Press, S. A. Teukolsky, W. T. Vettterling and B. P. Flannery, Cambridge University Press, 1989. The key parameters for the shown patterns are given in the table above. 33. The fitted B-DNA rosette is found interpolating the generic data set {(0,0,0), (1/2,1,1), (1,0,0)} using 61 = 62 = TT/5 and -r? = r{? = r? = r{22) = 0.99999999999. The pattern arises by building an iteration branch pasting together sev­ eral groups of 500 tosses each that correspond to alternative fair coin tosses, as generated via routine rani, Numerical Recipes: The Art of Scientific Computing, by W. H. Press, S. A. Teukolsky, W. T. Vettterling and B. P. Flannery, Cambridge University Press, 1989. The pathway of iterations contains 42,500 nodes and was obtained interacting with a template of the DNA rosette, selecting a suitable group of tosses among 100 groups per seed. The sequential list of seed and corresponding groups are as follows: seed —4668, group 48; seed -798, group 44; seed -367, groups 46, 90, 19, 27, 54, 69, 100, 7, 46, 51, 3, 22, 73, 7, 36, 31, 28, 29, 15, 3, 6, 12, 8, 23, 46, 19, 55, 9, 86, 47, 66, 77; seed -1675, group 54; seed -2678, groups 21, 84, 97, 64, 30, 76, 26, 43; seed -337, groups 38, 30, 11, 31, 22, 81, 28, 23, 22; seed -6715, groups 60, 83; seed —1212, group 99; seed —3026, group 16; seed 1697, groups 27, 83; seed -269, groups 80, 75, 70, 45, 51, 59, 91, 28, 42; seed -154, group 9; seed -672, groups 99, 21, 73, 29, 74; seed -1662, groups 21, 2, 16, 86, 31; seed -261157, group 38; seed —94477, groups 22, 60; seed —153, group 95; seed —64100, group 9; seed —126, groups 92, 83. The first two groups of 500 dots need to be deleted in order to get the inner hole on the rosette. 34. As expected, higher dimensional bells may be built in terms of affine mappings that have a higher number of coordinates. These bells sometimes are also decomposed in terms

85

Technical Notes

of non-trivial patterns. unexplored.

These patterns and the likely treasures they imply are largely

35. The bell also includes, as its members, patterns that match the probability distribution for some of the states of the hydrogen atom, as reported in Principles of Quantum Mechanics, by Hans C. Ohanian, Prentice Hall, 1990. 36. For lucid treatments on how simple ideas may perhaps explain the origins of order via the notion of self-organization, the reader is referred to At Home in the Universe, by S. Kaufman, Oxford University Press, 1995, to How Nature Works, by P. Bak, Copernicus, Springer-Verlag, 1996 and to A New Kind of Science, by S. Wolfram, Wolfram Media Inc., 2002. As the projection ideas hint at "intelligent" design inside the bell, the reader is also referred to Darwin's Black Box: The Biochemical Challenge to Evolution, by M. Behe, Free Press, 1996 and to No Free Lunch, by W. Dembski, Rowman k, Littlefield, 2002. 37. Although finding such results appears coincidental, it is certainly not trivial to have found that the exact pathway implicit in the first 40,000 binary digits of ir leads to the 20,000 dot rosette that closely resembles the one present in all life forms. For if, say, just the 10,000th binary digit of n is nipped from 1 to 0, then the rosette shown in Fig. 30 no longer appears. Because there is an extremely large number of possible iteration scenarios inside the bell (that is, in powers of two, yielding 2 40 ' 000 alternatives when considering the first two patterns in Fig. 29) and because added richness is obtained by varying the wire's interpolating points, finding the actual probability that the DNA rosette appears is, in the end, quite difficult. An estimate of such probability may be advanced following the construction of the rosette as reported in Fig. 24 and note 33, as follows. If one counts the number of cases which land inside the template and follow such a process stage by stage, that is, in groups of 500, one finds that the following number of groups (out of 100) could have been selected: 100, 3, 6, 9, 24, 11, 22, 49, 22, 25, 35, 62, 41, 63, 61, 4, 65, 84, 81, 82, 64, 89, 77, 91, 81, 84, 27, 65, 81, 34, 45, 61, 56, 52, 50, 49, 18, 47, 21, 18, 34, 52, 21, 40, 41, 52, 44, 14, 43, 54, 49, 19, 7, 13, 5, 21, 7, 5, 6, 10, 1, 4, 13, 10, 6, 10, 18, 5, 25, 1, 8, 13, 15, 8, 10, 17, 11, 9, 10, 12, 8, 7, 14, 10 and 3. Based on such numbers, and assuming that these are representative of any other path, one may find a probability for the rosette via their multiplication. The obtained probability is indeed exceedingly small — 3.64 x 10 - 5 9 . If the average acceptance probability of 0.32 from the above numbers is used to model the 40 groups of 500 points each making up the graph in Fig. 30, one gets another esti­ mate for the probability of the DNA rosette as (0.32)40 or 1.61 x 10~20. For additional details the reader is referred to "DNA, 7r and the Bell," by C. E. Puente, Complexity 6(2), 16 (2000). 38. General attractors that are not graphs of fractal interpolating functions may be defined via iV affine mappings:

Wr,

y

w

I On 0 0 \ (x\ cn un nn y

w

+

In \9n)

Technical Notes

86

with left-right end-data points {(xn(L),yn(L),zn(L)),

Wr,

/*i(L)\

(xn{L)\

yi(L)

Vn(L)

(xn(R),yn(R),zn(R))},

fxN(R)\

Vn(R)

\zN(R)J

\zn(R)J

where Xi(L) = mina; n (L), XN(R) = ma,xxn(R), n = 1,

K

(dnhn\(rWcoseU

\lnmn)

(xn(R)\

yN(R)

Wr,

1

Ir^sin^ )

that is

N, and such that -r^smtn r{n]cosLn

has L2-norm less than unity. As in note 16, a unique attractor G = Wi(G) U • • • U WN(G), a pseudo-wire, exists, whose fractal dimension D in [0, 3) depends on the coefficients an and the sub-matrix An. When D is maximal, either three when the left-right end-point intervals for the mappings overlap or less than three when they do not overlap yielding Cantorian domains, projec­ tions of unique measures generated in G via the chaos game generate bivariate Gaussian distributions that expound yet other kaleidoscopes of symmetric patterns. For further details on these generalizations that clearly include fractal interpolating func­ tions as a particular case, the reader is referred to "The exquisite geometric structure of a central limit theorem," by C. E. Puente, in press, Fractals, 2003. 39. A fortran program implementing the ideas in the previous note is as follows.

program branch-sw-pi c c c c c c c c c c c c c c c c c c

Computes dots located on the attractor of pseudo-wire denned by NF affine functions in three dimensions following a single branch of an NF-ary tree given by the binary digits of pi The mappings have end points: (xi(n),yi(n),zi(n)),

(xr(n),yr(n),zr(n))

The final parameters of the mappings are: (wa(n) 0 0) (we(n)) (wc(n) wd(n) wh(n)) + (wf(n)) (wk(n) wl(n) wm(n)) (wg(n)) Where the lower right matrix is: (rl(n)*cos(thl(n)) (rl(n)*sin(thl(n))

-r2(n)*sin(th2(n))) r2(n)*cos(th2(n)))

Technical Notes C

C+++++++++++++++++++++++++++++++++++++++ c c c c c c c

NOTES: currently the maximum number of functions is 8 angles are in degrees niter is the total number of iterations points stored are those between bstore and estore

c implicit double precision (a-h, o-z) parameter (Pi = 3.14159265358979324d0, NF = 8) c dimension xl(NF), yl(NF), zl(NF), xr(NF), yr(NF), zr(NF) dimension wa(NF), we(NF), wc(NF), wd(NF), wh(NF), wf(NF), wk(NF) dimension wl(NF), wm(NF), wg(NF), rl(NF), r2(NF), thl(NF), th2(NF) integer bstore, estore c character*70,filel c c

Open output files:

c print *, '? summary file' read *, filel open(10, file = filel, status = 'unknown') c print *, '? dots file' read *, filel open(ll, file = filel, status = 'unknown') c c

Open file containing the binary digits if pi

c open(12, file = 'filebipi', status = 'old') c

C+++++++++++++++++++++++++++++++++++++++++++ c

NOTE: This file may be replaced by pseudo-random numbers if desired

C+++++++++++++++++++++++++++++++++++++++++++ C

87

Technical Notes

88

c

Enter storage information

c print *, '? Number of iterations' read *, niter writeflO,*) '! Number of iterations ', niter c print *, '? Begining of storage' read *, bstore writeflO,*) '! Begining of storage ', bstore c print *, '? End of storage' read *, estore writeflO,*) '! End of storage ', estore c c

Enter initial geometric configuration

c print *, '? geometry file' read *, fuel open(14, file = fuel, status = 'old') c

C+++++++++++++++++ c

Read end-points

C+++++++++++++++++ c read (14,*) nfun do n = 1, nfun read (14,*) xl(n), xr(n) read (14,*) yl(n), yr(n) read (14,*) zl(n), zr(n) end do c write(10,*) ' ' write(10,*) ' ! Initial write (10,*) nfun do n = 1, nfun write(10,*) xl(n), write(10,*) yl(n), write(10,*) zl(n), end do c

Configuration: nfun, X, Y, Z'

xr(n) yr(n) zr(n)

Technical Notes

c

Enter scalings and rotations

c print *, '? scalings and rotations' print *, '? r l ' read *, (rl(n), n = 1, nfun) print *, '? t h l ' read *, (thl(n), n = 1, nfun) print *, '? r2' read *, (r2(n), n — 1, nfun) print *, '? th2' read *, (th2(n), n = 1, nfun) c write(10,*) ' ' write(10,*) ' ! Scalings and rotations by mapping: (rl, t h l , r2, th2)' do n = 1, nfun write(10,*) r l ( n ) , t h l ( n ) , r2(n), th2(n) end do c

C++++++++++++++++++++++++ c

Transform angles into radians

C++++++++++++++++++++++++ c do n = 1, nfun t h l ( n ) = Pi*thl(n)/180.0D0 th2(n) = Pi*th2(n)/180.0D0 end do c c

Define free affine mapping parameters

c do n = 1, nfun wd(n) = rl(n)*dcos(thl(n)) wh(n) = -r2(n)*dsin(th2(n)) wl(n) = rl(n)*dsin(thl(n)) wm(n) = r2(n)*dcos(th2(n)) end do c c c

Find other affine mapping parameters

89

Technical Notes

90

C+++++++++++++++++++++++++++++++++ c

first determine end points for contractions

c xendl yendl zendl xend2 yend2 zend2

= = = = = =

xl(l) yl(l) zl(l) xr(l) yr(l) zr(l)

c do n = 2, nfun if(xl(n).lt.xendl) then xendl = xl(n) yendl = yl(n) zendl = zl(n) end if if(xr(n).gt.xend2) then xend2 = xr(n) yend2 — yr(n) zend2 = zr(n) end if end do c

C++++++++++++++++++++++++ c

now find parameters

C++++++++++++++++++++++++ C

xrange = xend2 — xendl y range = yend2 — yendl zrange = zend2 — zendl c do n = 1, nfun wa(n) = (xr(n) — xl(n))/xrange wc(n) = (yr(n) — yl(n) — wd(n)*yrange — wh(n)*zrange)/xran wk(n) = (zr(n) — zl(n) — wl(n)*yrange — wm(n)*zrange)/xran we(n) = xl(n) — xendl*wa(n) wf(n) = yl(n) — wd(n)*yendl — wc(n)*xendl — wh(n)*zendl wg(n) = zl(n) — wl(n)*yendl — wk(n)*xendl — wm(n)*zendl end do c

c

Do iterations and store when requested

Technical Notes C

C++++++++++++++++++++++++++++++++++++++++++ c

points may be found sequentally or based on the same starting point

C++++++++++++++++++++++++++++++++++++++++++ c print *, '? Enter 1 if sequential patterns, 0 otherwise' read *, iseq write(10,*) '! Sequential patterns indicator ', iseq c print *, '? Enter number of points per frame' read *, nseq write(10,*) '! Points per frame ', nseq c

C++++++++++++++++++++++++++++++ c

Define the starting point

C++++++++++++++++++++++++++++++ C

xx = xr(l) yy = yr(l) zz = zr(l) c nstore = estore — bstore + 1 write (11,*) nstore c do i = 1, niter c if(iseq.eq.O) then if(mod(i,nseq).eq.O) then xx = xr(l) yy = yr(l) zz = zr(l) end if end if c read(12,*) ii n l = ii + 1 c xnew = wa(nl)*xx + we(nl) ynew = wc(nl)*xx + wd(nl)*yy + wh(nl)*zz + wf(nl) znew = wk(nl)*xx + wl(nl)*yy + wm(nl)*zz + wg(nl) c xx = xnew yy = ynew zz — znew

91

Technical Notes

92 C

C+++++++++++++++++++++++++++++++++ c

points stored in unit 11 may be plotted

C+++++++++++++++++++++++++++++++++ c if((i. ge. bstore).and.(i. le. estore)) then write(ll,*) nl, real(yy), real(zz) end if c end do c stop end 40. The color plates that follow and as included in the accompanying CD were obtained using the program in the previous note, using both wires (as defined in the text) and pseudo-wires (as defined in note 38). Such sets share an equal value for the magnitude of the scaling parameters of 0.99999999 and are found by iterating two suitable affine mappings according to chunks of binary digits starting the process at the second point of the first set of endpoints. The designs that follow are hence not sequential as those reported in the text, but they do reflect the implicit randomness of the binary digits of 7r. The sets use an alternative number of iterations as follows. Those portrayed by themselves on a page are made of 300,000 points, those used to portray evolutions (from left to right and bottom to top) contain a total of 90,000 points in increments of 15,000, and all others are made of 50,000 points. All designs are based on five alternative geometric set-ups, whose end-points are given by the following tables. The specific parameters of all the sets in the book (from left to right and bottom to top on a given page) share the following notation: (a) an asterisk to denote usage of pseudo-wires, (b) a number denoting the sign combination case on the mapping's scalings as given in note 20, (c) a number representing the end-points set-up used, (d) the angles 8i and #2 (in degrees) used in the iterations, and (e) the chunk of digits of TV that defines the set.

Wires Geometry 1 2 3 4 5

End-points {(0,0,0), (1/2,1,1)} {(0,0,0), (3/4,1,1)} {(0,0,0), (1/2,1,1)} {(0,0,0), (1/2,1,-5)} {(0,0,0), (1/2,1,-3)}

{(1/2,1,1), (1,0,0)} {(3/4,1,1), (1,0,0)} {(1/2,1,1), (1,3,0)} {(1/2,1,-5),(1,0,0)} {(1/2,1,-3),(1,5,0)}

Technical Notes

93

Pseudo-Wires Geometry

1 2 3 4 5

End-points {(0,0,0),(1/2,1,1)} {(0,0,0),(0.3,1,4)} {(0,0,0),(0.8,5,-5)} {(0,3,-3),(0.7,7,-6)} {(0,0,0),(0.2,1,1)}

{(0.4,2,2),(1,1,0)} {(0.6,0,1),(1,1,3)} {(0.2,1,1),(1,3,-2)} {(1/2,1/2,-0.6),(1,3,2)} {(0.7,1,1),(1,0,1)}

For example, the single set on Color Plate 1 made of 300,000 points is found via a wire with searing's signs — + + + (case 5 in note 20), one that interpolates {(0, 0, 0), (1/2,1, —3), (1, 5, 0)} (the fifth geometric set-up for wires) and uses 90 degrees on both angles. The set appears when the binary digits of ix from 900, 001 to 1, 200, 000 (the fourth set of 300,000 digits) guide the iterations, starting the procedure at (1/2,1, —3). All patterns shown are scaled so that they occupy the same space. The patterns with radial symmetries have been rotated for aesthetic purposes. The parameters for all the color plates are:

1 5 5 90 90 4

2 2 2 2 5

5 144 72 6 3 72 72 12 4 72 72 12 1 72 72 2

3* 5 5 5 5 5 3 5 5 5

1 120 60 14 3 180 60 1 1 60 60 1 4 120 60 14 3 0 60 14 5 60 60 14 1 180 60 14 4 60 60 14 5 120 60 17

4* 3 2 51.4286 51.4286 12 3 3 102.8571 51.4286 5 5 2 51.4286 51.4286 5

5 2 5 2 5 5 2 5 3

3 102.8571 51.4286 21 2 154.2857 51.4286 6 1 51.4286 51.4286 4 3 51.4286 51.4286 16 2 154.2857 51.4286 5 4 51.4286 51.4286 21 4 154.2857 51.4286 2 4 102.8571 51.4286 16 4 51.4286 51.4286 9

5 5 3 135 45 14 15 2 45 45 10 5 2 90 45 12 5 1 45 45 17 5 1 45 45 6 12 2 45 45 3 12 1 90 45 11 2 3 45 45 2 12 5 135 45 3 15 5 45 45 2 5 3 135 45 9 12 4 90 45 11

Technical Notes

94

6* 2 1 80 40 7 2 4 40 80 3 2 5 80 40 17 4 2 40 40 17 2 4 80 40 24 4 3 80 40 17 2 2 80 40 7 2 4 40 40 10 3 4 80 40 2 7 5 5 4 2

3 36 108 23 1 108 36 1 1 36 36 19 3 36 36 5

8* 2 2 163.6364 32.7273 21 5 5 98.1818 32.7273 19 5 2 65.4545 32.7273 1 4 4 32.7273 32.7273 10 9 5 4 5 2 5 2 5 5 5

3 30 150 8 1 30 120 6 1 30 150 17 4 30 30 22 1 30 150 4 4 30 60 6 1 30 150 8 1 90 30 23 5 90 30 7

10 2 5 27.6923 27.6923 16 5 4 51.4286 25.7143 20 5 4 72 24 14 2 4 22.5 22.5 6 2 4 63.5294 21.1765 8 2 4 20 80 9 5 1 18.9474 18.9474 8 2 4 18 36 13 5 5 17.1429 188.5714 9 5 4 16.3636 49.0909 14

2 4 78.2609 15.6522 15 5 4 15 105 18 11 2 4 55.3846 27.6923 15 2 4 25.7143 25.7143 12 2 1 24 24 13 5 4 67.5 22.5 12 2 4 63.5294 21.1765 21 2 4 20 80 2 2 4 94.7368 18.9474 2 2 4 18 36 5 5 5 17.1429 17.1429 16 5 4 16.3636 81.8182 16 2 4 15.6522 78.2609 1 5 4 15 105 12 12 1 1 90 180 1 13 1 1 0 72 8 1 2 144 72 6 1 2 72 72 6 16 1 36 36 9 14* 1 3 180 60 8 1 1 120 60 7 10 3 120 60 10 16 3 120 60 23 7 1 180 60 5 10 5 60 60 2 7 3 60 60 7 1 5 120 60 13 1 5 180 60 2 15* 1 2 51.4286 51.4286 10 1 5 51.4286 51.4286 23 1 5 102.8571 51.4286 21 1 4 51.4286 51.4286 16 1 3 154.2857 51.4286 10 1 2 102.8571 51.4286 15 1 5 51.4286 51.4286 4

Technical Notes

1 1 1 1 1

3 205.7143 51.4286 14 1 102.8571 51.4286 8 4 205.7143 51.4286 22 4 102.8571 51.4286 3 3 51.4286 51.4286 3

16* 16 4 45 90 7 1 4 45 135 12 10 1 45 0 10 1 1 45 90 6 10 3 45 135 15 7 5 45 225 14 10 4 45 0 7 10 4 45 90 8 7 1 45 225 17 10 1 45 135 23 10 2 45 225 18 10 2 45 0 22 17 1 5 80 40 3 1 1 0 40 5 1 1 120 40 16 1 1 40 120 7 1 5 40 160 8 1 1 40 0 8 1 5 40 40 10 1 1 160 40 15 1 1 40 160 20

1 1 1 1 1 1 1 1

1 25.7143 154.2857 9 1 24 168 12 1 22.5 90 1 1 63.5294 21.1765 4 1 80 20 6 1 75.7895 18.9474 11 1 18 18 6 4 68.5714 17.1429 15

22* 1 1 55.3846 27.6923 9 1 1 25.7143 25.7143 3 1 1 24 0 7 1 1 22.5 157.5 8 1 1 21.1765 21.1765 2 1 1 20 140 1 1 1 0 18.9474 5 1 1 72 18 11 1 1 34.2857 17.1429 15 1 1 49.0909 16.3636 5 1 1 15.6522 109.5652 10 1 1 45 15 15 23 15 1 60 30 1 24 1 1 90 45 8 25 2 4 32.7273 32.7273 1

18 10 5 72 36 11 16 2 144 72 9 16 3 72 36 4 1 5 36 72 6

26 1 1 72 24 5

19 1 1 32.7273 32.7273 2

28 5 3 90 90 9

20 1 1 0 30 4

29 10 1 60 60 17

21* 1 1 83.0769 27.6923 13

30 7 1 120 30 3

27

1 1 656.8421 18.9474 9

95

Index affine, 73, 78, 84, 85, 92 aleph, 20, 21 alternative universes, 21 angel wings, 76 attractor, 2-6, 11, 22, 29, 73, 74, 78, 85, 86

conduction, x, 11 contractile, 73, 78 coordinates, 2, 11, 14, 80, 84 correlation, 14, 79-81 covariance, 79 crossings, 8-11 crystals, ix, 15, 20, 22, 23, 83, 84

Barnsley, Michael, ix, 73, 74, 78, 81 bell, 1, 11-31 bell curve, ix, 9-11, 23, 77 biased coin, 6, 8, 10, 11, 17 binomial, 74, 77 biochemistry, ix, 23, 24, 84 biology, 23 Borges, Jorge Luis, 21, 82 branch, 2, 6, 20, 84

decompositions, ix, 6, 15-19, 21, 23, 25-29, 74, 80, 84 derived measure, 8, 9, 11, 12, 75-77, 79-81 designs, v, 21, 23, 31, 92 deterministic, ix, 2, 5, 8, 10, 11, 21, 73-75 diffuse measure, 77 disorder, 11 dissipation, 11 DNA molecule, ix, 23, 24, 26, 30, 84, 85

Cantor set, 74, 86 Cartesian coordinates, 13 central limit theorem, ix, 11, 77, 78, 86 chance, 1, 6, 20, 22, 23, 31 chaos, 74, 75, 83 chaos game, 1, 2, 5-11, 13-15, 20, 21, 73, 84, 86 chaperonin GroEl, 23, 24, 84 circle, 15, 23, 79, 81 circle map, 81 circular bell, ix, 16-29, 80, 81 cloud, 4 coefficient of variation, 76 complex, 8 complexity, 23, 75, 85

E. coli, 23, 24 eigenvalues, 78, 79 ellipse, 15, 79, 80 elliptical bell, 14, 20, 79-81 encode, 25, 26 ergodic, 74 eternity, v evolutions, ix, 22, 65-69, 73, 92 fair coin, 6, 8, 9, 12, 14-16, 18, 19, 83, 84 filtering, 11, 12, 22 fixed point, 73 foot-and-mouth virus, 23, 24 96

97

Index fractal, 5, 9-11, 73-75, 77-80, 82, 83, 85, 86 fractal dimension, 5, 9-12, 74, 86 function, 4, 8, 11, 29, 73-75, 77-79, 81, 82, 85, 86 Gaussian, ix, 14, 15, 20-23, 74-80, 86 God, v, 31

pollution, 11 probability, 23, 74-76, 78, 85 projection, 7-9, 11, 12, 14, 23, 26, 75, 77, 79, 86 pseudo-random number, 2, 8, 16-19, 21, 25, 80 pseudo-wire, 86, 92, 93 quilts, 70-72

hidden order, 22 histogram, 7-10, 13-15 hydrogen atom, 85 ice crystals, 22, 23, 83 induction, 75 infinity, 74-77, 81 intelligent design, 85 interpolating points, 5, 6, 8-10, 12-14, 16-19, 25-29, 79, 81, 85 irrational numbers, 23, 25, 29 iterated function system, 73 iterations, 2, 15 jigsaw puzzles, 22, 23 joint measure, 11, 12, 14 joint moments, 80 kaleidoscope, ix, 20-22, 86 mappings, ix, 2-5, 11, 13, 15, 20, 23, 25, 29, 73, 77, 78, 83-86, 92 marine microorganisms, 15 mean, 11, 75-77, 79, 81 measure, 6-10, 13, 22, 74-77, 86 moments, 75-77, 80 mountain, 4, 9, 76 multifractal, 6-8, 10, 11, 13, 15, 74, 75, 77-79 multiplicative process, 7, 74 normal, ix

radial, 20, 21, 35-47, 82, 83, 93 random, 25, 26, 74, 77, 92 rosette, ix, 15, 23, 26, 84, 85 rotational, 21, 49-61, 82, 83 Salmonella bacteria, 23, 24 scalings, 76, 81, 84, 92 self-organization, 85 sequential patterns, 16-19, 22, 25-29, 84 shadows, 7-9, 11, 12, 31, 74 Sierpinski triangle, 2-5, 73 sign combinations, 4, 13, 14, 21, 77, 79, 81-83, 92 silent bell, 29, 31 simplicity, 29, 31 singular, 6, 11, 22, 74 space-filling, 12, 14, 25, 81, 82 stable, 2, 6 stars, 81 statistics, 23 symmetry, ix, 8, 9, 11, 15, 20, 21, 25, 26, 76, 83, 86, 93 synchronized angles, 15, 21 texture, 6-8, 11, 12, 15, 74, 77 treasures, ix, 20, 23, 84 tree, 2-6, 20, 76, 81 turbulence, 11, 75, 78 uniform, 6-9, 11, 74, 77, 80 universal, ix, 11, 23

order, 11, 22, 23, 85 variance, 75-77, 79, 81 physics, 23 plane-filling, 11, 77 Plato, 74, 75 polar coordinates, 13

weather images, 11 wire, 3-12, 14, 15, 25, 26, 29, 73, 74, 76, 77, 80, 81, 83-85, 92, 93

I'll i: LSI lift INSIDE THE BELL H I D D E N ORDER IN C H A N C E

Generalized versions of the central limit t h e o r e m that lead t o Gaussian d i s t r i b u t i o n s over one and h i g h e r dimensions, via arbitrary iterations of simple mappings, have recently been discovered by the author of this book and his collaborators. Treasures Inside the Bell:

Hidden

Order in Chance reveals h o w these new constructions result in infinite exotic kaleidoscopic decompositions of t w o - d i m e n s i o n a l circular bells in t e r m s of b e a u t i f u l deterministic patterns possessing arbitrary

n-fold

symmetries.These are patterns that, while reminding us of the infinite structure previously found in the celebrated Mandelbrot set, turn out to contain natural shapes such as snow crystals and biochemical rosettes that include even the DNA structure of life.

World Scientific www.

worldscientific. com

5083 he

ISBN 981-238-140-6 mi 11inn11mi IIminimi

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