Other Title s i n Thi s Serie s

5 Davi d W . Farmer , Group s and symmetry : A guide to discovering mathematics, 199 6 4 V . V . Prasolov , Intuitiv e topology, 199 5 3 L . E . Sadovski Y and A . L . SadovskiY , Mathematic s an d sports , 199 3 2 Yu . A . Shashkin , Fixe d points , 1991 1 V . M . Tikhomirov , Storie s about maxim a and minima , 199 0

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Mathematical World • Volum e 5

Groups and Symmetr y A Guid e t o Discoverin g

Mathematics

David W. Farmer

American Mathematical Society

http://dx.doi.org/10.1090/mawrld/005

1991 Mathematics Subject Classification. Primar y 20-01, 51-01 .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Farmer, Davi d W. , 1963 -Groups an d symmetr y : a guid e t o discoverin g mathematic s / Davi d W . Farmer .

p. cm . — (Mathematica l world , ISS N 1055-9426 ; v . 5 ) Includes bibliographica l reference s (p . - ) . ISBN O-8218-045O- 2 (acid-free ) 1. Grou p theory . 2 . Patter n perception . I . Title . II . Series .

QA174.2.F37 199 5 511.3'3—<lc20 95-2197 6

CIP

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10 9 8 7 6 5 4 0 2 0 1 0 0 9 9

Table o f Content s

Chapter 1 . Squares , Hexagons , an d Triangle s 1 1.1 Th e squar e gri d 1.2 Th e hexago n gri d 1.3 Th e triangl e gri d 1.4 Puttin g i t al l togethe r 1.5 Note s

Chapter 2 . Th e Rigi d Motion s o f the Plan e 1 5 2.1 Translatio n an d rotatio n 2.2 Combinin g transla t ions an d rotation s 2.3 Mirro r reflectio n 2.4 Glid e reflectio n 2.5 Combinin g rigi d motion s 2.6 Note s

Chapter 3 . Finit e Figure s 2 7 3.1 Symmetr y 3.2 Combinin g symmetrie s 3.3 Multiplicatio n table s 3.4 Inverse s 3.5 Th e finite symmetr y type s 3.6 Note s

Chapter 4 . Stri p Pattern s 3 9 4.1 Symmetrie s o f strip s 4.2 Classifyin g stri p pattern s 4.3 Note s

Chapter 5 . Wallpattern s 4 3 5.1 Rotatio n symmetr y 5.2 Mirror s an d glide s 5.3 Classifyin g wallpattern s 5.4 Basi c unit s 5.5 Group s 5.6 Note s 5.7 Sampl e pattern s 5.8 Wallpatter n flowchart

VI TABLE O F CONTENT S

Chapter 6 . Finit e Group s 5 9 6.1 Finit e figure s 6.2 CN an d D/v , agai n 6.3 Additio n 6.4 Multiplicatio n 6.5 Rearrangement s 6.6 Permutation s 6.7 Note s

Chapter 7 . Cayle y Diagram s 7 3 7.1 Generator s 7.2 Rearrangin g basi c unit s 7.3 Stri p pattern s 7.4 Wallpattern s

Chapter 8 . Symmetr y i n th e Rea l World 8 7 8.1 Analyzin g pattern s 8.2 Pattern s i n ar t an d architectur e

Bricks Decorative floors an d wall s The ar t o f M.C. Ecshe r The ar t o f William Morri s Islamic ar t African weaving s Indian potter y Rugs an d carpet s Amish quilt s

8.3 Mathematica l project s The 15-puzzl e More arithmeti c mo d N Generators, relations , an d Cayle y diagram s 3-dimensional symmetr y Magie squar e wallpattern s

8.4 Rando m project s Kinship structure s Chemistry Tiling a wal l Make your ow n pattern s

Bibliography 9 9

Index 10 1

Preface

This boo k i s a guide t o discoverin g mathematics .

Every mathematic s textboo k i s fille d wit h result s an d technique s whic h once were unknown. Th e result s were discovered b y mathematicians wh o exper -iment ed, conjectured , discusse d thei r wor k wit h others , an d the n experimente d some more. Man y promisin g idea s turned ou t t o b e dead-ends, an d lot s of har d work resulted i n littl e output . Ofte n th e first progres s was the understandin g o f some special cases. Continue d work led to greater understanding , an d sometime s a complex picture began to be seen as simple and familiar . B y the time the work reaches a textbook , i t bear s n o resemblance t o it s early form, an d th e detail s of its birth an d adolescenc e have been lost . Th e precis e and methodica l expositio n of a typica l textboo k ofte n lead s peopl e t o mistakenl y thin k tha t mathematic s is a dry , rigid , an d unchangin g subject .

The mos t excitin g par t o f mathematics i s the proces s o f invention an d dis -covery. Th e ai m o f this boo k i s to introducé tha t proces s to you. B y means o f a wide variety o f tasks, this book will lead you to discover some real mathematics . There are no formulas t o memorize. Ther e are no procedures to follow. B y look-ing a t examples , searchin g fo r pattern s i n thos e examples , an d the n searchin g for th e reason s behin d thos e patterns , yo u wil l develo p your ow n mathematica l ideas. Th e book i s only a guide; its job i s to star t yo u in the righ t direction , an d to brin g yo u bac k i f you stra y to o far . Th e discover y i s left t o you .

This book i s suitable fo r a one semester cours e at th e beginnin g undergrad -uate level . Ther e ar e n o prerequisites . An y colleg e student intereste d i n discov -ering th e beaut y o f mathematics ca n enjo y a course taugh t fro m thi s book . A n interested hig h schoo l studen t wil l find thi s boo k t o b e a pleasan t introductio n to som e moder n area s o f mathematics .

I than k Dav e Baye r fo r showin g m e hi s metho d o f drawin g th e Cayle y diagrams o f wallpatter n groups . Whil e preparin g thi s boo k I wa s fortunat e t o have acces s t o excellen t note s taken b y Hui-Chu n Le e and b y Eli e Levine . I t i s a pleasur e t o than k Benj i Fisher , Klau s Peters , Sand y Rhoades , Te d Stanford , John Sullivan , an d Gretche n Wrigh t fo r helpfu l comment s o n earlie r version s of this book .

David W. Farmer September, 199 5

Vil

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Bibliography

[AC] Africa Counts, b y Claudi a Zaslavsky , Lawrenc e Hil l Books , 1990 . Interest ing accoun t o f th e mathematic s o f th e nativ e Africa n peoples . Section 5 , "Patter n an d Shape, " i s relevant t o thi s book .

[AWM] The Art of William Morris, b y Ayme r Vallance , Dover , 1988 . Description o f hi s lif e an d work . 4 0 color plate s o f hi s wallpatterns .

[C] Connections, b y Ja y Kappraff , McGra w Hill , 1990 . An interestin g boo k whic h touche s o n man y o f th e topic s discusse d i n the presen t book .

[E] Ethnomathematics: A Multicultural View of Mathematical Ideas, b y Mar -cia Ascher , Brooks/Cole , 1991 .

Presents a n interestin g accoun t o f th e mathematica l sophisticatio n o f 'primitive' people . Th e chapter , "Th e Logi c of Kin Relations " i s fasci -nating. "Symmetri e Stri p Decorations " i s a nic e introduction .

[EAS] M.C. Escher: Art and Science, H.S.M . Coxeter , ed. , Nor t h-Holland, 1986 . A collectio n o f 3 5 paper s o n mathematica l symmetry . Mos t relat e t o Escher's work , an d mos t hav e nic e pictures . A goo d plac e t o se e ho w others hav e analyze d real-worl d symmetry . Som e o f th e paper s ar e very mathematical .

[FS] Fivefold Symmetry, Istva n Hargittai , ed. , Worl d Scientific , 1992 . Various paper s o n pentagona l symmetry . Th e paper , "800-Year-Ol d Pentagonal Tiling... " suggest s tha t Penros e tiling s wer e invente d i n 12^-century Iran .

[FSE] Fantas y and Symmetry: The Periodic Drawings of M.C. Escher, b y Carolin e MacGillavery, Harr y N . Abrams , 1976 .

[GCIA] Geometrie Concepts in Islamic Art, b y I . El-Said an d A . Parman, Worl d of Islam Festival , 1976 .

Mathematics an d Islami c Art .

[HRP] Handbook ofRegular Patterns, b y Pete r Stevens , MI T Press , 1981. A comprehensiv e boo k o n regula r patterns . Million s o f examples , an d some reasonabl e guide s o n ho w t o construc t interestin g patterns .

99

100 BIBLIOGRAPHY

[KS] Knots and Surfaces: a guide to discovering mathematics, b y Davi d Farme r and Theodor e B . Stanford , America n Mathematica l Society , 1996 .

A boo k i n th e sam e styl e a s Groups and Symmetry.

[MS] Mathematical Snapshots, b y Hug o Steinhaus , variou s publishers . Nice chapter s o n a variet y o f mathematics , writte n fo r a genera l au -dience. Al l o f i t i s interesting , an d tw o o r thre e o f th e chapter s ar e relevant t o thi s book .

[S] Symmetry, b y Herman n Weyl , Princeto n Universit y Press , 1989 . Interesting essay s o n real-worl d symmetry .

[SC] Symmetries of Culture, b y Doroth y Washbur n an d Donal d Crowe , Univer -sity o f Washington Press , 1988 .

An excellen t book . Designe d fo r anthropologist s wh o wan t t o analyz e patterns. Lot s o f details . Man y pictures . Usefu l flow-charts t o classif y patterns. Plent y o f references .

[SSS] Shapes, Space, and Symmetry, b y Ala n Holden , Dover , 1991 . Pictures o f hundred s o f 3-dimensiona l symmetri e shapes . Lot s o f reg -ular an d semi-regula r solids . Discussio n o f thei r symmetries .

[SS2] Symmetry an d Symmetry 2, Istva n Hargittai , ed. , Pergamo n Press , 1986 , 1989.

Two hug e book s o f papers o n al l aspect s o f symmetry .

[TP] Tilings and Patterns, b y Brank o Grünbau m an d G.C . Shephard , Freeman , 1987.

Very mathematica l an d no t tha t eas y t o jus t pie k u p an d read . I t is comprehensive . Interestin g exercise s i n Chapte r 5 . Lot s o f goo d references.

[VM] Th e Visual Mind, Michel e Emmer , ed. , MI T Press , 1993 . A collectio n 3 6 papers dealin g wit h mathematica l aspect s o f art . Sev -eral paper s ar e relevan t t o thi s book . Th e paper , "Interlac e pattern s in Islami c an d Mooris h art " include s Cayle y diagram s o f wallpape r groups.

[VS] Visions of Symmetry, b y Dori s Schattschneider , W.H . Freema n an d Co. , 1990.

The comprehensiv e sourc e o f Esche r plan e tilings . Page s an d page s of fascinatin g pictures . Descriptio n o f Escher' s ow n classificatio n o f patterns.

Index

1 stands fo r do-nothing , 3 2 15-puzzle, 9 0 2-cycle, 7 1

addition mo d n , 6 3 alternating group , 9 0 AJV, alternatin g group , 9 0 associative operation , 5 3

basic unit , 8 , 4 9 bilateral symmetry , 3 8

Cayley diagram , 7 3 Cayley, Arthur , 6 7 closed path , 9 3 CN

cyclic group , 6 1 symmetry type , 3 6

Conway, Joh n H , 4 2 crystallographic restriction , 5 2 cycle, 68 cyclic group , C/v , 6 1

dihedral group , DN, 6 1 DN

dihedral group , 6 1 symmetry type , 3 7

do-nothing denoted b y 1 , 32 is a rotation an d a translation , is always a symmetry , 2 7

dual, 9 5

equivalent mirror lines , 4 7 rotocenter, 4 4

Escher, M.C. , 4 8 even permutation , 9 0

exponents rules of , 3 2

Fedorov, E.S. , 48 finite figure, 2 8 fixed point , 2 3 footprints, 4 2 fundamental domain , 1 1

generator, 73 , 92 glide line , 2 2 glide reflection , 2 2

phantom figure, 2 2 group, 51 , 53

AN, alternatin g group , 9 0 CN, cycli c group , 6 1 DN, dihedra l group , 6 1 generators, 7 3 presentation, 9 2 relation, 9 2 SN, symmetri e group , 68

identity matrix , 9 2 inverse, 35 , 38 isomorphism, 7 1

Kali, 53 , 98

Lloyd, Sam , 9 1

magie square , 9 6 matrix, 9 1 mirror, 1 9 mirror line , 1 9 mod, 6 3 modular arithmetic , 9 1

addition, 6 3 multiplication, 6 5

101

102 INDEX

modulo, see mod multiplication mo d n , 6 5 multiplication tabl e

of a triangle , 3 5

n-gon, 3 0

odd permutation , 9 0 order

of a rotation , 4 6

Penrose tiles , 5 3 pentagon

easy wa y t o mak e one , 3 1 regular, 2 9

permutation, 6 7 even o r odd , 9 0

phantom figure, 2 2 Pölya, George , 4 8 polygon

symmetries of , 3 3 presentation, 9 2

QuasiTiler, 5 3

fl, 19 r, smalles t rotatio n symmetry , 3 2 reflection, 1 9 regular polygon , 2 9

symmetries of , 3 0 regular solid , 9 5 relation, 9 2 rigid motio n

glide reflection , 2 2 reflection, 1 9 rotation, 1 6

measured counterclockwise , 2 8

translation, 1 5 rigid motio n o f the plane , 1 5 rotation, 1 6

measured counterclockwise , 2 8 rotocenter, 1 6

rotocenter, 1 6 equivalent, 4 4

rules o f exponents , 3 2

silly walks , 4 2 SN, symmetri e group , 6 8 spokes o f a wheel , 1 7 square

symmetries of , 29 , 33 strip pattern , 3 9 subgroup, 6 1 symmetrie group , 68 symmetries

of a polygon , 3 3 of a square , 29 , 33

symmetry, 2 7 symmetry type , 36 , 7 1

CN, cyclic , 3 6 DN, dihedral , 3 7

Timbanidis, Nicolas , 3 4 translation, 1 5 transposition, 71 , 90 triangle

multiplication table , 3 5 trivial symmetry , 2 7

wallpaper pattern , see wallpatter n wallpattern, 4 3

fl, 19