The Symmetries of Things

T h e S y m m e t r i e s o f

T h i n g s

J o h n H . C o n w a y

H e i d i B u r g i e !

C h a i m G o o d m a n - S t r a u s s

;

A K P e t e r s . L t d .

W e l l e s l e y . M a s s a c h u s e t t s

w w w . e b o o k 3 0 0 0 . c o m

w w w . e b o o k 3 0 0 0 . c o m

Editorial, Sales, and Customer Service Office

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All rights reserved. No part of the material protected by this copyright notice may be reproduced or ut ilized in any form, electronic or mechani-cal, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner.

Library of Congress Cataloging-in-Publication Data

Conway, John Horton. The symmetries of things / John H. Conway, Heidi Burgie!, Chaim Goodman-

Strauss. p. cm.

Includes bibliographical references and index. ISBN-13: 978-1-56881-220-5 (alk. paper) ISBN-10: 1-56881-220-5 (alk. paper)

1. Symmetry (Mathematics) 2. Geometry. 3. Shapes. I. Burgie!, Heidi, 1968- II. Goodman-Strauss, Chaim, 1967- III. Title.

QAl 74.7.S96C66 2008 516' .l- dc22

Printed in India 12 11 10 09 08

2007046446

10 9 8 7 6 5 4 3 2 1

www.ebook3000.com

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'tJUtJfG 'lp.8.AtJ:) OJ_

w w w . e b o o k 3 0 0 0 . c o m

C o n t e n t s

P r e f a c e

F i g u r e A c k n o w l e d g m e n t s

2

3

S y m m e t r i e s o f F i n i t e O b j e c t s a n d P l a n e R e p e a t i n g P a t t e r n s

S y m m e t r i e s

K a l e i d o s c o p e s

G y r a t i o n s . .

R o s e t t e P a t t e r n s

F r i e z e P a t t e r n s

R e p e a t i n g P a t t e r n s o n t h e P l a n e a n d S p h e r e

W h e r e A r e W e ? . . . . . . . . . . . . . .

P l a n a r P a t t e r n s

M i r r o r L i n e s . .

D e s c r i b i n g K a l e i d o s c o p e s .

G y r a t i o n s . . . . . . . .

M o r e M i r r o r s a n d M i r a c l e s

W a n d e r i n g s a n d W o n d e r - R i n g s

T h e F o u r F u n d a m e n t a l F e a t u r e s !

W h e r e A r e W e ? . . .

T h e M a g i c T h e o r e m

E v e r y t h i n g H a s I t s C o s t !

F i n d i n g t h e S i g n a t u r e o f a P a t t e r n

J u s t 1 7 S y m m e t r y T y p e s . . . . .

H o w t h e S i g n a t u r e D e t e r m i n e s t h e S y m m e t r y T y p e

v i i

x i i i

x v i i

7

7

9

1 0

1 1

1 2

1 3

1 5

1 5

1 7

1 9

2 3

2 6

2 7

2 7

2 9

2 9

3 0

3 3

4 0

w w w . e b o o k 3 0 0 0 . c o m

viii

4

Interlude: About Kaleidoscopes . Where Are We? Exercises ... . .. .

The Spherical Patterns The 14 Varieties of Spherical Pattern T he Existence Problem: Proving t he Proviso . . . . Group Theory and All the Spherical Symmetry Types All t he Spherical Types Where Are We? Examples . ..

5 Frieze Patterns Where Are We? Exercises ...

6 Why the Magic Theorems Work Folding Up Our Surface ... . . Maps on the Sphere: Euler 's T heorem Why char = ch . . . . . . . . . . . T he Magic T heorem for Frieze P atterns T he Magic T heorem for P lane Pat t erns Where Are We? . . .

7 Euler's Map Theorem Proof of Euler's T heorem T he Euler Characteristic of a Surface . . . . . T he Euler Charact eristics of Familiar Surfaces . Where Are We? . . .. . .

8 Classification of Surfaces Caps, Crosscaps, Handles, and Cross-Handles We Don 't Need Cross-Handles . Two crosscaps make one handle That 's All , Folks! Where Are We? Examples

9 Orbi folds

Contents

41 41 41

51 53 57 57 59 59 60

67 69 70

75 75 76 78 79 80 81

83 83 85 87 90

93 93

101 101 102 104 104

109

Contents

II Color Symmetry. Group Theory. and Tilings

10 Presenting Presentations Generators Corresponding to Features The Geometry of the Generators Where Are We? ....... .

11 Twofold Colorations Describing Twofold Symmetries Classifying Twofold Plane Colorings Complete List of Twofold Color Types Duality Groups Where Are We?

12 Threefold Colorings of Plane Patterns A Look at Threefold Colorings . Complete List for Plane Patterns Where Are We? ..... .

13 Other Primefold Colorings Plane Patterns . . . . . . . The Remaining Primefold Types for Plane Patterns The "Gaussian" Cases . . The "Eisensteinian" Cases Spherical Patterns and Frieze Patterns Where Are We? . . . . .

14 Searching for Relations On Left and Right . . . . Justifying the Presentations The Sufficiency of the Relations The General Case Simplifications . Alias and Alibi Where Are We? Exercises . . . Answers to Exercises .

ix

117

125 125 126 132

135 135 136 139 149 150

153 154 155 158

161 161 165 165 166 166 169

171 171 171 173 175 178 179 180 181 182

x Contents

15 Types of Ti lings 185 Heesch Types 185 Isohedral Types 187 Where Are We? 197

16 Abstract Groups 199 Cyclic Groups, Direct Products, and Abelian Groups 199 Split and Non-split Extensions 200 Dihedral, Quaternionic, and QuasiDihedral Groups 200 Extraspecial and Special Groups . 201 Groups of the Simplest Orders . 202 The Group umber Function gnu(n) 206 The gnu-Hunting Conjecture: Hunting moas 209 Appendix: The Number of Groups to Order 2009 209

Il l Repeating Patterns in Other Spaces 215

17 Introducing Hyperbolic Groups 219 No Projection Is Perfect! . 219 Analyzing Hyperbolic Patt erns 224 What Do Negative Characteristics Mean? 226 Types of Coloring, Tiling, and Group Presentations 227 Where Are We? 229

18 More on Hyperbolic Groups 231 Which Signatures Are Really t he Same? . 231 Inequivalence and Equivalence Theorems . 232 Existence and Construction 237 Enumerating Hyperbolic Groups . 239 Thurston's Geometrization Program . 240 Appendix: Proof of the Inequivalence Theorem . 241 Interlude: Two Drums That Sound t he Same 246

19 Archimedean Ti lings 251 The Permutation Symbol . 253 Exist ence 256 Relative versus Absolute 256

C o n t e n t s

E n u m e r a t i n g t h e T e s s e l l a t i o n s

A r c h i m e d e s W a s R i g h t ! . . .

T h e H y p e r b o l i c A r c h i m e d e a n T e s s e l l a t i o n s

E x a m p l e s a n d E x e r c i s e s . . .

2 0 G e n e r a l i z e d S c h l a A i S y m b o l s

F l a g s a n d F l a g s t o n e s . . .

M o r e P r e c i s e D e f i n i t i o n s .

M o r e G e n e r a l D e f i n i t i o n s .

I n t e r l u d e : P o l y g o n s a n d P o l y t o p e s

2 1 N a m i n g A r c h i m e d e a n a n d C a t a l a n P o l y h e d r a a n d T i l i n g s

T r u n c a t i o n a n d " K i s " i n g . . . .

M a r r i a g e a n d C h i l d r e n . . . . .

C o x e t e r ' s S e m i - S n u b O p e r a t i o n .

E u c l i d e a n P l a n e T e s s e l l a t i o n s .

A d d i t i o n a l D a t a . . . . . . . .

A r c h i t e c t o n i c a n d C a t o p t r i c T e s s e l l a t i o n s

2 2 T h e 3 5 " P r i m e " S p a c e G r o u p s

T h e T h r e e L a t t i c e s

D i s p l a y i n g t h e G r o u p s . . . .

T r a n s l a t i o n L a t t i c e s a n d P o i n t G r o u p s

C a t a l o g u e o f P l e n a r y G r o u p s

T h e Q u a r t e r G r o u p s . . . .

C a t a l o g u e o f Q u a r t e r G r o u p s

W h y T h i s L i s t I s C o m p l e t e .

A p p e n d i x : G e n e r a t o r s a n d R e l a t i o n s

2 3 O b j e c t s w i t h P r i m e S y m m e t r y

T h e T h r e e L a t t i c e s

V o r o n o i T i l i n g s o f t h e L a t t i c e s

S a l t , D i a m o n d , a n d B u b b l e s

I n f i n i t e P l a t o n i c P o l y h e d r a .

T h e i r A r c h i m e d e a n R e l a t i v e s

P s e u d o - P l a t o n i c P o l y h e d r a .

T h e T h r e e A t o m i c N e t s a n d T h e i r S e p t a

N a m i n g P o i n t s . . . . . . . . . . . . .

x i

2 5 9

2 6 0

2 6 1

2 6 4

2 6 9

2 6 9

2 7 3

2 7 4

2 7 5

2 8 3

2 8 4

2 8 5

. 2 8 7

. 2 8 8

. 2 8 9

. 2 9 2

3 0 1

3 0 3

3 0 4

3 0 5

3 0 6

3 1 5

3 1 6

3 1 9

3 2 0

3 2 7

3 2 7

. 3 2 9

. 3 3 1

. 3 3 3

. 3 3 6

. 3 4 0

. 3 4 4

. 3 4 5

xii

Polystix . . . . . . ....... . Checkerstix and t he Quarter Groups Hexastix from Checkerstix . . . . . Trist akes, Hexastakes, and Tetrastakes Understanding t he Irish Bubbles . . . The Triamond Net and Hemistix . . . Furt her Remarks about Space Groups

24 Flat Universes Compact Platycosms . Torocosms .... . . The Klein Bottle as a Universe The Ot her Platycosms Infinite Platycosms Where Are We? ...

25 The 184 Composite Space Groups The Alias P roblem .. . Examples and Exercises

26 Higher Still Four-Dimensional Point Groups Regular Polytopes . . . . . . . Four-Dimensional Archimedean Polytopes . Regular Star-Polytopes .... . Groups Generated by Reflections Hemicubes T he Gosset Series . . . . . . . The Symmetries of St ill Higher T hings Where Are We? .... ... . .. .

A Other Notations for the Plane and Spherical Groups

Bibliography

Index

Contents

. 346

. 348

. 349

. 349

. 351

. 351

. 353

355 . 356 . 357 . 358 . 359 . 363 . 365

367 . 368 . 375

383 . 383 . 387 . 389 . 404 . 409 . 409 . 411 . 413 . 414

415

419

423

P r e f a c e

T h i s b o o k h a s b e e n g e r m i n a t i n g f o r a l o n g t i m e . J o h n C o n w a y h a s

a l w a y s b e e n i n t e r e s t e d i n g e o m e t r i c a l g r o u p s , f o r m a n y o f w h i c h h e

d e v i s e d p a r t i c u l a r n o t a t i o n s w h e n h e w a s t e a c h i n g a t C a m b r i d g e

U n i v e r s i t y . H o w e v e r , a f t e r h e m o v e d t o P r i n c e t o n U n i v e r s i t y i n 1 9 8 5

a n d B i l l T h u r s t o n t o l d h i m o f t h e o r b i f o l d i d e a , h e d r o p p e d t h o s e

n o t a t i o n s f o r e v e r a n d d e v i s e d t h e s i g n a t u r e n o t a t i o n u s e d i n t h i s

b o o k . H e t h e n b e c a m e T h u r s t o n ' s m o s t a v i d p r o p h e t , l e c t u r i n g o n

t h e t h e o r y t o s c o r e s o f a u d i e n c e s - r a n g i n g f r o m t h e P r i n c e t o n R u g

S o c i e t y t o t h e I n t e r n a t i o n a l C o n g r e s s o f M a t h e m a t i c i a n s !

O n e o f t h o s e a u d i e n c e s c o n t a i n e d t h e y o u n g g r a d u a t e s t u d e n t

H e i d i B u r g i e l , w h o w a s t a k i n g n o t e s o n t h e t a l k f o r d i s t r i b u t i o n d u r -

i n g t h e c o n f e r e n c e . H e i d i w e n t o n t o c o m p l e t e a g r a d u a t e p r o g r a m i n

c o m b i n a t o r i c s a n d d i s c r e t e g e o m e t r y . Y e a r s l a t e r , w h e n J o h n s p e n t

s o m e t i m e a t N o r t h w e s t e r n U n i v e r s i t y , H e i d i o f f e r e d t o " w r i t e s o m e -

t h i n g e l s e u p " w i t h J o h n , b u t i n t h e e n d t h e y d e c i d e d t o w r i t e t h e

s a m e t h e o r y i n m o r e d e t a i l a s a p r o p o s e d b o o k . T h a t b o o k h a s b e e n

g r o w i n g e v e r s i n c e .

A l l t h e y h a d i n t e n d e d t o w r i t e w a s t h e c o n t e n t o f w h a t i s n o w

P a r t I - a n e l e m e n t a r y i n t r o d u c t i o n t o t h e o r b i f o l d s i g n a t u r e n o t a -

t i o n . B u t t h e n c a m e t h e i d e a o f w r i t i n g a s e c o n d p a r t t h a t w o u l d

e x t e n d t h e s i g n a t u r e t o c o l o r s y m m e t r y . A t t h i s p o i n t i t b e c a m e

c l e a r t h a t C h a i m G o o d m a n - S t r a u s s w o u l d m a k e a n e x c e l l e n t a d d i -

t i o n t o t h e t e a m o f a u t h o r s . C h a i m h a d b e e n p r e a c h i n g t h e g o s p e l

o f t h e o r b i f o l d s i g n a t u r e o n h i s o w n a n d w a s k n o w n f o r h i s g o r g e o u s

i l l u s t r a t i o n s .

M o r e t o p i c s b u r s t i n t o b l o o m a t v a r i o u s s e a s o n s . W h e n C o n w a y ,

D e l g a d o , H u s o n , a n d T h u r s t o n u s e d t h e s i g n a t u r e t o r e - e n u m e r a t e

t h e t h r e e - d i m e n s i o n a l s p a c e g r o u p s , i t s e e m e d a g o o d i d e a t o i n c o r -

p o r a t e t h i s a l s o i n t h e s e c o n d p a r t . T h a t " s e c o n d p a r t " i s n o w P a r t s

I I a n d I I I .

x i i i

xiv Preface

Much of the book was written in hectic three-day sessions on the few occasions when all three of us could get together- this paragraph is being finalized on the way to the Tampa airport, days before the book is sent to press. We usually managed to write several chapters in each session, often including one that only arose just then. For example, at one session, Chaim said "we could perhaps do Heesch types,'' and an hour later Chapter 15 was complete. Just after com-pleting the next section of this introduction (which describes what 's new to this book) , the three of us celebrated at a restaurant , dis-cussed "Archimedean tilings,'' and Chaim and John discovered the "Archifold notation" that characterizes such things as they walked home after the meal. The next day this too was in the book. Of course, it often took Chaim years to catch up with the illustrations.

What's New in This Book? Many of the results and proofs in this book are new, or nearly new, in the sense that their only previous appearances have been in the scholarly papers (often involving one of us) that are cited in the appropriate chapters. These new things are

the orbifold signature,

the statement of our Magic Theorem,

its use to enumerate symmetry types (however , we should point out that a few decades ago, MacBeath introduced his own signature that is in fact equivalent to ours-but more complicated- and used it in the same way) ,

Conway's "zip proof" of the classification of surfaces,

uniform presentations for all the groups ,

their proof,

our analysis and notation for color symmetry,

the p-color types for all primes p ,

the simplified enumeration of Heesch types ,

P r e f a c e

x v

t h e B e s c h e - E i c k - O B r i e n t a b l e o f g r o u p n u m b e r s ,

t h e e x t e n s i o n o f a l l o f t h e a b o v e g e o m e t r i c a l t h e o r y t o h y p e r -

b o l i c g r o u p s ,

a n e w p r o o f o f t h e a b s t r a c t d i s t i n c t n e s s o f i n f i n i t e g r o u p s w i t h

c o m p a c t o r b i f o l d s ,

t h e e x p l a n a t i o n o f i s o s p e c t r a l " d r u m s " v i a h y p e r b o l i c g r o u p s ,

t h e c l a s s i f i c a t i o n o f A r c h i m e d e a n t i l i n g s i n t h e h y p e r b o l i c p l a n e ,

g e n e r a l i z e d S c h l i i f i i s y m b o l s ,

A r c h i t e c t o n i c a n d C a t o p t r i c 3 - t e s s e l l a t i o n s ,

t h e n e w s p a c e g r o u p n o t a t i o n s a n d a p a n o p l y o f o b j e c t s w i t h

p r i m e s p a c e s y m m e t r i e s ,

n a m e s a n d e n u m e r a t i o n o f p l a t y c o s m s ,

a l i s t o f A r c h i m e d e a n 4 - p o l y t o p e s .

E v e n w h e n t h e r e s u l t s a r e o l d , o u r e x p o s i t i o n i s n e w .

W e a r e a l s o p r o u d o f o u r e x p o s i t i o n a n d i l l u s t r a t i o n s . C h a i m

G o o d m a n - S t r a u s s a s s u r e s o u r r e a d e r s t h a t h i s s o f t w a r e a n d i l l u s t r a -

t i o n s a r e a v a i l a b l e f o r s a l e a n d l i c e n s i n g .

W e a r e r e l i e v e d t h a t n o w t h e b o o k i s i n p r i n t , b r i n g i n g t h e o r b -

i f o l d s i g n a t u r e t o t h e w o r l d . T h i s w o u l d n o t h a v e h a p p e n e d w i t h -

o u t h e l p f r o m m a n y p e o p l e i n c l u d i n g R o b e r t S t r a u s s , T r o y G i l b e r t ,

M a r c C u l l e r , T o m M o o r e , C h a r l o t t e H e n d e r s o n , A l i c e a n d K l a u s

P e t e r s , B i l l T h u r s t o n , S i l v i o L e v y , P e t e r D o y l e , N a t a s h a J o n a s k a ,

D a n i e l H u s o n , O l a f D e l g a d o F r i e d r i c h s , D o r i s S c h a t t s c h n e i d e r , M a r -

j o r i e S e n e c h a l , J a v i e r B r a c h o , o u r s t u d e n t s , a n d o u r c o l l e a g u e s ; a n d

t h e p a t i e n c e a n d s y m p a t h y o f o u r p a r t n e r s D i a n a , K e n d a l l , a n d

R a c h e l . W e t h a n k t h e i n s t i t u t i o n s t h a t s u p p o r t e d o u r w o r k , i n -

c l u d i n g P r i n c e t o n U n i v e r s i t y , t h e U n i v e r s i t y o f A r k a n s a s , t h e U n i -

v e r s i d a d N a c i o n a l A u t o m a t a d e M e x i c o , N o r t h w e s t e r n U n i v e r s i t y ,

t h e U n i v e r s i t y o f I l l i n o i s a t C h i c a g o , B r i d g e w a t e r S t a t e C o l l e g e , a n d

t h e N a t i o n a l S c i e n c e F o u n d a t i o n .

F i g u r e A c k n o w l e d g m e n t s

P a g e 7

P a g e 8

P a g e 9

P a g e 1 0

P a g e 1 1

P a g e 1 1

P a g e 1 2

P a g e 1 2

P a g e 6 0

P a g e 6 0

P a g e 6 2

G r y p h o n s f r o m B r o o k s B r o t h e r s s t o r e i n C h i c a g o ,

p h o t o g r a p h b y S u s a n M c B u r n e y .

T h r e e t r a c e r y f r o m N e w Y o r k C i t y , p h o t o g r a p h b y

C h a i m G o o d m a n - S t r a u s s .

D r a w i n g s o f s n a k e s a n d g o t h i c t r a c e r y f r o m 1 , 1 0 0

D e s i g n s a n d M o t i f s f r o m H i s t o r i c a l S o u r c e s b y

J o h n L e i g h t o n ( D o v e r P r e s s , N e w Y o r k , 1 9 9 5 ) .

M i l a n C a t h e d r a l w i n d o w , c o u r t e s y o f V a l e r i a G i b -

e r t o n i a n d G i o v a n n i P e t r i s .

F r i e z e p a t t e r n s i n d o w n t o w n C h i c a g o , p h o t o g r a p h s

b y S u s a n M c B u r n e y : 2 0 8 S L a S a l l e , N e a r M i c h i g a n

A v e o n C r o s s S t , R e d R o o f I n n , R o o k e r y , T r i b u n e

T o w e r .

D a r t a n d e g g f r i e z e , p h o t o g r a p h b y S u s a n M c B u r -

n e y .

P a v e m e n t i n S i e n n a , I t a l y , p h o t o g r a p h b y O t t m a r

L i e b e r t .

S o c c e r b a l l P o v - r a y f i l e b y R e m c o d e K o r t e .

F i r s t f i v e p o l y h e d r a o n t h e p a g e , m o d e l s a n d p h o -

t o g r a p h s b y C h a i m G o o d m a n - S t r a u s s .

L a s t p o l y h e d r o n o n t h e p a g e , m o d u l a r o r i g a m i c o n -

s t r u c t i o n b y J u d y P e n g , p h o t o g r a p h b y C h a i m

G o o d m a n - S t r a u s s .

T e m a r i b a l l s , f i r s t f i v e m o d e l s b y G . T h o m p s o n ,

p h o t o g r a p h s b y C h a i m G o o d m a n - S t r a u s s .

x v i i

xviii Figure Acknowledgments

Pages 64- 65 Sculptures and photographs by Bathsheba Gross-man.

Page 70 Frieze patterns in downtown Chicago, photographs by Susan McBurney: 208 S LaSalle , Near Michigan Ave on Cross St, Red Roof Inn, Rookery, Tribune Tower .

Page 82 Map on a sphere, image and software by Ken Stephenson, combinatorics by Jim Cannon, Bill Floyd, and Walter Parry.

Page 134 M. C. Escher 's Symmetry Work 67 2007 The M. C. Escher Company-Holland. All rights re-served. www.mcescher.com




Page 180 John, Jane, and baby, clip art by anonymous artist from copyright-free CD packaged with Adobe CS 1, arranged by Chaim Goodman-Strauss.

Page 184 Paving stones in Zakopane, Poland, photograph by David Harvey.

Page 224 M. C. Escher 's Circle Limit IV 2007 The M. C. Escher Company-Holland. All rights re-served. www.mcescher.com

Page 318 Birhombohedrille, models and photograph by Chaim Goodman-Strauss.

Page 348 Pencil hexastix, construction by John H. Conway, photograph by Chaim Goodman-Strauss.

Page 366 Dragonfly, clip art by anonymous artist from copyright-free CD packaged with Adobe CS 1, ar-ranged by Chaim Goodman-Strauss.

All other illustrations by Chaim Goodman-Strauss.

suJd)Jed 6u!Jeddd(J due1d pue S)Jdf qQ d)!Ll!:I jO Sd!l)dWWAS

I JJed

I n t r o d u c t i o n t o P a r t I

S y m m e t r i e s a n d s y m m e t r i c p a t t e r n s s u r r o u n d u s t h r o u g h o u t o u r

l i v e s . T h e a i m o f t h e f i r s t p a r t o f t h i s b o o k i s t o d e s c r i b e a n d

e n u m e r a t e a l l t h e s y m m e t r i e s f o u n d i n r e p e a t i n g p a t t e r n s o n s u r -

f a c e s . T o p r o v e t h a t o u r e n u m e r a t i o n i s a c c u r a t e , w e t h e n e x p l a i n

t h e b e a u t i f u l i d e a s f r o m t o p o l o g y a n d a l g e b r a t h a t f o r m t h e b a s i s

f o r o u r c o n c l u s i o n s .

W e s t a r t w i t h a p r o b l e m - e n u m e r a t i n g s y m m e t r i c p a t t e r n s . W e

t h e n i n t r o d u c e t o o l s f o r s o l v i n g t h i s p r o b l e m a n d c o m p l e t e t h e e n u -

m e r a t i o n . B u t t h e n w e a r e p r e s e n t e d w i t h a s e c o n d p r o b l e m -

d e m o n s t r a t i n g t h a t t h e s e t o o l s w o r k t h e w a y w e c l a i m , t h a t t h e r e

i s a s o l i d m a t h e m a t i c a l f o u n d a t i o n b e n e a t h o u r r e s u l t s . A g a i n , w e

s o l v e t h i s p r o b l e m w i t h s o m e t o o l s , t h e n p r e s e n t t h e m a t h e m a t i c s

s u p p o r t i n g t h e u s e o f t h o s e t o o l s . I n t h i s w a y , e a c h c h a p t e r r e d u c e s

t h e p r o b l e m s l e f t b y t h e p r e c e d i n g c h a p t e r t o a n o t h e r p r o b l e m w h o s e

s o l u t i o n i s p o s t p o n e d t o t h e f o l l o w i n g c h a p t e r .

T h i s i s a d e p a r t u r e f r o m t h e t r a d i t i o n a l p r a c t i c e o f b u i l d i n g a t h e -

o r y s t a r t i n g w i t h b a s i c p r i n c i p l e s a n d w o r k i n g t o w a r d t h e u l t i m a t e

g o a l o f p r o v i n g s o m e f i n a l t h e o r e m . W e b e l i e v e t h a t o u r b a c k w a r d

a p p r o a c h w i l l b e s u c c e s s f u l b e c a u s e i t a l l o w s u s t o p r e s e n t o n e c o n -

c e p t a t a t i m e , a t t h e c o s t o f a l w a y s p o s t p o n i n g t h e p r o o f o f j u s t

o n e t h i n g t o t h e n e x t c h a p t e r . W e h o p e a l s o t h a t t h e a r g u m e n t

w i l l b e c l e a r e r w h e n p r e s e n t e d i n a s i n g l e l o g i c a l t h r e a d , o f t h e f o r m

A = B = C = . . . = Z .

T h e f i r s t c h a p t e r i s a g e n t l e i n t r o d u c t i o n t o s y m m e t r y . C h a p -

t e r 2 i n t r o d u c e s t h e f o u r f u n d a m e n t a l f e a t u r e s t h a t w e u s e t o c l a s -

s i f y s y m m e t r y . I n C h a p t e r 3 w e s t a t e o u r M a g i c T h e o r e m a n d a p p l y

i t t o f i n d t h e 1 7 p o s s i b l e t y p e s o f r e p e a t i n g p l a n a r p a t t e r n s , w h i l e

C h a p t e r s 4 a n d 5 p e r f o r m a s i m i l a r s e r v i c e f o r s p h e r i c a l a n d f r i e z e

p a t t e r n s , r e s p e c t i v e l y . T h e M a g i c T h e o r e m i s d e d u c e d i n C h a p t e r 6

3

4 Introduction to Part I

from Euler 's Theorem, which is itself proved in Chapter 7. Finally, Chapter 8 gives our new proof of the classification of surfaces, and Chapter 9 illustrates the orbifolds that underlie our theory.

- 1

S y m m e t r i e s

E v e r y d a y w e a r e s u r r o u n d e d b y s y m m e t r i c o b j e c t s a n d p a t t e r n s .

F r o m f u r n i t u r e t o f l o o r i n g , s y m m e t r y i s t h e r u l e . I n a r t , s y m m e t r y

i s p l e a s i n g t o t h e e y e , a n d t h e i n t r i c a c i e s o f e x t r e m e l y s y m m e t r i c

p a t t e r n s c a n e n t r a n c e a n a u d i e n c e . I n a r c h i t e c t u r e , s y m m e t r i c d e -

s i g n s a r e a t t r a c t i v e f o r y e t a n o t h e r r e a s o n - r e p e t i t i o n o f a d e s i g n

e l e m e n t m e a n s r e - u s e , w h i c h u l t i m a t e l y r e q u i r e s l e s s p l a n n i n g a n d

t e s t i n g . I n m a n u f a c t u r i n g , i t i s s i m p l e r , c h e a p e r a n d m o r e e f f i c i e n t

t o r e p e a t a p a t t e r n a t r e g u l a r i n t e r v a l s . E v e n N a t u r e h a s r e a s o n s t o

u s e s y m m e t r y i n h e r w o r k .

R e c e n t l y , J o h n H . C o n w a y a n d W i l l i a m T h u r s t o n a d a p t e d M u r -

r a y M a c B e a t h ' s m a t h e m a t i c a l l a n g u a g e f o r d i s c u s s i n g s y m m e t r y .

N o w , t h e s y m m e t r i e s o f a p a t t e r n c a n b e d e f i n e d b y a s i n g l e s y m b o l

t h a t w e c a l l i t s s i g n a t u r e : f o r e x a m p l e , 3 * 3 , f o r t h e p a t t e r n o n t h e

l e f t . W i t h s o m e p r a c t i c e , a l m o s t a n y o n e w i t h s o m e k n o w l e d g e o f

h i g h - s c h o o l g e o m e t r y c a n r e a d t h i s s i g n a t u r e a n d i d e n t i f y t h e s y m -

m e t r i e s i t d e s c r i b e s .

K a l e i d o s c o p e s

T h e s i m p l e s t s i g n a t u r e i s j u s t * ( s t a r ) . A * d e n o t e s a m i r r o r o r

k a l e i d o s c o p i c s y m m e t r y , a n d a * a l o n e m e a n s t h a t t h e r e a r e n o o t h e r

s y m m e t r i e s t o t h e f i g u r e . T h e p a i r o f g r y p h o n s ( r i g h t ) h a s a s i n g l e

l i n e o f m i r r o r s y m m e t r y r u n n i n g b e t w e e n t h e m .

( o p p o s i t e p a g e ) T h i s p a t t e r n - w h i c h t o a m a t h e m a t i c i a n e x t e n d s f o r e v e r i n e v e r y d i r e c t i o n ! -

h a s r e A e c t i o n s a n d g y r a t i o n s .

7

E t y m o l o g y

T h e w o r d s y m m e t r y i s a

c o m b i n a t i o n o f t h e w o r d s

s y m ( t o g e t h e r ) a n d m e t r o n

( m e a s u r i n g ) . T h e m e a n i n g

o f b i l a t e r a l i s . l i t e r a l l y . t w o -

s i d e d .

' I t

"Vesica piscis" (fish bladder) is the traditional architec-tural name for patterns of this shape.

WAVYTUMIMUTYVAW

BOECK BOECK

OXIH HIXO OXIH HIXO

Many letters of the Ro-man alphabet have mir-ror symmetry (or approxi-mately sol! Symmetry will vary from typeface to type-face.

8 I. Symmetries

This vesica piscis (left) has signature *2 , pronounced "star two point symmetry" or, more formally, "period two kaleidoscopic sym-metry fixing a point." We use stars for kaleidoscopes to suggest the star formed by the mirrors through a kaleidoscopic point. The pe-riod of a kaleidoscopic point is the number of mirror lines through it. In this case two lines of mirror symmetry- one vertical, the other horizontal- meet at the center of the flower. Finally, the point ( ) indicates that all the symmetries fix a point.

You can probably guess that in a figure with signature *3 , three lines of mirror symmetry meet at its center, and similarly for signa-tures *4 , *5 , *6 , and so on. Mark the mirror lines and find the signatures of the tracery shown above.

For your first quiz. identify the mirror lines and signatures of these lovely cut-paper snowflakes.

G y r a t i o n s

9

G y r a t i o n s

T h i s t r i s k e l i o n ( r i g h t ) a p p e a r s o n t h e c o a t o f a r m s o f t h e I s l e o f

M a n . T h i s f i g u r e l o o k s t h e s a m e i n t h r e e o r i e n t a t i o n s ; t h e r o t a t i o n

t h r o u g h 1 2 0 d e g r e e s i s a c o n g r u e n c e t h a t t a k e s t h e f i g u r e t o i t s e l f . A

t r i s k e l i o n h a s p e r i o d 3 g y r a t i o n a l p o i n t s y m m e t r y a n d s i g n a t u r e 3 .

T h e s n a k e s i n t h e m i d d l e o f t h e a b o v e f i g u r e e n t w i n e w i t h a

p e r i o d 2 g y r a t i o n a l p o i n t s y m m e t r y a n d s o h a v e s i g n a t u r e 2 . T h e

g o t h i c t r a c e r y p a t t e r n s t o t h e l e f t a n d r i g h t h a v e s i g n a t u r e s 4 a n d

6 , r e s p e c t i v e l y .

T h e s e h u b c a p s h a v e g y r a t i o n a l s y m m e t r i e s . w h o s e s i g n a t u r e s y o u m a y i d e n t i f y f o r y o u r s e c -

o n d q u i z .

,~i,, 1'~''

I~

e r . , ,

. . . . ~ . . . ~

, .

: ; ;

~

N s z , z s N

T h r e e r o m a n l e t t e r s h a v e

g y r a t i o n a l s y m m e t r y .

10 1. Symmetries

Rosette Patterns Obviously, we could keep going like this, generating pictures with period 37 kaleidoscopic point symmetry or period 42 gyrational point symmetry. But what else can we do?

For the finite rosette patterns like those on the last two pages, there are no other signatures. In a finite pattern, all symmetries of the pattern must fix (i.e., cannot move) the center of the pattern. Reflections across the center of the rosette and rotations about its center are the only symmetries that do this, so t hey're t he only symmetries such a pattern can have.

By experimenting with different combinations of rotational and reflective symmetries, you can easily convince yourself that the types * , *2 , *3 , *4 , ... , *N and 2 , 3, 4 , 5 , ... , N are the only signatures possible for rosettes, to which we add le = for the case of no symmetry.

Milan Cathedral window. (Courtesy of Valeria Gibertoni and Giovanni Petris.)

F r i e z e P a t t e r n s 1 1


A f t e r i s o l a t e d p i c t u r e s o n a p a g e , t h e e a s i e s t p a t t e r n s t o u n d e r s t a n d

a r e t h o s e m a d e b y r e p e a t i n g p i c t u r e s i n a r o w . W e s e e p a t t e r n s l i k e

t h i s i n f r i e z e s , r i b b o n s , a n i m a l t r a c k s a n d f e n c e s .

- - - - - - - - - - - - -

- - - - - - - - - - . -

. r i i / \ / , , ,

1

. 0 ) ~~

1

"' 3i1). - ~, 0'),::~ . . \ ! J . . . 0

-~ ::,~ ' - , . - ~~-- ~~ ~-~ -~,._, ~~~.:.J..:.::..-

f\V!)~A!~~t~~~l':V(\f.\ "

1

.

F r i e z e p a t t e r n s p h o t o g r a p h e d i n d o w n t o w n C h i c a g o .

T h e d i f f e r e n c e b e t w e e n f r i e z e p a t t e r n s a n d i s o l a t e d f i g u r e s i s t h a t ,

i n a d d i t i o n t o a n y r e f l e c t i v e a n d r o t a t i o n a l s y m m e t r i e s o f t h e f i g u r e s

t h a t m a k e u p t h e p a t t e r n , a f r i e z e p a t t e r n h a s a t r a n s l a t i o n a l s y m -

m e t r y t h a t t a k e s t h e f i g u r e t o a n e i g h b o r i n g f i g u r e . T h e f i r s t h a l f o f

t h e b o o k c o n c e r n s i t s e l f w i t h p a t t e r n s o f t h i s s o r t , c a l l e d r e p e a t i n g

p a t t e r n s .

T h e " d a r t a n d e g g " f r i e z e p a t t e r n i s t r u l y a n c i e n t : l i k e a l l f r i e z e p a t t e r n s w i t h t h i s t y p e o f

s y m m e t r y . i t i s c r e a t e d b y r e f l e c t i n g a m o t i f a c r o s s a l i n e o f k a l e i d o s c o p i c s y m m e t r y . t h e n

r e p e a t i n g t h e p a i r o f i m a g e s F o r w a r d a n d b a c k w a r d a l o n g t h e k a l e i d o s c o p e .

M a k e Y o u r O w n


Y o u c a n e a s i l y g e n e r a t e

f r i e z e p a t t e r n s u s i n g s y m -

m e t r i c l e t t e r s ! H e r e a r e

s o m e e x a m p l e s : c a n y o u

m a k e s o m e o t h e r s ?

p p p p p p p

b b b b b b b b b

p d p d p d p

p p p p p p p p p

b d b d b d b d b

p q p q p q p

p q p q p q p q p

p b p b p b p

b d b d b d b d b

q p q p q p q p q

12 I. Symmetries

Repeating Patterns on the Plane and Sphere Frieze patterns have "forward and back" translational symmetry. Plane patterns add translational symmetry in another direction. These patterns can extend to cover an ent ire page, or beyond. We see t hem every day on t he floors and walls around us.

In order to study the symmetries of common objects like hair-brushes and furniture, we will also need to learn about the symme-tries of patterns on spheres . Basketballs have two planes of reflective symmetry, as do tennis balls. But t hese balls also have a 2-fold rota-t ional symmetry. A cube has nine planes of mirror symmetry, while some soccer balls have fifteen! In order to classify such patterns we will study repeating patterns on spheres.

/

W h e r e A r e W e ? 1 3

W h e r e A r e W e ?

A t t h e b e g i n n i n g o f t h i s c h a p t e r w e f o u n d a l l t h e p o s s i b l e t y p e s o f

s y m m e t r y f o r r o s e t t e s - n a m e l y = l , * = * h , 2 , * 2 , 3 , * 3 ,

4 , * 4 , . . . . W e ' v e a l s o i n t r o d u c e d t h r e e c a t e g o r i e s o f r e p e a t i n g

p a t t e r n - r e p e a t i n g p a t t e r n s i n t h e E u c l i d e a n p l a n e , f r i e z e p a t t e r n s ,

a n d p a t t e r n s o n t h e s p h e r e . T h e f o c u s o f t h i s b o o k i s t o c l a s s i f y

t h e d i f f e r e n t t y p e s o f s y m m e t r y t h a t o b j e c t s i n t h e s e c a t e g o r i e s c a n

h a v e . W e ' v e t o l d y o u r o u g h l y w h a t i t m e a n s t o s a y t h a t t w o t h i n g s

h a v e t h e s a m e t y p e o f s y m m e t r y , b u t w e ' l l h a v e t o p o s t p o n e a p r e c i s e

d e f i n i t i o n o f o u r p r o b l e m u n t i l w e ' v e n e a r l y s o l v e d i t .

I n f a c t , o u r b o o k w i l l h a v e a b o u t a s m a n y p o s t p o n e m e n t s a s

c h a p t e r s ! F o r e x a m p l e , i n t h e n e x t c h a p t e r w e ' l l i n t r o d u c e f o u r f e a -

t u r e s t h a t i n f a c t d e t e r m i n e t h e n o t i o n o f s y m m e t r y t y p e , b u t w i l l

p o s t p o n e t h e p r o o f t h a t t h e y d o s o . T h e s e f e a t u r e s d e t e r m i n e t h e

s i g n a t u r e s t h a t w e u s e i n C h a p t e r s 2 - 5 a n d 1 7 t o l i s t a l l p o s s i b l e

t y p e s f o r e a c h o f o u r t h r e e c a t e g o r i e s . T o d o s o , w e e m p l o y a " M a g i c

T h e o r e m " w h o s e p r o o f i s p o s t p o n e d t o C h a p t e r 6 . I n t h a t c h a p t e r

w e a l s o s e e t h a t t h e s i g n a t u r e r e a l l y d e s c r i b e s a t o p o l o g i c a l s u r f a c e

c a l l e d a n o r b i f o l d t h a t e n c a p s u l a t e s a l l t h e s y m m e t r i e s o f a p a t t e r n .

T h e M a g i c T h e o r e m i s t h e n r e v e a l e d t o d e s c r i b e a s i m p l e i n v a r i a n t ,

t h e E u l e r c h a r a c t e r i s t i c , o f t h i s o r b i f o l d ; a d e t a i l e d i n v e s t i g a t i o n o f

t h e E u l e r c h a r a c t e r i s t i c i s i n t u r n p o s t p o n e d t o C h a p t e r 7 . A n o r b -

i f o l d i s a s p e c i a l k i n d o f s u r f a c e , a n d o u r l a s t p o s t p o n e m e n t i s t h e

f a c t t h a t E u l e r ' s c h a r a c t e r i s t i c r e a l l y d o e s c h a r a c t e r i z e t h e d i f f e r e n t

p o s s i b l e t o p o l o g i c a l t y p e s o f s u r f a c e . O u r n e w " z i p p r o o f " o f t h i s

w r a p s u p t h e p r o o f o f a l l o u r r e s u l t s , a n d c l o s e s t h e f i r s t p a r t o f o u r

b o o k .

- 2 -

P l a n a r P a t t e r n s

I n t h i s b o o k w e h e l p y o u u n d e r s t a n d t h e s y m m e t r i e s o f t h i n g s . I n

t h i s c h a p t e r w e l o o k a t s o m e r e p e a t i n g p a t t e r n s a n d i n t r o d u c e y o u

t o t h e w a y w e t h i n k a b o u t t h e m . W e d e s c r i b e t h e f o u r f u n d a m e n t a l

f e a t u r e s o f a r e p e a t i n g p a t t e r n i n t h e p l a n e ( o r o n a n y s u r f a c e ! ) a n d

i n t r o d u c e t h e s i g n a t u r e w e u s e t o r e c o r d t h e s e f e a t u r e s o f t h e p a t t e r n .

M i r r o r L i n e s

T h e f l o r a l p a t t e r n t o t h e l e f t h a s m a n y s y m m e t r i e s . F o r e x a m p l e ,

t h e p a t t e r n i s l e f t - r i g h t s y m m e t r i c : i t h a s t h e v e r t i c a l m i r r o r l i n e

s h o w n o n t h e l e f t b e l o w .

T h e f i g u r e i n t h e m i d d l e s h o w s a n o t h e r m i r r o r l i n e , w h i c h i s a

d i f f e r e n t k i n d b e c a u s e , u n l i k e t h e f i r s t o n e , i t r u n s b e t w e e n , r a t h e r

t h a n a l o n g , t h e p e t a l s . D r a w i n g a l l t h e m i r r o r l i n e s w e c a n , w e g e t

t h e f i g u r e o n t h e r i g h t , w h i c h i s a t f i r s t s i g h t r a t h e r c o n f u s i n g .

F o r t u n a t e l y , t h e s m a l l p a r t w e ' v e h i g h l i g h t e d i n t h e m a r g i n c o n -

t a i n s e n o u g h i n f o r m a t i o n t o r e c o n s t r u c t t h e w h o l e p a t t e r n . T h i s i s

b e c a u s e i f w e s u r r o u n d t h i s s m a l l t r i a n g l e b y m i r r o r s , a s i n a k a l e i -

d o s c o p e , t h e r e f l e c t i o n s o f t h e o r i g i n a l t r i a n g l e w i l l f i l l i n t h e n e i g h -

b o r i n g t r i a n g u l a r r e g i o n s . T h e r e f l e c t i o n s o f t h e s e r e f l e c t i o n s w i l l f i l l

15

16 2. Planar Patterns

in the neighbors of these neighbors, and so on, until the entire pat-tern is restored. With three small pieces of mirror (available at most hardware stores) and a little dexterity, you can try this yourself!

The patterns of Figures 2.1 and 2.2 are less ornate. The new patterns are somewhat simpler but have all the symmetries of the original; for our purposes all three patterns are identical.

Figure 2.1. A simpler pattern.

Figure 2.2. Another simple pattern.

Repeating patterns like the ones studied in this book are made up of many symmetric copies of a motif. What we are studying here are the symmetries relating each motif to each other motif in the pattern.

D e s c r i b i n g K a l e i d o s c o p e s

1 7

D e s c r i b i n g K a l e i d o s c o p e s

P a t t e r n s w h o s e s y m m e t r i e s a r e d e f i n e d b y r e f l e c t i o n s a r e c a l l e d k a l e i -

d o s c o p i c b e c a u s e o f t h e i r s i m i l a r i t y t o t h e p a t t e r n s s e e n i n k a l e i d o -

s c o p e s . T h e y a r e c l a s s i f i e d b y t h e w a y t h e i r l i n e s o f m i r r o r s y m m e t r y

i n t e r s e c t . S o , f o r i n s t a n c e , i n F i g u r e 2 . 3 t h e r e a r e t h r e e p a r t i c u l a r l y

i n t e r e s t i n g k i n d s o f p o i n t , o n e w h e r e s i x m i r r o r s m e e t , o n e w h e r e

t h r e e m i r r o r s m e e t , a n d o n e w h e r e t w o m i r r o r s m e e t . W e c a l l t h e s e

6 - f o l d , 3 - f o l d , a n d 2 - f o l d k a l e i d o s c o p i c p o i n t s , r e s p e c t i v e l y , b e c a u s e

t h e l o c a l s y m m e t r i e s ( r i g h t ) a r e * 6 , * 3 , a n d * 2 . T h e w h o l e p a t -

t e r n h a s k a l e i d o s c o p i c s y m m e t r y o f s i g n a t u r e * 6 3 2 , w h e r e t h e r e i s

n o f i n a l p o i n t ( ) b e c a u s e t h e s y m m e t r i e s d o n ' t a l l f i x a p o i n t .

F i g u r e 2 . 3 . A k a l e i d o s c o p e o f t y p e * 6 3 2 .

T h e n u m b e r s d e f i n i n g t h e t y p e ( o r s i g n a t u r e ) o f a k a l e i d o s c o p e

c a n b e c y c l i c a l l y p e r m u t e d , s o t h a t * 6 3 2 , * 3 2 6 , a n d * 2 6 3 m e a n t h e

s a m e , o r a l s o r e v e r s e d , e q u a t i n g t h e s e w i t h * 2 3 6 , * 3 6 2 , a n d * 6 2 3 .

..l>~'l

.. +t-2


Figure 2.4. Decorated square tiling.

Patterns with a squarish sort of symmmetry, such as in Figure 2.4 are more common. The symmetry of this pattern is kaleidoscopic with signature *442. There are two 4's in the symbol because there are two different kinds of 4-fold kaleidoscopic points. The 2 in the symbol refers to the 2-fold kaleidoscopic point.

The fact that there can be several different kinds of kaleido-scopic points of the same order forces us to make it clear what same kind means for such points. We say, more generally, that any two features of a pattern are of the same kind only if they are re-lated by a symmetry of the whole pattern. The points shown in the top two marginal figures are both 4-fold kaleidoscopic points but are obviously different . We will say that two points P and Q are the same if P can be moved to Q without changing the pattern's appearance in any way. (This "move" could include a reflection.)

G y r a t i o n s 1 9

G y r a t i o n s

T h e t y p e o f t h e k a l e i d o s c o p e i n F i g u r e 2 . 5 i s o n l y * 3 r a t h e r t h a n

* 3 3 3 , b e c a u s e a l l t h e k a l e i d o s c o p i c p o i n t s i n t h a t f i g u r e a r e o f t h e

s a m e k i n d . H o w e v e r , t h e s y m m e t r i e s o f t h i s p a t t e r n a r e n o t p u r e l y

k a l e i d o s c o p i c . T h e r e i s a n e w f e a t u r e - a 3 - f o l d r o t a t i o n a l s y m m e t r y

s h o w n a t r i g h t b e l o w .

L e t ' s l o o k a t t h i s m o r e c l o s e l y . T h e p a t t e r n w o u l d b e u n d i s t u r b e d

i f t h e w h o l e p l a n e w e r e t o b e r o t a t e d t h r o u g h 1 2 0 d e g r e e s a r o u n d

t h e p o i n t m a r k e d 3 i n t h e m i d d l e o f t h e f i g u r e . T h e s a m e i s a l s o

t r u e o f t h e p o i n t 3 i n t h e t o p f i g u r e , b u t w e ' v e a l r e a d y a c c o u n t e d f o r

t h i s b y c a l l i n g i t a 3 - f o l d k a l e i d o s c o p i c p o i n t - t h i s r o t a t i o n i s " d o n e

b y m i r r o r s . " S i n c e t h e p a t t e r n h a s o n e k i n d o f 3 - f o l d g y r a t i o n p o i n t

a n d a k a l e i d o s c o p e w i t h o n e k i n d o f 3 - f o l d k a l e i d o s c o p i c p o i n t , i t s

s i g n a t u r e i s 3 * 3 .

F i g u r e 2 . 5 . A p a t t e r n w i t h s i g n a t u r e 3 * 3 .

- 3 . . 3


Figure 2.6. Pattern with signature 2* 22.

The pattern in Figure 2.6 has two kinds of 2-fold kaleidoscopic points and one kind of 2-fold gyration point. The signature of this pattern is 2*22.

The * designating the presence of a kaleidoscope separates the digit representing the gyration point from those describing the kalei-doscopic points, which are read around the kaleidoscope.

G y r a t i o n s

2 1

O n c e y o u a r e f a m i l i a r w i t h t h i s n o t a t i o n , y o u c a n t e l l i m m e d i a t e l y

t h a t t h e s y m b o l 4 * 2 d e s c r i b e s a p a t t e r n w i t h o n e k i n d o f 4 - f o l d

g y r a t i o n p o i n t a n d o n e k i n d o f 2 - f o l d k a l e i d o s c o p i c p o i n t . F i g u r e 2 . 7

a n d t h e m a r g i n a l f i g u r e s s h o w a n e x a m p l e o f s u c h a p a t t e r n .

F i g u r e 2 . 7 . P a t t e r n w i t h s i g n a t u r e 4 * 2 .

+. . . - 2


In Figure 2.8 we see a pattern that has only gyration points and no kaleidoscopes. Since there are three kinds of 3-fold gyration point, t he symmetry is of type 333.

Figure 2.8. Pattern w ith signature 333.


2 3


S o f a r w e h a v e d i s c u s s e d t w o f e a t u r e s o f p a t t e r n s i n t h e p l a n e : k a l e i -

d o s c o p e s a n d g y r a t i o n p o i n t s . I t i s n a t u r a l t o a s k i n w h a t w a y s t h e s e

c a n o c c u r i n p l a n a r p a t t e r n s . F o r i n s t a n c e , c a n a p a t t e r n h a v e m o r e

t h a n o n e k a l e i d o s c o p e ?

F i g u r e 2 . 9 . M o r e t h a n o n e k i n d o f m i r r o r s i g n a t u r e * *

A l l t h e k a l e i d o s c o p e s t h a t w e ' v e s e e n s o f a r h a v e b e e n d e f i n e d b y

p o l y g o n s e n c l o s i n g p a r t o f o u r p a t t e r n , b u t t h a t ' s n o t t h e o n l y t y p e

t h e r e i s . A s i n g l e m i r r o r l i n e t h a t h a s n o o t h e r m i r r o r l i n e s c r o s s i n g

i t i s a k a l e i d o s c o p e w i t h s i g n a t u r e * F i g u r e 2 . 9 s h o w s a p a t t e r n w i t h

t w o o f t h i s k i n d o f k a l e i d o s c o p e i n i t , a n d i t s s i g n a t u r e i s * * ( Y o u

s h o u l d c h e c k t h a t t h e s e t w o m i r r o r l i n e s r e a l l y a r e d i f f e r e n t ! )

W e ' r e a l s o s e e i n g s o m e t h i n g e l s e f o r t h e f i r s t t i m e h e r e . T h e

s m a l l e s t s u b r e g i o n m a r k e d o f f b y m i r r o r l i n e s i n F i g u r e 2 . 9 i s i n f i n i t e !

T h e r e a r e s e v e r a l n e w f e a t u r e s t o b e f o u n d i n p a t t e r n s l i k e t h i s , w h i c h

w i l l b e p r e s e n t e d i n t h i s s e c t i o n a n d t h e n e x t .

~-M-


Figure 2.10 . A pattern with a mirror and a miracle: signature *X .

At first , Figure 2.10 looks very much like Figure 2.9. None of its mirror lines intersect, and the smallest subregion bounded by mirror lines is again infinite. But in t his figure there is only one kind of mirror line!

And, t here's a miracle here! There is a path from a left-handed spiral to a right-handed spiral t hat does not go t hrough a mirror line. We will record t he presence of such a path by a red cross ( x ) in t he signature. We call t his a "mirrorless crossing,'' or , for short, a miracle, and indicate it in figures by a red dotted line and cross.

Figure 2.10 has both mirrors and miracles, but only one kind of each , so its signature is * x .


2 5

W e c a n h a v e t w o m i r a c l e s , j u s t a s w e c a n h a v e t w o d i f f e r e n t k i n d s

o f m i r r o r . T h i s h a p p e n s i n F i g u r e 2 . 1 1 , w h i c h h a s s i g n a t u r e x x .

( T h e r e a r e m o r e t h a n t w o p a t h s f r o m l e f t - h a n d e d t o r i g h t - h a n d e d

s p i r a l s , b u t a l l o f t h e m c a n b e m a d e u p o f c o m b i n a t i o n s o f i d e n t i c a l

c o p i e s o f t h e o n e s w e ' v e m a r k e d i n t h e m a r g i n . )

F i g u r e 2 . 1 1 . M o r e t h a n o n e k i n d o f m i r a c l e : s i g n a t u r e x x .

(

X ' ) (

0


Wanderings and Wonder-Rings Just as a miracle is a repetit ion-wit h-reflection of a fundamental region t hat 's not "explained by" mirrors, it 's possible to have a fun-damental region repeated without reflection in a way t hat's not ex-plained by gyrat ions, mirrors, or miracles. In fact, such repetit ions always come in pairs. We call such a pair of paths a "wonderful wan-dering" and denote it by a blue "wonder-ring,'' o . As in t he figure in t he margin, we draw such a pair of paths wit h blue dotted lines and wit h a blue ring nearby. The signature for F igure 2.12 is just o .

Figure 2.12. A wonderful wonder-ring: signature o.


2 7


I t i s a r e m a r k a b l e f a c t t h a t w o n d e r s , g y r a t i o n s , k a l e i d o s c o p e s , a n d

m i r r o r s s u f f i c e t o d e s c r i b e a l l t h e s y m m e t r i e s o f a n y p a t t e r n w h a t -

s o e v e r , a s w e s h a l l s h o w i n C h a p t e r 3 . W e t h e r e f o r e c a l l t h e m t h e

f o u r f u n d a m e n t a l f e a t u r e s . Y o u g e t t h e s i g n a t u r e o f a p a t t e r n j u s t

b y w r i t i n g d o w n w h i c h e v e r o f t h e s e f e a t u r e s i t h a s . U p t o t h i s p o i n t ,

w e ' v e u s e d b l u e f o r w o n d e r s a n d g y r a t i o n s , s i n c e t h e s e p r e s e r v e t h e

t r u e o r i e n t a t i o n o f a f u n d a m e n t a l r e g i o n , a n d r e d f o r k a l e i d o s c o p e s

a n d m i r a c l e s , s i n c e t h e s e r e f l e c t . H o w e v e r , y o u c a n w r i t e t h e s e i n

b l a c k i n k i f y o u a l w a y s w r i t e t h e m i n t h e s a m e o r d e r , s i n c e t h e n

y o u ' l l b e a b l e t o w o r k o u t w h i c h c o l o r s t h e y s h o u l d b e .

T a b l e 2 . 1 l i s t s t h e f o u r f u n d a m e n t a l f e a t u r e s i n t h e a p p r o p r i a t e

o r d e r a n d t h e c o d e s w e u s e t o r e p r e s e n t t h e m i n t h e s i g n a t u r e .

w o n d e r s

o . . . o

g y r a t i o n s

A B . . . C

k a l e i d o s c o p e s

* a b . . . c * d e . . . f . . .

T a b l e 2 . 1 . F e a t u r e s o f a p a t t e r n .


m i r a c l e s

x . . . x

I n t h i s c h a p t e r , w e h a v e d e s c r i b e d t h e f o u r f e a t u r e s o f r e p e a t i n g p l a n e

p a t t e r n s a n d i n t r o d u c e d t h e s i g n a t u r e t h a t d e s c r i b e s w h i c h o f t h e m

a p p e a r i n a g i v e n p a t t e r n . I n t h e n e x t c h a p t e r , w e l e a r n h o w t h e s e

s i g n a t u r e s c a n b e u s e d t o d e t e r m i n e w h a t c o m b i n a t i o n s o f f e a t u r e s

a r e p o s s i b l e f o r p l a n e p a t t e r n s .

G y r a t i o n s : W h a t ' s i n a

N a m e ?

W e c h o o s e t h e t e r m g y -

r a t i o n t o s u g g e s t m o t i o n

a b o u t a p o i n t . T h e r o t a -

t i o n a l l y s y m m e t r i c p a t t e r n s

c r e a t e d b y c r o s s e d m i r r o r

l i n e s a r e t h e s a m e i n t h e

c l o c k w i s e d i r e c t i o n a s t h e y

a r e i n t h e c o u n t e r c l o c k -

w i s e d i r e c t i o n . I n a p a t -

t e r n w i t h g y r a t i o n a l s y m -

m e t r y . t h e r e i s a c l e a r d i s -

t i n c t i o n b e t w e e n t h e c l o c k -

w i s e a n d c o u n t e r c l o c k w i s e

d i r e c t i o n s a t t h e g y r a t i o n

p o i n t .

- 3 -

T h e M a g i c T h e o r e m

I n t h e l a s t c h a p t e r w e i n t r o d u c e d t h e f o u r f u n d a m e n t a l f e a t u r e s t h a t

c o m p l e t e l y d e s c r i b e t h e t y p e s o f s y m m e t r y f o r r e p e a t i n g p a t t e r n s .

F r o m n o w o n w e s h a l l o f t e n s p e c i f y t h e s y m m e t r i e s o f a p a t t e r n j u s t

b y g i v i n g i t s s i g n a t u r e ( w h i c h l i s t s i t s f e a t u r e s ) . W e h a v e n ' t y e t s a i d

w h y j u s t t h e s e p a r t i c u l a r f e a t u r e s a r e s o f u n d a m e n t a l - a n d w e w o n ' t ,

u n t i l C h a p t e r 8 - n o r h a v e w e f o u n d j u s t w h i c h s i g n a t u r e s a r i s e .

I n t h i s c h a p t e r w e ' l l i n t r o d u c e y o u t o t h e " M a g i c T h e o r e m " [ 4 ] ,

u s e i t t o s h o w t h a t j u s t 1 7 s i g n a t u r e s a r e p o s s i b l e f o r p l a n e r e p e a t i n g

p a t t e r n s , a n d t h e n d e d u c e t h a t s u c h p a t t e r n s c o m e i n j u s t 1 7 t y p e s .

T h e p r o o f o f t h e M a g i c T h e o r e m i t s e l f i s s o m e t h i n g e l s e y o u ' l l h a v e

t o w a i t f o r !

E v e r y t h i n g H a s I t s C o s t !

I t t u r n s o u t t o b e a g o o d i d e a t o a s s o c i a t e a c o s t t o e v e r y s y m b o l i n

t h e s i g n a t u r e , a s s h o w n i n T a b l e 3 . 1 .

S y m b o l

C o s t ( $ )

S y m b o l

C o s t ( $ )

0

2

* o r x

1

2

1

2

1

2 4

3

2

3

1

3

3

4

3

4

3

4

8

5

4

5

2

5

5

6

5

6

5

6 1 2

N

N - 1

N

N - 1

; ; ; -

2 N

0 0

1

0 0

1

?

T a b l e 3 . 1 . C o s t s o f s y m b o l s i n s i g n a t u r e s .

o p p o s i t e p a g e ) T h e m a g i c t h e o r e m n o t o n l y c l a s s i f i e s s i g n a t u r e s . b u t h e l p s u s c a l c u l a t e t h e

s i g n a t u r e o f a p a t t e r n . T h e s i g n a t u r e 2 2 x o f t h i s p a t t e r n . l i k e t h a t o f a l l p l a n a r p a t t e r n s .

c o s t s e x a c t l y s 2 .

2 9

*632 costs $2

~ :, -~~ . T . .

* x costs $2

30 3. The Magic Theorem

Why is t his? Because, as we shall see in the next few chapters, t here are Magic Theorems that describe the possible signatures in terms of their costs. Here is the one we'll use in t his chapter:

Theorem 3.1 (The Magic Theorem for plane repeating patterns) The signatures of plane repeating patterns are precisely those with total cost$ 2.

For example, t he first pat tern we analyzed (F igure 2.3) had sig-nature *632, which has cost

$1 + ~ + ~ + ~ = $2. 12 3 4

(Normally, we only put the dollar sign on t he first of several terms to be summed.) Figure 2.5 has signature 3*3, which costs

$ ~ + 1 + ~ = $2. 3 3

The patt ern in Figure 2.6 has a kaleidoscope wit h two different 2-fold kaleidoscopic points and a 2-fold gyration point. Its signature is 2*22, wit h cost

$ ~ + 1 + ~ + ~ = $2. 2 4 4

Finally, the signature of Figure 2.9 is **, wit h cost

1 + 1 = $2.

This is the same as t he cost for t he pattern of type * x in Figure 2.10.

The proof of t he Magic Theorem is quite easy, but we'll postpone it unt il later in our book. In this chapter we just use the theorem to help find the possible signatures for repeating patterns.

Finding the Signature of a Pattern We can now exactly identify t he signature of any repeating pattern on t he plane by the following steps. As we proceed , we write down t he symbols in t he signature, start ing from t he middle and working outward. If we list larger numbers before smaller ones (using *632

F i n d i n g t h e S i g n a t u r e o f a P a t t e r n

3 1

r a t h e r t h a n * 2 3 6 ) , w e c a n t e l l a t a g l a n c e w h i c h p a t t e r n s h a v e t h e

s a m e t y p e .

1 . M a r k a n y k a l e i d o s c o p e s i n r e d . I f t h e r e a r e m i r r o r l i n e s , r e s t r i c t

a t t e n t i o n t o o n e o f t h e r e g i o n s i n t o w h i c h t h e y c u t t h e p l a n e .

P u t a r e d * n e a r a n y o n e k a l e i d o s c o p e ; t h e n , f i n d j u s t o n e

c o r n e r o f e a c h t y p e ( a s i n C h a p t e r 2 ) , a n d w r i t e t h e n u m b e r s

o f m i r r o r s t h r o u g h e a c h o f t h e s e c o r n e r s , a l s o i n r e d .

2 . L o o k f o r g y r a t i o n p o i n t s . I n b l u e , m a r k j u s t o n e g y r a t i o n p o i n t

o f e a c h t y p e w i t h a s p o t a n d i t s o r d e r .

3 . A r e t h e r e m i r a c l e s ? C a n y o u w a l k f r o m s o m e p o i n t t o a c o p y

o f i t s e l f w i t h o u t e v e r t o u c h i n g a m i r r o r l i n e ? I f s o , a m i r a c l e

h a s o c c u r r e d . M a r k j u s t o n e s u c h p a t h w i t h a b r o k e n r e d l i n e

a n d a r e d c r o s s n e a r b y .

4 . I s t h e r e a w o n d e r ? I f y o u ' v e f o u n d n o n e o f t h e a b o v e , t h e n

t h e r e i s : m a r k i t w i t h a b l u e w o n d e r - r i n g .

I f y o u e n c o u n t e r a t r i c k y p a t t e r n , t h e r e a r e s o m e t h i n g s y o u

h o u l d d o t o m a k e y o u r w o r k e a s i e r . I f t w o f e a t u r e s a r e t h e s a m e y o u

m u s t o n l y m a r k o n e o f t h e m ; s o m e t i m e s i t h e l p s t o l a b e l g y r a t i o n

p o i n t s b e f o r e l a b e l i n g k a l e i d o s c o p e s . B e s u r e t h e r e a r e n ' t a n y m i r r o r

l i n e s i n s i d e t h e r e g i o n b o u n d e d b y a k a l e i d o s c o p e , a n d d o n ' t f o r g e t

t h a t g y r a t i o n p o i n t s n e v e r l i e o n m i r r o r l i n e s !

T h e r u l e s a b o v e w o r k f o r a n y r e p e a t i n g p a t t e r n ; h e r e a r e s o m e

m o r e h i n t s t h a t w o r k j u s t f o r p a t t e r n s i n t h e p l a n e . T h e r e i s o n e t y p e

o f p l a n e p a t t e r n w i t h t w o k a l e i d o s c o p e s a n d o n e w i t h t w o m i r a c l e s ; i f

y o u ' r e w o r k i n g w i t h o n e o f t h e s e , y o u s h o u l d b e a b l e t o s e e d i f f e r e n c e s

b e t w e e n t h e s e f e a t u r e s b y l o o k i n g c a r e f u l l y a t y o u r p a t t e r n . Y o u

k n o w t h a t t h e t o t a l c o s t i s $ 2 ; y o u c a n u s e t h i s i n s e v e r a l w a y s . Y o u

c a n s t o p w h e n i t r e a c h e s $ 2 ( f o r i n s t a n c e , i f y o u f i n d a w o n d e r ) , o r i f

_ : . - o u h a w n o t y e t r e a c h e d

8

2 . y o u w i l l k n o w t h a t t h e r e m u s t b e m o r e

: e a u r e s t o f i n d .

11 I

I ' I

1 1 J

11


The fact that the signature of a plane pattern always costs $2 can help us check that the signature we have found for a pattern is correct: it can also help to complete it! For example, all we can see at first is that there are two kinds of 2-fold gyration points in Figure 3.1. But , 22 would only cost$~+~= $1 , so there should be an extra dollar's worth to be discovered. Indeed there is! Figure 3.1 is the same as its mirror image although it has no mirror line, so there must be a miracle instead! We look at this more closely in the figure in the margin: there's a symmetry that takes a leaf to a backwards copy of itself, and the path joining these is the required mirrorless crossing, giving us the signature 22 x .

Figure 3.1. What type is this?

J u s t 1 7 S y m m e t r y T y p e s 3 3

W h a t i s t h e s i g n a t u r e o f t h e p a t t e r n i n F i g u r e 3 . 2 ? H e r e t h e r e a r e

a l s o t w o k i n d s o f 2 - f o l d g y r a t i o n p o i n t s , w h i c h d o n o t b y t h e m s e l v e s

c o s t $ 2 . T h e p a t t e r n i s a g a i n t h e s a m e a s i t s m i r r o r i m a g e , b u t a

m i r r o r , n o t a m i r a c l e , e x p l a i n s t h i s , a n d t h e t y p e i s 2 2 * . ( S e e t h e

m a r g i n a l f i g u r e . )

F i g u r e 3 . 2 . W h a t s i g n a t u r e d o e s t h i s h a v e ?

J u s t 1 7 S y m m e t r y T y p e s

W h y a r e t h e r e j u s t 1 7 t y p e s o f s y m m e t r y f o r p l a n e p a t t e r n s ? W e ' l l

d e d u c e t h i s u s i n g o n l y t h e M a g i c T h e o r e m a n d s o m e s i m p l e a r i t h -

m e t i c . T h e c a l c u l a t i o n s i n t h e n e x t f e w s e c t i o n s a r e v e r y s i m i l a r

t o t h o s e t h a t a n s w e r t h e q u e s t i o n , " H o w m a n y d i f f e r e n t w a y s c a n I

m a k e c h a n g e f o r a d o l l a r i f I u s e o n l y q u a r t e r s a n d d i m e s ? " I f t h e

r e s u l t s a t f i r s t s e e m m y s t i c a l , t r y w o r k i n g t h r o u g h a f e w e x a m p l e s

f o r y o u r s e l f .

I'

I;

Exercise

Check the types on these two pages.


The Five 'True Blue" Types If all symmetries of a pattern are obtainable by t rue motions, as in t he patterns on these two pages , t he signature will be ent irely blue. If a blue string of digits AB ... C is to cost $2, t here must be more t han two of t hem, since each costs less than $1. If t here are exactly t hree, t he values in Table 3.1 show t hat t he signature can only be one of 632 , 442 , or 333 . If t here are more, it can only be 2222 , since each digit costs at least $ ~. Finally if there's a wonder-ring, the signature must be o , since t he ring already costs us $2.

T he following figures illustrate t he five t rue blue types: 632 , 442 , 333 , 2222 , and o .

We get type 333 if all characters have the mean cost of $~ . Otherwise. one character must be 2.

J u s t 17 S y m m e t r y T y p e s

I f t h e r e m a i n i n g t w o c h a r a c t e r s h a v e t h e i r m e a n c o s t of$~

w e g e t 4 4 2 .

T h e o n l y E u c l i d e a n t y p e w i t h f o u r k i n d s o f g y r a t i o n p o i n t s i s

2 2 2 2 . s i n c e $ ! + ! + ! + ! i s a l r e a d y $ 2 .

3 5

I f n o t . a s e c o n d c h a r a c t e r m u s t b e 3 . a n d 6 3 2 i s f o r c e d . s i n c e

$~+~+!=$2

I f t h e r e ' s a w o n d e r r i n g 0 ( c o s t i n g $ 2 ) . t h e r e c a n ' t b e a n y -

t h i n g e l s e .

Exercise

Check the t two pages. ypes on these

36 3. The Magic Th eorem

The Five "R eAecting Red" T' Now consid ypes Th er the si ey correspond gnatures t hat are e . and only if if AB t oNthe previous cas~tibrely red and have no c

... does: ecause *AB rosses. 11 A-1 .. . C costs'2if

+ 211 + .. . +N -l 2N = $2 {::::::::> $A - 1 while t here can onl --:;\ + ... + N - 1 ~ ' star. T his y ld y be one such . N 2, ie s t he five signature ( **) . reflecting red t with more th ypes. an one

J u s t 17 S y m m e t r y T y p e s 3 7

T h e a l l - r e d s i g n a t u r e s , * 3 3 3 , * 4 4 2 , * 6 3 2 , * 2 2 2 2 , a n d * * , c o r r e -

s p o n d e x a c t l y t o t h e a l l - b l u e s i g n a t u r e s 3 3 3 , 4 4 2 , 6 3 2 , 2 2 2 2 , a n d

o , s i n c e e a c h r e d d i g i t c o s t s h a l f a s m u c h a s t h e c o r r e s p o n d i n g b l u e

d i g i t a n d a k a l e i d o s c o p e ( * ) c o s t s h a l f o f $ 2 .

I

1,

I 11

I

I I

I I ,

Exercise

Check the types on these pages.


The Seven "Hybrid" Types The remaining signatures eit her mix blue and red or involve x sym-bols. To help us enumerate t hese "hybrid" types, we note t hat t he "demotions"

replace Il* by *nn replace x by *

don't change t he cost and must event ually lead to one of t he five previous cases. So, we can recover all these mixed signatures by making t he inverse "promot ions"

in all possible ways:

*632

replace *nn by n * replace a final * by x

*442 *333 *2222 l l l

4*2 3*3 2*22 l

22* l

22 x

**

l *X l

xx

The following seven figures represent t he mixed types 3*3, 4*2, 2*22 , 22 x , 22*, xx , and *X .

2J rr/6 rr/2


We conclude t hat there are just 17 possibilities for the signature, and so just 17 symmetry types for repeating patterns on the plane. (See Table 3.2.)

*632 *442 *333 *2222 ** 2*22

*X 4*2 3*3 22* xx

22 x 632 442 333 2222 0

Table 3.2. The 17 symmetry types of plane patterns

So indeed the Magic Theorem does imply t hat there are at most 17 symmetry types for a plane repeating pattern. These are tradition-ally called the 17 plane crystallographic groups.1

How the Signature Determines the Symmetry Type We have ignored some details. To what extent can we recover the symmetry of a pattern from its signature? This is a real problem, as we shall see in the spherical case, but the answers in t he plane case are easy. In the end, they depend only on t he existence of rectangles and triangles wit h given angles, provided t hat t hose angles have the correct sum of 7r .

For instance, a pattern with signature *632 must be generated by reflections in t he sides of a triangle with angles i, 1, and ~ . All triangles that satisfy t his condition will be the same up to size, so up to similarity there's just one possibility for the symmetries of a pattern with signature *632.

For 4*2, four copies of a fundamental region combine to form a square. Then reflections in t he sides of that square generate the rest of t he pattern, as shown to the left , so there's really only one set of symmetries corresponding to 4*2 as well. Case-by-case arguments like these work for all 17 types; you can confirm for yourself that t he argument given for 4*2 is easily adapted to the types 3*3 and 2*22.

1The nonreflecting elements of any of these groups form its rotation subgroup, at the bottom of t he column.

I n t e r l u d e : A b o u t K a l e i d o s c o p e s

4 1

I n t h e s a m e v e i n , t h e s y m m e t r i e s o f a p a t t e r n w i t h s i g n a t u r e

* 2 2 2 2 a r e g e n e r a t e d b y t h e r e f l e c t i o n s i n t h e s i d e s o f a q u a d r i l a t e r a l

w h o s e f o u r a n g l e s a r e ~-that i s t o s a y , a r e c t a n g l e . H e r e t h e s e t

o f s y m m e t r i e s i s n o l o n g e r u n i q u e u p t o s c a l e ; a n y o n e v e r s i o n c a n

b e c o n t i n u o u s l y r e s h a p e d i n t o a n y o t h e r b y g r a d u a l l y v a r y i n g t h i s

r e c t a n g l e .

T h e r e s u l t i s t h a t o n e s e t o f s y m m e t r i e s c a n b e c o n t i n u o u s l y

t r a n s f o r m e d i n t o t h e o t h e r w h i l e c o n s i s t e n t l y m a i n t a i n i n g i t s t y p e .

I n t e c h n i c a l l a n g u a g e t h i s k i n d o f d e f o r m a t i o n i s c a l l e d a n i s o t o p y .

S o , w e ' l l s a y t h a t t h e s y m m e t r i e s o f a n y o n e p a t t e r n w i t h a g i v e n

s i g n a t u r e c a n b e i s o t o p i c a l l y r e s h a p e d t o b e c o m e t h o s e o f a n y o t h e r

p a t t e r n w i t h t h e s a m e s i g n a t u r e .

I n t e r l u d e : A b o u t K a l e i d o s c o p e s

K a l e i d o s c o p e s - t h e p h y s i c a l k i n d f o u n d i n t o y s t o r e s - w e r e i n v e n t e d

b y S i r D a v i d B r e w s t e r i n 1 8 1 6 . I n a r e a l k a l e i d o s c o p e , w i t h a p r o p e r l y

r e p e a t i n g , p l a n a r p a t t e r n s e e n a t t h e e n d , t h e m i r r o r s c a n o n l y b e

a r r a n g e d a s s h o w n o n t h e r i g h t .

T h a t i s , t h e s y m m e t r y s i g n a t u r e i s j u s t t h a t o f o n e o f t h e r e f l e c t -

i n g r e d t y p e s * 3 3 3 , * 4 4 2 , * 6 3 2 , o r * 2 2 2 2 . O b t a i n s o m e m i r r o r s a n d

m a k e a k a l e i d o s c o p e y o u r s e l f !


W h a t w e ' v e s h o w n ( u s i n g t h e M a g i c T h e o r e m , o f c o u r s e ! ) i s t h a t u p

t o i s o t o p i c r e s h a p i n g t h e r e a r e j u s t 1 7 p l a n e c r y s t a l l o g r a p h i c g r o u p s .

A s w e s a i d , y o u ' l l h a v e t o w a i t t o s e e w h y t h e M a g i c T h e o r e m i s

t r u e .

T h e n e x t t w o c h a p t e r s w i l l d i s c u s s t h e v e r s i o n s o f i t t h a t a p p l y

t o p a t t e r n s o n t h e s p h e r e a n d t o p l a n a r f r i e z e p a t t e r n s .

E x e r c i s e s

W e ' v e t o l d y o u h o w t o f i n d t h e s i g n a t u r e o f a p a t t e r n , b u t m o s t

p e o p l e n e e d s o m e p r a c t i c e t o g e t i t r i g h t . F o l l o w t h e s t e p s o n p a g e 3 1

t o i d e n t i f y t h e t y p e s o f t h e p a t t e r n s o n p a g e s 4 2 - 4 9 .

zj~

\ > O

I I

I !

42

l. Repeating patterns on brick walls.

(a) Running bond

(cl Flemish bond

(el Spiral bond

(g) An unusual bond. seen on the o ld sec-t ion of Princeton's Fr ist Student Center

3. The Magic Theorem

(bl English bond

(d) Dutch bond

(fl Zigzag running bond

-

-(h) Another unusual bond. seen on the new section o f the Frist Student Center

E x e r c i s e s

4 3

C h e c k y o u r a n s w e r s .

( a ) R u n n i n g b o n d h a s t y p e 2 * 2 2

( b ) E n g l i s h b o n d h a s t y p e * 2 2 2 2

~mr

( c ) F l e m i s h b o n d h a s t y p e 2 * 2 2

( d ) D u t c h b o n d a l s o h a s t y p e 2 * 2 2

'

"-''"~''-' " ' ~,.,, . . . . . . , , . _ ; > . ' < I - ( . - , , . ; " " - ' ' ' ' ' - 1 " - w - -

~-'rO ~ o~ ~;..

t i .

( e l S p i r a l b o n d h a s t y p e 2 2 2 2

( f ) Z i g z a g r u n n i n g b o n d h a s t y p e 2 2 *

( g ) O l d F r i s t b o n d h a s t y p e 2 2 *

( h ) N e w F r i s t b o n d a l s o h a s t y p e 2 2 *


2. The placement of the dots changes the symmetry types of these patterns. Identify them.

(el (f)

E x e r c i s e s

4 5


I

~~

2 * 2 2

( c l

* X

( d )

0

( e l

2 2 2 2 ( f )

2 2 *


3. Find the signatures of these patterns.

Cal (b)

Ccl

Ce) (fl

E x e r c i s e s 4 7


~ ~ L \ ~' ' - .~---.,

~1 ~~,~,~

,f.J~ , , . . . . ~

_1 ,~ ~~ . . . . \ Li\~,

,,,-~~v ~ ,'&,~

~ ~ \ [ \ ~ ~~ . / 1 1 ; 1

, . . . . . , ~~~ ~~,~,.!~~

'tf.J~;~~~

~ -~~::)'....,, (\'~''

( a )

6 3 2

* X

( e l

( f )


4. Even more!

.-.-- ~~ di~

(el (f)

E x e r c i s e s 4 9


3 * 3 C b l 2 2 x

6 3 2

( e l 2 2 x ( f )

2 2 x

- 4 -

T h e S p h e r i c a l P a t t e r n s

S o f a r , w e h a v e d i s c u s s e d o n l y s y m m e t r i c p a t t e r n s o n p l a n a r s u r f a c e s .

H o w e v e r , m o s t o f t h e s y m m e t r i c t h i n g s w e e n c o u n t e r i n o u r e v e r y d a y

l i v e s a r e n ' t p l a n a r s u r f a c e s . C h a i r s , d e s k s , b o x e s , a n d e v e n p e o p l e

( r o u g h l y ) a r e s y m m e t r i c , b u t n o n - p l a n a r .

T o f i n d t h e f e a t u r e s d e s c r i b i n g t h e s y m m e t r i e s o f a n o b j e c t l i k e

a c h a i r o r t a b l e w e i m a g i n e i t a s r e s t i n g i n s i d e t h e " c e l e s t i a l s p h e r e " .

~

( o p p o s i t e p a g e ) T h r e e s p h e r i c a l p a t t e r n s . w i t h s i g n a t u r e s * 5 3 2 . * 2 2 1 1 . a n d * 4 3 2 .

5 1

52 4. The Spherical Pat terns

For t he chair there is a single plane of reflection that intersects t he sphere in a single mirror line-in other words, it has bilateral symm etry. The signature for the bilateral type of symmetry is *, because we see one mirror line on the surface of the sphere and it meet s no ot her mirror lines.

We see from Table 3.1 t hat t his only costs $1, so it is cheaper than the plane crystallographic groups, which all cost $2.

More complicated objects can have kaleidoscopic points, gyration points and miracles. For t he rectangular table above, the mirror lines are two great circles t hat meet at right angles. On the sphere they have two intersection points, both of angle ~, so the symmetries of this table has type *22. They cost

1 1 3 $1 + - + - = $_ 4 4 2 '

again less t han $2. It turns out that an important quant ity is t he change we get from

$2, for which we will use t he abbreviation ch . T hus,

In particular,

3 1 ch( *22) = $2 - cost( *22) = $2 - - = $_ .

2 2

T h e 1 4 V a r i e t i e s o f S p h e r i c a l P a t t e r n 5 3

T h e s i g n a t u r e s o f t h e E u c l i d e a n p l a n e p a t t e r n s a l l c o s t e x a c t l y

$ 2 , s o i f y o u p u r c h a s e d a n y o n e o f t h e m w i t h a $ 2 b i l l , y o u w o u l d

g e t n o c h a n g e a t a l l . B u t f o r s p h e r i c a l p a t t e r n s , w h i c h h a v e o n l y

f i n i t e l y m a n y s y m m e t r i e s , t h e r u l e i s d i f f e r e n t : t h e c h a n g e y o u g e t i s

p r e c i s e l y $~vided b y t h e n u m b e r o f s y m m e t r i e s .

T h e o r e m 4 . 1 ( T h e M a g i c T h e o r e m f o r s p h e r i c a l p a t t e r n s ) T h e s i g n a -

t u r e o f a s p h e r i c a l p a t t e r n c o s t s e x a c t l y $ 2 - ~, w h e r e g i s t h e t o -

t a l n u m b e r o f s y m m e t r i e s .

I n p a r t i c u l a r , t h e c h a n g e i s a l w a y s p o s i t i v e , s o t h e c o s t i s a l w a y s

l e s s t h a n $ 2 . W e ' l l p r o v e t h i s i n C h a p t e r 6 . I n t h i s c h a p t e r , w e ' l l u s e

i t t o d e r i v e t h e l i s t o f p o s s i b l e t y p e s o f s p h e r i c a l p a t t e r n .

O u r E u c l i d e a n M a g i c T h e o r e m i s r e a l l y j u s t a p a r t i c u l a r c a s e o f

t h i s , b e c a u s e t h e r e g = o o a n d s o t h e c h a n g e i s c h = $ c ! , o r 0 . T h u s ,

w e d o n ' t r e a l l y h a v e t w o m a g i c t h e o r e m s b u t o n l y o n e .

T h e 1 4 V a r i e t i e s o f S p h e r i c a l P a t t e r n

H e r e t h e c o n c l u s i o n f r o m t h e M a g i c T h e o r e m i s o n l y t h a t t h e s p h e r -

i c a l t y p e s a r e a m o n g

* 5 3 2

* 4 3 2

5 3 2 4 3 2

* 3 3 2

3 * 2

3 3 2

* 2 2 N

2 * N

2 2 N

* M N

N *

N x

M N

b u t i t t u r n s o u t t h a t t h e r e i s a p r o v i s o : t h e t y p e s * M N a n d M N o n l y

h a p p e n w h e n M = N . H e r e M a n d N r e p r e s e n t a r b i t r a r y p o s i t i v e

i n t e g e r s . W e a l l o w t h e s e i n t e g e r s t o b e 1 , w i t h t h e c o n v e n t i o n t h a t

d i g i t s 1 c a n b e o m i t t e d . T h i s m a k e s s e n s e - a g y r a t i o n p o i n t o f o r d e r

1 o r a k a l e i d o s c o p i c p o i n t w i t h e x a c t l y 1 m i r r o r p a s s i n g t h r o u g h i t

i s u n i n t e r e s t i n g t o u s , s o w e l e t 1 * = * 1 1 = * .

A s i n C h a p t e r 2 , w e p r o c e e d b y f i r s t c o u n t i n g t h e a l l - b l u e s p h e r i -

c a l s i g n a t u r e s , t h e n t h e r e d o n e s , a n d f i n a l l y t h o s e t h a t i n v o l v e b o t h

c o l o r s .

II .

l t

Signature 332

54 4. The Spherical Patterns

The Five 'True Blue" Types

Since the total cost of the signature must be less than $2, we cannot afford a wonder ring ( o ) or to have more than three digits (distinct from 1). The most general signature with fewer than three digits may be written MN by inserting l 's if necessary. Every such sig-nature does cost less than $2, but according to the proviso it only corresponds to a symmetry type if M = N .

Finally, if there are exactly three digits , then one must be a 2, because $ ~ + ~ + ~ = $2. Then the symbol is 22N if there are two or more 2's, and just 332, 432 or 532 if there is only one.

First note that if the signature contains two 2's, it must be 22N for some N .

-N -N

Signature NN Signature 22N

If there is just one 2, then some other digit must be 3 since $ ! + i + i = $2; then, the remaining digit must be 3, 4, or 5 since $! + ~ + ~ = $2.

Signature 432 Signature 532

T h e 1 4 V a r i e t i e s o f S p h e r i c a l Pa t t e r n

5 5

T h e F i v e " R e f l e c t i n g R e d " T y p e s

T h e a l l - r e d s i g n a t u r e s f o r s p h e r e p a t t e r n s m u s t h a v e t h e f o r m * A B . . . N

s i n c e w e c a n n o l o n g e r a f f o r d t w o * ' s . T h e o n e s f o r w h i c h c h i s p o s -

i t i v e a r e i n p e r f e c t c o r r e s p o n d e n c e w i t h t h e t r u e b l u e t y p e s , s i n c e

c h ( * A B . . . N) i s e x a c t l y h a l f o f c h ( A B . . . N ) , a s w e s e e f r o m t h e f o l -

l o w i n g :

- - - - - - -

$ ( A- 1 N - 1 )

c h ( * A B . . . N) = 2 - 1 - 2 } 1 + + - - - y : ; - ,

$ ( A- 1 N- 1 )

c h ( A B . . . N ) = 2 - -y + +~ .

B u t r e m e m b e r t h e p r o v i s o : * M N e x i s t s o n l y i f M = N .

/

S i g n a t u r e * N N

S i g n a t u r e * 2 2 N

S i g n a t u r e * 4 3 2

S i g n a t u r e * 5 3 2

II

II

11

11

S i g n a t u r e * 3 3 2

56 4. The Spherical Pat terns

The Four Hybrid Types As in t he plane case, t hese must all be obtainable by promot ion from the red reflective cases. Here are all t he possibilities:

*532 *432 *332 *22N *NN 1 1 1

3*2 2*N N * 1

N x

- N

T h e E x i s t e n c e P r o b l e m : P r o v i n g t h e P r o v i s o 5 7

T h e E x i s t e n c e P r o b l e m : P r o v i n g t h e P r o v i s o

A l l 1 7 p o s s i b i l i t i e s t h a t w e e n u m e r a t e d f o r p l a n e p a t t e r n s a c t u a l l y

a r o s e . I n t h e s p h e r i c a l c a s e , t h e c o r r e s p o n d i n g s t a t e m e n t i s n o t q u i t e

t r u e ; t h e t y p e s M N a n d * M N o n l y e x i s t i f M = N . T h e o t h e r c a s e s

c a u s e n o p r o b l e m .

F o r e x a m p l e , * 4 4 2 w a s g e n e r a t e d b y r e f l e c t i o n s i n a t r i a n g l e o f

a n g l e s ~, ~, ~, a n d a p l a n e p a t t e r n w i t h t h i s t y p e o f s y m m e t r y e x i s t s

b e c a u s e s u c h a t r i a n g l e e x i s t s i n t h e E u c l i d e a n p l a n e . S i m i l a r l y ,

* 5 3 2 i s g e n e r a t e d b y r e f l e c t i o n s i n a t r i a n g l e o f a n g l e s ~, i , ~, a n d

a s p h e r i c a l p a t t e r n w i t h t h i s s y m m e t r y e x i s t s b e c a u s e t h e r e i s a

s p h e r i c a l t r i a n g l e w i t h t h e s e a n g l e s .

1

1 N o w f o r t h e p r o v i s o ! T h e t y p e * M N , w h e n i t e x i s t s , i s g e n e r a t e d

b y t h e r e f l e c t i o n s i n t h e s i d e s o f a t w o - s i d e d p o l y g o n w i t h a n g l e s I a

a n d N T h i s d o e s e x i s t w h e n M = N ; i t ' s t h e l u n e b o u n d e d b y t w o

g r e a t s e m i c i r c l e s a t a n g l e N ( a t r i g h t ) , b u t d o e s n o t w h e n M i - N .

( F o r t h e s a m e r e a s o n * M, w h i c h e q u a l s * Ml , f a i l s t o e x i s t f o r M > 1 . )

A h y p o t h e t i c a l p a t t e r n o f t y p e M N w i t h M i - N w o u l d c o n t a i n

j u s t t w o t y p e s o f g y r a t i o n p o i n t . B u t t h e n , b y s u p e r p o s i n g i t w i t h

i t s i m a g e u n d e r a r e f l e c t i o n f i x i n g a g y r a t i o n p o i n t o f e a c h t y p e , w e

s h o u l d o b t a i n o n e o f t y p e * M N , w h i c h i s i m p o s s i b l e . T h e r e f o r e , M N

a l s o f a i l s t o e x i s t i f M i - N , a n d M f a i l s t o e x i s t i f M i - 1 .

G r o u p T h e o r y a n d A l l t h e S p h e r i c a l S y m m e t r y T y p e s

G r o u p t h e o r y i s n o t d i s c u s s e d i n d e t a i l h e r e , a t l e a s t n o t u n t i l m u c h

l a t e r i n t h i s b o o k ; t h e r e a r e m a n y t e x t s a v a i l a b l e t h a t t e a c h g r o u p

t h e o r y b e t t e r t h a n w e a r e a b l e t o i n t h e s p a c e a v a i l a b l e h e r e . I n

b r i e f , t h e s y m m e t r i e s o f _ a p a t t e r n f o r m a g r o u p ; s u p p o s e A a n d

B a r e s y m m e t r i e s o f a p a t t e r n , d e s c r i b e d b y s o m e m o t i o n o f t h e

p a t t e r n t h a t t a k e s a f u n d a m e n t a l r e g i o n t o a c o p y o f i t s e l f . T h e n , t h e

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