Continuou s Symmetr y Fro m Eucli d to Klei n This page intentionally left blank http://dx.doi.org/10.1090/mbk/047

Continuou s Symmetr y Fro m Eucli d to Klei n

Willia m Barke r Roge r How e

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2000 Mathematics Subject Classification. Primar y 51-01 , 20-01 .

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Library o f Congres s Cataloging-in-Publicatio n Dat a Barker, William . Continuous symmetr y : fro m Eucli d t o Klei n / Willia m Barker , Roge r Howe . p. cm . Includes bibliographica l reference s an d index . ISBN-13: 978-0-8218-3900- 3 (alk . paper ) ISBN-10: 0-8218-3900- 4 (alk . paper ) 1. Geometry, Plane . 2 . Group theory . 3 . Symmetry groups . I . Howe , Roger , 1945 -

QA455.H84 200 7 516.22—dc22 200706079 5

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© 200 7 b y the authors . Al l right s reserved . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 2 1 1 1 0 09 0 8 0 7 To Su e and Lyn , for lov e and suppor t This page intentionally left blank Contents

Instructor Prefac e i x Student Prefac e xii i Acknowledgments xi x I. Foundation s o f Geometr y i n th e Plan e LI. Th e Rea l Number s 1 1.2. Th e Incidenc e Axiom s 6 1.3. Distanc e an d th e Rule r Axio m 1 7 1.4. Betweennes s 2 2 1.5. Th e Plan e Separatio n Axio m 2 7 1.6. Th e Angula r Measur e Axiom s 3 4 1.7. Triangle s an d th e SA S Axiom 4 6 1.8. Geometri c Inequalitie s 5 6 1.9. Parallelis m 6 2 1.10. Th e Paralle l Postulat e 7 0 1.11. Directe d Angl e Measur e an d Ra y Translatio n 8 4 1.12. Similarit y 9 4 1.13. Circle s 11 0 1.14. Bolzano' s Theore m 11 5 1.15. Axiom s fo r the Euclidea n Plan e 11 9 II. Isometrie s i n the Plane : Product s o f Reflection s ILL Transformation s i n the Plan e 12 1 11.2. Isometrie s i n the Plan e 13 5 11.3. Compositio n an d Inversio n 14 6 11.4. Fixe d Point s an d th e Firs t Structur e Theore m 15 6 11.5. Triangl e Congruenc e an d Isometrie s 16 1 III. Isometrie s i n the Plane : Classificatio n an d Structur e 111.1. Tw o Reflections : Translation s an d Rotation s 16 5 111.2. Glid e Reflection s 18 1 111.3. Th e Classificatio n Theore m 18 8 111.4. Orientatio n 19 1 111.5. Group s o f Transformations 19 9 111.6. Th e Secon d Structur e Theore m 20 6 111.7. Rotatio n Angle s 21 1 Vlll Contents

IV. Similaritie s i n th e Plan e IV. 1. Elementar y Propertie s o f Similarities 21 7 IV.2. Dilation s a s Similaritie s 22 4 IV.3. Th e Structur e o f Similarities 23 1 IV.4. Orientatio n an d Rotatio n Angle s 23 5 IV.5. Fixe d Point s fo r Similaritie s 24 0

V. Conjugac y an d Geometri c Equivalenc e V.l. Congruenc e an d Geometri c Equivalenc e 25 1 V.2. Geometri c Equivalenc e o f Transformations: Conjugac y 25 6 V.3. Geometri c Equivalenc e unde r Similaritie s 26 6 V.4. Euclidea n Geometr y Derive d fro m Transformation s 27 6

VI. Application s t o Plan e Geometr y VI. 1. Symmetr y i n Early Geometr y 28 7 VI.2. Th e Classica l Coincidence s 29 2 VI.3. Dilatio n b y Minu s Two aroun d th e Centroi d 29 8 VI.4. Reflections , Light , an d Distanc e 30 9 VI.5. Fagnano' s Proble m an d th e Orthi c Triangl e 31 5 VI.6. Th e Ferma t Proble m 32 2 VI.7. Th e Circl e o f Apollonius 34 0

VII. Symmetri c Figure s i n th e Plan e VII. 1. Symmetr y Group s 34 7 VII.2. Invarian t Set s an d Orbit s 35 6 VII.3. Bounde d Figure s i n the Plan e 36 3

VIII. Friez e an d Wallpape r Group s VIII. 1. Poin t Group s an d Translatio n Subgroup s 37 6 VIII.2. Friez e Group s 39 9 VIII.3. Two-Dimensiona l Translatio n Lattice s 41 6 VIII.4. Wallpape r Group s 43 9

IX. Area , Volume , an d Scalin g IX. 1. Lengt h o f Curves 45 9 IX.2. Are a o f Polygonal Regions : Basi c Properties 46 7 IX.3. Are a an d Equidecomposabilit y 48 2 IX.4. Are a b y Approximation 48 7 IX.5. Are a an d Similarit y 50 5 IX.6. Scalin g an d Dimensio n 52 0

References 53 1 Index 53 3 Instructor Prefac e

This text i s intended fo r a one-semester cours e o n geometry. W e have trie d to write a book that honor s the Greek tradition o f synthetic geometry and at the same time takes Feli x Klein's Erlanger Program m seriously . Th e pri- mary focu s i s on transformations o f the plane , specificall y isometrie s (rigi d motions) an d similarities , bu t ever y effor t i s mad e t o integrat e transfor - mations wit h th e traditiona l geometr y o f lines , triangles , an d circles . O n one hand, w e discuss i n detail the concrete properties o f transformations as geometric objects; on the other hand, w e try to show by example how trans- formations ca n b e use d as tools t o prov e interestin g theorems , sometime s with greate r insigh t tha n traditiona l method s provide . We have bee n surprise d an d please d a t ho w fa r thi s ide a ca n b e taken. W e hope w e hav e mad e concret e th e usuall y abstrac t dictu m o f th e Erlanger Programm: a geometry is determined by its symmetry group. For example , w e have tried t o sho w the intimate relationshi p betwee n Fag- nano 's Problem (inscrib e in a given triangle a triangle o f minimal perimeter ) and th e problem o f computing the product o f three reflections . (Thi s latte r problem is natural since by the First Structure Theorem o f §11. 4 every isome- try i s the product o f at mos t three reflections.) Fro m this, one can prove the concurrency o f the altitudes o f a triangle usin g only reflections, no t similar - ities as does the traditional proof . A s a consequence, on e can later conclud e that th e concurrency o f altitudes hold s equally wel l in elliptic geometry an d (to th e exten t possible ) hyperboli c geometry . I n th e othe r direction , tra - ditional geometri c reasonin g i s used i n showin g that ever y stric t similarit y transformation ha s a fixed point an d the computation o f the product o f two rotations i s interpreted i n terms o f traditional geometry . The Erlanger Programm i s often treate d a s a component o f the larger topi c of grou p theory . W e first introduc e th e grou p concep t i n Chapte r II I an d use it constantly throughout th e subsequent material ; i n particular, i t play s a centra l rol e i n Chapter s VI I an d VIII . But group s ar e neve r investigate d for themselves ; the y ar e alway s subservien t t o th e geometry . Nevertheless , students who have taken a course from this book will have a store of examples to mak e copin g with grou p theory i n a subsequen t abstrac t algebr a cours e easier an d mor e meaningful .

ix X Instructor Prefac e

We have expende d considerabl e effor t t o giv e a self-contained developmen t and to mak e explanations a s clear a s possible. Ou r students hav e found th e text t o b e quite readable ; w e hope the sam e wil l be true fo r you r students . Our tex t is , i n fact , th e firs t o f a projecte d two-volum e work . Th e sec - ond volum e wil l g o beyon d traditiona l Euclidea n geometr y b y introducin g coordinates, discussing different geometrie s — affine an d non-Euclidean (hy - perbolic an d spherical/elliptical) — in a projective setting , an d ending wit h an interpretatio n o f Einstein's Special Theory of Relativity a s a n analo g i n higher dimensions o f hyperbolic plane geometry. Unti l the completion o f the second volume we hope this text will stand on its own as a treatment o f plane Euclidean geometr y tha t deepen s th e reader' s understandin g o f symmetr y and it s natural plac e i n geometry . This boo k ca n suppor t severa l differen t type s o f courses. Chapte r I give s a fairly complet e axiomatic developmen t o f plane Euclidean geometry , largel y inspired b y Moise , Elementary Geometry from an Advanced Standpoint. This ca n b e mad e a substantia l par t o f th e cours e o r b e use d entirel y a s a referenc e o r something i n between. Th e hear t o f the boo k i s composed o f Chapters I I t o V . Her e w e develop th e basi c facts abou t isometrie s (Chap - ters I I an d III ) an d similaritie s (Chapte r IV ) an d attemp t t o integrat e th e Kleinian transformational viewpoin t wit h geometry a s formulated tradition - ally (Chapte r V) . Chapters V I throug h I X presen t application s o f th e materia l fro m Chap - ters I I to V . Differen t choice s o f material fro m thes e chapter s wil l result i n courses o f quit e differen t flavors . I n particula r Chapte r VI , Chapter s VI I and VIII , an d Chapte r I X ar e essentiall y independen t o f each other . Chapter V I uses the transformational approac h to establish ver y traditiona l theorems o f plan e geometry . Th e chapte r begin s wit h th e classica l coin - cidences — th e circumcenter , incenter , centroid , an d orthocenter . I n th e treatment o f the centroid , w e emphasize the rol e o f the media l triangle (th e triangle forme d b y th e midpoint s o f sides ) an d i n particula r th e fac t tha t it i s similar t o th e origina l triangl e b y a similarit y tha t stretche s b y 2 and rotates b y 180° . Thi s immediatel y lead s t o th e Eule r lin e an d provide s re - lationships tha t carr y u s o n to th e nine-poin t circle . W e attempt t o revea l the dept h o f this remarkabl e configuration . Severa l othe r topic s whic h ca n be approached naturall y vi a transformations, includin g Fagnano's Theorem and it s relationshi p t o th e orthi c triangle , Napoleon's Theorem, an d th e Fermat Problem, ar e als o discussed .

Chapters VI I an d VII I ar e devote d t o understandin g symmetri c figures . They describ e th e classificatio n o f discret e group s o f plan e isometrie s — rosette groups , friez e groups , an d wallpape r groups . W e have attempted t o Instructor Prefac e XI give a carefu l treatment . Especiall y i n the cas e o f the wallpape r groups , w e hope th e ingredient s tha t g o into th e classificatio n ar e brough t ou t clearl y and tha t th e argument , a s wel l a s th e final result , wil l b e memorabl e (i n the sens e tha t you r student s wil l actuall y b e abl e t o remembe r it!) . Th e group theoreti c notio n o f a split extensio n i s the basi c tool, bu t w e treat i t in a concrete fashion . I n particular, thi s concep t help s to organiz e both th e classification an d it s justification . Chapter I X studies scalin g and dimension an d make s a brief fora y int o frac - tals. I t discusse s th e are a o f plan e figures, fro m familia r are a formula s t o Jordan conten t fo r mor e exotic shapes. Th e treatment i s relatively abbrevi - ated an d provides many opportunities fo r students to fill in the details (wit h generous hints) . Throughout th e book the approach i s concrete, an d w e try t o giv e complet e explanations. Th e nee d fo r justification an d proo f i s taken fo r granted , an d there are many opportunities fo r students to construct thei r ow n arguments. Because o f thi s an d becaus e o f th e concret e an d accessibl e natur e o f th e material, thi s tex t migh t for m th e basi s fo r a bridg e cours e t o introduc e students to mathematica l reasoning . This page intentionally left blank Student Prefac e

This book i s about Euclidea n geometry , the same subject yo u studied i n hig h school. However , th e viewpoin t i s probably ver y different . Th e goa l o f thi s text i s to present geometr y i n a way that honor s the idea s o f transformatio n and symmetr y tha t hav e s o profoundly shape d th e moder n scientifi c vie w o f the world . T o se t th e stag e fo r th e book , w e offe r yo u a thumbnail sketc h o f its historica l roots .

Geometry a s a deductive scienc e was invented b y the Greeks . Thei r wor k o n the subject , th e wor k o f many thinker s ove r severa l centuries , wa s collecte d and wove n togethe r brilliantl y b y Euclid o f Alexandria aroun d 30 0 B.C. Fo r nearly tw o thousand years , Euclid' s Elements , expoundin g Gree k geometry , was th e hig h poin t o f human intellectua l achievement .

Though universall y revere d an d admired , ther e wer e aspect s o f Euclid' s Elements tha t worrie d som e readers . Hi s syste m o f geometr y wa s buil t o n several "commo n notions " — easil y accepte d principle s o f reasonin g whic h applied t o man y area s — togethe r wit h five "postulates, " accepte d fact s that wer e specificall y abou t geometry . Fou r o f thes e postulate s wer e short , simple, an d eas y t o accept . Th e fifth, however , wa s troublesom e t o thos e who though t seriousl y abou t th e subject . Her e i s a n Englis h translatio n o f what i t said .

If a straigh t lin e fallin g o n tw o straigh t line s make s th e interio r angles o n the sam e sid e les s than tw o righ t angles , the tw o straigh t lines, i f produced indefinitely , mee t o n that sid e on which the angle s are les s than tw o righ t angles .

The statemen t i s rathe r long , abou t a s lon g a s th e othe r fou r postulate s put together . Bu t i t wa s no t it s lengt h tha t unnerve d thoughtfu l student s of Euclid . I t wa s th e "i f produce d indefinitely" . I t assert s tha t tw o line s which cros s a thir d lin e i n a certai n wa y mus t eventuall y intersect . Bu t where? I f the interio r angle s o f intersectio n ar e muc h smalle r tha n 90° , th e lines wil l intersec t nearby . However , i f the angle s ar e clos e t o 90° , the line s may intersec t fa r awa y — possibl y far , fa r away : i n th e nex t stat e o r i n Europe o r beyon d th e Moo n o r outsid e ou r sola r syste m o r eve n ou r galax y (of whos e existenc e Eucli d an d everyon e els e befor e th e twentiet h centur y was blissfull y unaware) .

xiii XIV Student Prefac e

This i s the difficult y wit h th e Fifth Postulate: i t make s a n assertio n abou t the structur e o f space in the large; in fact , infinitel y large . (Perhap s suc h a statement wa s easier to make in ancient times, when people had little inklin g of ho w larg e thing s coul d be. ) Ther e i s no wa y w e can physicall y chec k th e truth o f Euclid' s Fifth Postulate becaus e w e ca n never physicall y confir m that tw o line s d o no t intersect ; n o matte r ho w fa r w e g o withou t finding an intersection point , w e can neve r rul e out tha t suc h a point exist s "jus t a little further " alon g the lines . So people who were inclined to worry about suc h logical and aestheti c issue s were unhapp y wit h th e Fifth Postulate. Man y attempte d t o eliminat e th e need fo r it b y turning it into a theorem, i.e. , they tried to show it followe d a s a logical necessity fro m th e first fou r postulates . However , n o one succeeded in doin g s o althoug h man y purporte d "proofs " wer e constructe d ove r th e centuries. Thoug h alway s flawed i n som e way , severa l o f thes e "proofs " carried thei r reasonin g quit e fa r an d establishe d importan t result s that do , in fact , follo w fro m denyin g the Fifth Postulate. After s o man y faile d attempt s ove r tw o millennia , th e suspicio n bega n t o grow i n the earl y nineteent h centur y tha t ther e migh t indee d b e "othe r ge - ometries" i n whic h th e Fifth Postulate wa s false . Th e first t o publis h a description o f a non-Euclidea n geometr y (1829 ) wa s Nikola i Lobachevsky , professor a t th e Universit y o f Kaza n i n Russia . Lobachevsky' s wor k ap - peared i n th e Bulleti n o f Kaza n University . I n tha t pre-interne t (indeed , pre-theory o f electricit y an d magnetism! ) era , communication s coul d b e rather slow , an d thu s mathematician s furthe r wes t remaine d unawar e o f Lobachevsky's work . I n 1832 , a n independen t accoun t o f a non-Euclidea n geometry wa s publishe d b y Jano s Bolyai , a youn g Hungarian . Thes e tw o papers inaugurate d th e post-Euclidea n er a i n geometry . These wer e th e first leak s i n a da m tha t ha d bee n holdin g bac k huma n thought. Th e nex t decade s sa w a flood o f researc h i n geometr y an d th e creation o f a great profusio n o f geometric systems . Th e traditional meanin g of geometry coul d no t encompas s th e ne w wealth o f phenomena. A revised understanding o f the natur e o f geometry wa s urgently needed . The basis for this new understanding came from a subject invente d a t nearl y the sam e tim e a s non-Euclidea n geometr y — the theor y o f groups. Unlik e the question s surroundin g th e Fifth Postulate whic h wer e i n th e mind s o f many mathematician s a t th e time , grou p theor y wa s th e inventio n o f on e extraordinary individual , Evarist e Galois. 1

1 Galois wa s inspire d b y a proble m whic h ha d als o attracted muc h attentio n — th e proble m of givin g formula s fo r th e solution s o f polynomial equations . However , h e applie d hi s ow n uniqu e and revolutionar y approac h t o th e problem . Student Prefac e xv

Galois wa s to o smar t fo r hi s ow n good . A studen t revolutionar y wh o onc e toasted th e kin g o f Franc e b y buryin g a knif e i n a table , h e die d i n a due l over a woma n a t th e ag e o f twenty-two . Tha t wa s i n 1832 , th e sam e yea r as Bofyai' s publicatio n o n non-Euclidea n geometry . Fortunatel y fo r math - ematics, Galoi s spen t th e nigh t befor e th e due l furiousl y writin g dow n hi s ideas abou t group s an d thei r application s t o solvin g polynomia l equations .

Galois' ideas were hard fo r hi s contemporary mathematician s t o understand . However, the y di d ge t studie d an d appreciatio n fo r thei r powe r graduall y percolated throug h th e mathematica l communit y durin g th e sam e perio d when geometri c researc h wa s roarin g ful l throttle . Indeed , a realizatio n o f the relevance o f group theory to geometry began to grow, with groups arisin g from symmetries — transformations o f a system which preserve the essentia l characteristics o f the system .

The semina l connectio n betwee n geometr y an d grou p theory wa s discovere d by Feli x Klein . I t wa s th e custo m i n Germa n universitie s o f tha t er a fo r new professor s t o giv e a n inaugura l lectur e o n thei r researc h t o th e ful l faculty.2 I n 1872 , a t th e Universit y o f Erlangen , Feli x Klein , the n onl y twenty-three year s old , presente d t o hi s colleague s strikin g idea s abou t ho w to unif y geometr y b y mean s o f symmetr y vi a grou p theory . Thi s proposa l has becom e know n a s Klein's Erlange r Programm. 3

Klein's firs t observatio n wa s tha t geometr y i s no t abou t individua l figures but abou t classes of equivalent figures. Fo r example , i n Euclidean geometr y there i s a notio n o f congruence. An y tw o congruen t figures ar e "th e same " from th e poin t o f vie w o f Euclidea n geometr y — the y hav e th e sam e geo - metric properties . Furthermore , yo u ca n tel l i f two figures ar e congruen t b y transforming o r movin g on e s o tha t i t become s (mor e correctly , coincide s with) th e other . Th e transformatio n yo u us e shoul d b e a rigid motion, i.e. , it shoul d preserv e distance s an d angles .

Thus, a t th e cor e o f Euclidea n geometry , a s wel l a s a t th e cor e o f mos t other geometrie s constructe d afte r 1830 , ther e wa s a notio n o f geometric equivalence, define d b y a specified collectio n o f transformations know n a s th e symmetries o f the geometry . (I n Euclid's Elements, thi s ide a was somewha t hidden an d neve r explicitl y acknowledged , bu t th e attentiv e reade r ca n se e it use d a t certai n critica l places. ) Klei n furthe r observe d tha t

2 It wa s a muc h bigge r dea l t o b e a professo r i n those day s — i n mos t universitie s ther e wa s often jus t on e professo r i n a subject . The y wer e addresse d a s "Her r Professo r Doktor, " th e titl e a bi t lon g but admirabl y distinguished . 3The "r " o n the en d o f "Erlanger " i s not a misprint . It' s ho w Germa n gramma r works . XVI Student Prefac e

(i) th e se t o f symmetries forme d a group i n the sens e o f Galois an d (ii) yo u ca n reconstruc t th e geometr y fro m it s symmetry group. 4 In short, the fundamental ide a in geometry i s that o f symmetry, an d a given geometry i s governed b y the natur e o f its symmetries . Klein's ideas and related work sparked a second wave of remarkable discovery that produced , amon g othe r things , a classificatio n o f th e buildin g block s of al l possibl e symmetr y group s o f geometrie s tha t ar e continuous i n tha t they allo w continuou s movement . Whil e thi s classificatio n liste d familia r objects suc h a s Euclidea n an d non-Euclidea n geometrie s an d thei r higher - dimensional cousins , i t als o include d a fe w exoti c system s o f symmetrie s whose associated geometrie s ar e stil l onl y partially understood . Through th e en d o f the nineteent h centur y thi s wa s al l pure mathematics , inspired b y nagging questions i n the field and divorce d fro m practica l goals . In 1905 , however , Alber t Einstei n introduce d hi s Special Theory of Rela- tivity tha t explaine d th e troublin g result s o f som e experiment s (e.g. , th e Michelson-Morley experiment ) mad e t o prob e Maxwell' s theor y o f electro - magnetism. A yea r later , Herman n Minkowski , wh o ha d bee n Einstein' s mathematics teache r a t th e University o f Koenigsberg an d was embarrasse d by th e unsophisticate d leve l o f th e mathematic s i n Einstein' s paper , rein - terpreted Einstein' s result s i n term s o f a non-Euclidea n geometr y o f four - dimensional space-time . Transformation s ha d playe d a key role in the inter - pretation o f th e Michelson-Morle y experimen t an d i n Einstein' s theor y — Minkowski foun d th e four-dimensiona l geometr y fo r whic h the y comprise d the grou p o f symmetries . Since the appearanc e o f Einstein' s paper , symmetrie s an d transformation s have playe d a n ever-greate r rol e i n theoretica l physics . I t i s no t to o muc h of an exaggeration t o sa y that symmetr y an d group s hav e been a dominan t theme i n modern physics . I n particular, th e structure o f atoms, which lead s to th e chemistr y tha t shape s al l o f biology , ha s a t it s bas e a structur e o f exquisite symmetry . Thu s grou p theory an d symmetry , whic h wer e first in - troduced i n response to questions about solution s o f equations, prove d late r to b e fundamenta l fo r understandin g geometry , an d stil l later , fo r under - standing th e deepe r truth s o f ou r rea l physica l world . Thi s histor y i s a beautiful exampl e o f ho w mathematica l ideas , pursue d fo r thei r ow n sake , can hav e a dramatic impac t an d practica l consequence s i n domain s fa r be - yond thei r origina l birthplace. 5

4 Actually th e reconstructio n o f th e geometr y require s a littl e additiona l informatio n alon g with th e symmetr y group . Bu t th e symmetr y grou p remain s th e centra l object . 5 The sam e poin t ca n b e mad e i n a n eve n stronge r wa y fo r th e investigation s tha t le d t o th e discovery o f non-Euclidea n geometry . A t fac e value , thes e seeme d t o b e archetypicall y useles s academic pursuit s — they wer e not eve n goin g to produc e ne w theorems, onl y tidy u p the syste m Student Prefac e xvn

Our book takes Klein's Erlanger Programm seriously , while still retaining the flavor o f a classica l stud y o f geometry i n which triangles, circles , quadrilat - erals, and other simpl e shapes ar e the primary object s o f investigation. Th e study o f transformations i s integrated wit h serious attention to the beautifu l results o f synthetic geometr y W e do this i n two ways : transformation s ar e studied as geometric objects, emphasizin g thei r concret e geometri c proper - ties, an d transformation s ar e als o use d as tools t o understan d interestin g concepts i n geometr y suc h a s the circumcircle , incircle , centroid , orthocen - ter, Eule r line , an d nine-poin t circle . Th e author s hav e bee n surprise d a t the exten t t o whic h this unifie d viewpoin t ca n succeed . We hop e thi s boo k present s th e philosoph y o f th e Erlanger Programm — that symmetr y i s the basi s o f geometry — not merel y a s an abstract , orga - nizational viewpoint, but a s a practical approac h that enhance s your under - standing an d highlight s the beaut y o f this timeles s subject . Above all , w e hope yo u enjo y th e journey yo u ar e about t o begin .

Cross-Reference Conventions . Tex t cross-reference s t o theorems , fig - ures, equations , an d othe r labele d item s ar e handle d a s follows . Withi n Chapter V , fo r example , cross-reference s suc h a s Theore m 1.2 , Figur e 5.3 , and (3.3 ) refe r t o items with those labels contained in Chapte r V . However , within Chapte r V , cross-reference s suc h a s Theore m II . 1.2, Figur e II.5.3 , and (II.3.3 ) refe r t o item s wit h thos e label s i n Chapte r II , henc e outside of Chapter V .

of postulates . A s i t turne d out , the y brough t u s t o a grea t watershe d i n thought , producin g massive reverberations i n mathematics, science , an d philosoph y that hav e shaped an d continu e t o shape th e natur e o f ou r thinking . This page intentionally left blank Acknowledgments

When a book has been in preparation fo r eleven years, there are many people who have contributed helpfu l advic e and who have supported the authors i n numerous important ways . Thoug h w e can list onl y a small number o f them — students, colleagues , friends, an d famil y — they hav e all earned ou r dee p appreciation. This boo k originate d fro m a cours e firs t pilote d b y th e author s a t Yal e University i n the fal l o f 199 6 with fundin g fro m th e Nationa l Scienc e Foun - dation. W e thank the NSF fo r the encouragement an d support the y gav e to the project durin g that formativ e initia l period (NS F Grant DUE-9555134) . Jim Rei d an d Rober t Rosenbau m o f Wesleya n Universit y wer e highl y in - volved the firs t year , attendin g the class and preparin g extensiv e comment s on ou r preliminar y note s — w e ar e gratefu l fo r thei r aid . Ji m remaine d involved ove r the years an d ha s contributed man y valuable suggestions tha t have entered ou r expositio n i n significant ways ; his unwavering support an d advice hav e bee n deepl y appreciated . We als o thank Thoma s Berge r o f Colb y Colleg e an d Anit a Sale m o f Rock - hurst College , both o f whom taught course s fro m earl y version s o f our man - uscript o n multiple occasion s — it wa s reassuring to have confirmation tha t the manuscrip t worke d outsid e o f the hand s o f it s creators ! To m als o con - tributed man y insightfu l critique s o n ou r material s tha t hav e affecte d th e final product . Hearty thank s t o Zalma n Usiski n o f th e Universit y o f Chicag o an d Jame s King o f the University o f Washington fo r sharing their insights o n geometry . Their comment s hav e bee n incorporate d i n severa l exercises . We greatly appreciate d th e invitatio n fro m Rober t Bryan t o f Duke Univer - sity an d Joh n Polkin g o f Ric e Universit y t o teac h Continuous Symmetry at th e IAS/Par k Cit y Mathematic s Institut e i n the summe r o f 1998 . Thi s gave ou r material s wid e exposur e t o a n intereste d grou p o f colleague s an d students. We have taught Continuous Symmetry ove r the past te n year s a t bot h Yal e and Bowdoi n — we gratefully than k al l our many students fo r the insightfu l comments they gave , fo r the carefu l proofreadin g the y provided, an d fo r th e

xix XX Acknowledgments patience they displayed i n dealing with a sometimes rough work-in-progress . Their enthusiasm fo r the material wa s a great suppor t durin g the long hours spent preparin g the manuscript. W e cannot lis t them all , but a fe w deserv e special mention . Anthony Philippakis , Yale , 1998 , a studen t whe n w e pilote d Continuous Symmetry, wa s a n unendin g sourc e o f enthusiasm , commentary , sugges - tions, an d support . H e continue d workin g o n thi s materia l lon g afte r th e course ended, producin g a lovely articl e [13 ] for the American Mathematical Monthly. Andre w Shaw , Bowdoin , 2002 , an d Sa m Kolins , Bowdoin , 2006 , went abov e an d beyon d th e norma l cours e work , enthusiasticall y providin g commentary an d suggestion s that le d to improvement s i n the text. W e ar e grateful fo r thei r insight , help , and support . We also wish to thank Ree d Hastings , Bowdoin , 1983 , whose generous sup - port o f th e Bowdoi n Colleg e Mathematic s Departmen t helpe d allo w on e author a n extr a semester' s leav e a t Yale . Thi s wa s o f immens e valu e i n producing the curren t text . We mus t expres s ou r dee p appreciatio n t o th e editor s an d technica l staf f at th e AMS . Barbar a Beeto n an d Stephe n Moy e hav e spen t man y hour s helping u s tam e T^ X an d produc e a n acceptabl e manuscrip t — they wer e generous with their time and accurate with their advice . Arlen e O'Sean, th e AMS Cop y Editor, wa s insightful an d meticulou s i n he r excellen t editin g o f our manuscript. W e are particularly thankful t o AMS Editors Ed Dunne and Ina Mett e fo r thei r unfailin g encouragemen t an d suppor t fo r thi s project , their gentl e reminders about deadlines , their understandin g an d patienc e a s we misse d thos e deadlines , an d thei r unflappabl e goo d humo r an d soun d advice a t ever y poin t durin g the productio n o f this book .

We als o enjoye d th e fines t administrativ e suppor t possibl e fro m th e staf f o f our respectiv e mathematic s departments . Me l Delvecchi o a t Yal e an d Su e Theberge at Bowdoi n spent many hours these past years helping the author s cope with al l the administrative issue s involved wit h this writing project, al l with efficiency , skill , and goo d humor . Finally, the greatest thanks o f all goes to our wives, Sue and Lyn, fo r havin g to endur e th e lon g hour s an d lat e night s thi s projec t ha s consume d thes e many month s an d years . W e hav e relie d o n thei r lov e an d suppor t an d patience an d understanding , especiall y whe n i t seeme d lik e th e en d wa s never goin g to arrive . Acknowledgments xxi

Acknowledgments fo r Graphics . Th e flo w char t o f Figur e VIII.2.1 9 i s adopted fro m Georg e E . Martin' s "Transformatio n Geometry : A n Intro - duction t o Symmetry, " Springer , 1982 : Figur e 10.11 , pag e 83 . Th e friez e patterns o f Figur e VIII. 2.20, on e o f whic h i s repeate d i n Figur e VIII. 2.1, are base d o n Thoma s Sibley' s "Th e Geometri c Viewpoint, " Addiso n Wes - ley, 1998 , Figur e 5.18 , pag e 193 . Mathematica® wa s use d t o generat e Fig - ure IX.6.4. Freehand ® wa s used to dra w al l the othe r figures . This page intentionally left blank This page intentionally left blank References

[1] Coxeter , H . S . M . Introduction to Geometry, 2n d Edition . Wile y an d Sons , New York, 1969 . [2] Coxeter , H . S . M., and Greitzer , S . L . Geometry Revisited. Th e Mathematica l Association o f America, Washington , D.C. , 1967 . [3] Hartshorne , R . Geometry: Euclid and Beyond. Springer-Verlag , Ne w York , 2000. [4] Heilbron , J . L . Geometry Civilized: History, Culture, and Technique. Claren - don Press , Oxford , 1998 . [5] Hilbert , D. , an d Cohn-Vossen , S . Geometry and the Imagination. Translate d by P . Nemenyi. AM S Chelse a Publishing, Providence , 1999 . [6] Honsberger , R . Episodes in Nineteenth and Twentieth Century Euclidean Ge- ometry. Th e Mathematica l Associatio n o f America, Washington , D.C. , 1995 . [7] Klein , F . Elementary Mathematics from an Advanced Standpoint: Geometry. Translated b y E. R . Hedric k an d C . A . Noble. Macmillan , Ne w York, 1939 . [8] Kline , M . Mathematical Thought from Ancient to Modern Times. Oxfor d University Press , Ne w York, 1972 . [9] Martin , G . E . The Foundations of Geometry and the Non-Euclidean Plane. Springer-Verlag, Ne w York, 1982 . [10] Martin , G . E . Transformation Geometry. Springer-Verlag , Ne w York, 1982 . [11] Moise , E. E. Elementary Geometry from an Advanced Standpoint, 3r d Edition. Addison-Wesley, Reading , MA , 1990 . [12] Pedoe , D . Circles: A Mathematical View. Th e Mathematica l Associatio n o f America, Washington , D.C. , 1995 . [13] Philippakis , A . The Orthic Triangle and the O.K. Quadrilateral. Th e American Mathematical Monthl y 109 , No . 8 (Oct. 2002) , pp. 704-728 . [14] Sibley , T. Q . The Geometric Viewpoint. Addison-Wesley , Reading , MA , 1998 . [15] Weyl , H . Symmetry. Princeto n Universit y Press , Princeton, 1989 .

[16] Yale , P. B. Geometry and Symmetry Dover , Mineola , NY , 1988 .

531 This page intentionally left blank Index

AAA (Angle-Angle-Angle ) Angle Construction , 3 5 Thales's statemen t of , 28 9 angle measure, 3 5 AAA an d SS S criteria fo r similarity , directed, 38-4 5 270 Angle Measur e Axioms , 34-4 5 altitude, 106 , 29 6 statement of , 35 , 12 0 angle antipodal point , 1 2 acute, 3 7 Appolonius o f Perga, 34 0 central, 80 , 465 Archimedean Order , 2 complementary, 3 7 area, 45 9 congruence, 3 6 computation vi a Jordan measure , corresponding, 7 8 490-505 definition of , 2 4 Euclid's approach , 475-47 6 directed, definitio n of , 24-2 5 formulas fo r familie s o f simila r equal modul o 360 , 3 9 figures, 507 , 515-51 6 exterior, o f triangle, 56 , 80 of a disk, 488-49 0 initial ra y of , 3 9 of a parallelogram, formul a for , inscribed i n a circle, 80-8 1 473 intercepted ar c o f an inscribed , 8 0 of a rectangle, formul a for , 47 3 interior, 3 0 of a sector o f a disk, 49 0 linear pair , 3 5 of a square, formul a for , 471-47 2 measure, 3 5 of a trapezoid, 47 7 obtuse, 3 7 of a triangl e of incidence, 31 2 in terms o f ASA, 47 8 of reflection, 31 2 in terms o f base an d height , 47 4 opposite a side o f a triangle, 3 3 in terms o f SAS , 478 remote interior , o f triangle, 5 6 in terms o f SSS , 477 right, 3 7 of polygonal regions , 467-48 7 straight, 3 9 scales under dilatio n wit h th e sum o f angles o f triangle, 7 1 square o f the dilatio n factor , supplementary, 3 5 489, 50 6 terminal ra y of , 3 9 area formulas , 45 9 trivial, 3 9 area functio n o n polygonal region s vertical pair , 3 6 axioms fo r an , 47 0 Angle Addition, 3 5 constructed fro m Jorda n measure , angle bisecto r 503 as locus o f points equidistan t fro m existence an d uniquenes s of , 47 0 sides, 6 1 uniqueness vi a dissection, 47 1 definition of , 3 6 ASA (Angle-Side-Angle ) existence an d uniquenes s of , 3 7 statement of , 4 8 Angle Bisector Theorem , 34 0 Thales's statemen t of , 28 8

533 534 Index axiom syste m definition of , 55 , 11 2 axioms o r assumption s in , 6 diameter of , 56 , 11 2 model for , 7 inscribed angl e of , 8 0 undefined objec t in , 6 intercepted ar c of , 8 0 axiomatic method , 6 intersection criterion , 11 4 Axioms fo r Euclidea n Geometry , intersection wit h a line , 11 4 collected, 119-12 0 length o f a n ar c o f a , 464-46 5 line tangent to , 11 4 Basic Similarit y Principle , 9 9 Line-Circle Theorem , 11 4 basis fo r discret e plana r translatio n of Appolonius , 340-34 6 group, 417-42 2 orthogonal, 34 4 definition of , 41 8 point o f contac t o f tangent line , existence of , 418 , 43 8 betweenness, 2 2 114 for thre e point s o n a line , 2 2 possible intersectio n wit h another , billiards, 32 2 112 Bisector/Fixed Poin t Relation , 15 6 radius of , 56 , 11 2 Bolzano's Theorem , 115-11 9 radius segmen t of , 11 4 for Euclidea n plane , 11 8 Theorem o f Thales, 8 0 boundary Two Circl e Theorem , 11 2 of a triangula r region , 46 8 circumcenter, 292-29 3 bounded figure, 35 4 lies o n th e Eule r line , 30 1 of a triangl e i s the orthocente r o f Cantor set , 52 6 the media l triangle , 30 1 and bas e b expansions, 52 9 circumcircle, 29 3 generalized, 52 8 circumference unsymmetric, 52 9 arbitrarily larg e relativ e t o center o f a group, 41 5 diameter, 51 3 centers o f similitude , 230 , 239-240 , of a circle , 459-46 4 344 of a se t i n th e plane , 51 0 centralizer classical coincidences , 292-29 8 of a transformation , 204 , 22 9 Classification o f Plane Isometries , centrally symmetri c figure, 511-51 2 188 centroid, 292 , 294-29 6 classification procedur e fo r a triangl e an d it s media l triangl e symmetry groups , 393 , 40 0 have th e same , 30 0 aka cente r o f gravity , 29 5 Classification Theore m fo r dilation b y — 2 around the , Similarities, 24 2 298-309 closed region , 46 8 divides eac h media n i n a ratio o f closed se t i n the plane , 50 7 1:2, 29 5 coincidence i n a triangl e lies o n th e Eule r line , 30 1 of altitudes (orthocenter) , change o f scal e 300-301, 31 8 change o f length unde r a , 46 2 of angl e bisector s (incenter) , chirality, 30 9 293-294 circle, 8 0 of median s (centroid) , 294-29 5 arc o f a , 45 9 of perpendicula r bisector s center of , 56 , 11 2 (circumcenter), 292-29 3 central angl e of , 8 0 collinear, 18 , 2 2 Index 535 commutativity equivalence o f transformationa l and fixed points , 15 4 and forma l definitions , fo r examples, 15 5 triangles, 25 2 compatibility for lin e segments , 2 5 of point grou p an d translatio n for triangle s i s provided b y orbit, 386-38 7 isometries, 16 1 of poin t group s an d translatio n of angles , 3 6 subgroup, 38 1 of triangles, 4 6 completeness oriented, 26 7 of Euclidean plane , 11 7 transformational definition , 25 1 of rea l numbers , 2- 3 conjugacy composition as geometri c equivalenc e fo r definition of , 14 6 transformations, 256-26 6 inversion of , 15 3 classes o f isometries , 263 , 27 5 classes o f similarities , 273 , 27 5 is an algebrai c operation , 14 8 criteria for , 259-26 5 is associative , 14 9 is a n equivalenc e relation , 25 8 is not commutative , 14 8 of rotations an d orientation , 26 5 juxtaposition notatio n for , 15 0 of subgroups , 35 2 of a rotatio n an d a reflection , 17 8 of two-fol d product s take n i n bot h of a rotation an d a translation , orders, 26 5 201, 26 6 conjugation of a n eve n numbe r o f reflection s i s and commutativity , 17 9 orientation-preserving, 19 4 by similarities , 273 , 27 5 of an od d numbe r o f reflection s i s definition of , 25 8 orientation-reversing, 19 4 examples of , 179-18 0 of fou r reflection s i s a compositio n of a glid e reflectio n b y a similarity , of two reflections , 19 2 187, 27 3 of orientation-preservin g of a reflectio n b y a similarity , 179 , isometries i s 260, 27 3 orientation-preserving, 19 5 of a rotation b y a similarity , of three reflections , singula r case , 179-180, 265 , 27 3 188 of a stric t similarit y b y a of two mappings , 14 6 translation, 27 5 of two reflections , 165-18 0 of a translation b y a similarity , of two rotations , 17 8 179, 27 3 of two translations i s a translation , of a unifor m dilatio n b y a 176, 17 8 similarity, 235 , 27 3 of unifor m dilations , 226 , 23 9 of a n isometr y b y a similarit y preserves isometries , 14 7 gives a n isometry , 23 9 preserves one-to-oneness , 14 7 of on e transformatio n b y another , preserves ontoness , 14 7 definition of , 179 , 25 8 preserves transformations , 14 7 takes inverse s t o inverses , 26 1 congruence takes product s t o products , 26 1 and geometri c equivalence , Contraction Mappin g Theorem , 25 0 251-256 convex quadrilateral , 3 0 as equivalenc e relation , 25 4 convex set , 3 2 criteria for , 254-25 5 definition of , 2 7 536 Index

coordinate system s discrete grou p o f isometries , 375 , definition of , 1 7 414, 45 6 equivalence of , 1 9 dissection, 467 , 48 2 coordinates transitive o n polygona l regions , of a poin t i n the Euclidea n plane , 484 109 distance of a poin t o n a line , 1 7 between point s o n a line , 1 8 Crossbar Theorem , 30 , 3 4 from poin t t o line , 6 1 crystallographic notation , standard , properties of , 1 8 455 triangle inequalit y for , 18-1 9 crystallographic restriction , 44 1 cyclic group , 35 4 ellipse, 31 4 definition, 35 0 equality modul o a number , 3 9 of infinit e order , 35 4 equidecomposable of orientation-preservin g a rectangl e an d a squar e o f th e isometries, 36 5 same are a are , 482-48 4 Desargues' Littl e Theorem , 77 , 8 3 a triangl e an d a parallelogra m diameter with th e sam e bas e an d hal f th e of a se t i n the plane , 50 8 height are , 47 5 of a square , 50 9 definition of , 48 2 of a triangle , 51 7 equivalence relatio n o n polygona l of a n equilatera l triangle , 50 9 regions, 48 4 properties of , 517-51 8 two parallelogram s wit h th e sam e relationship wit h radius , 511-51 2 base an d heigh t are , 47 4 dihedral group , 35 0 two polygona l region s o f the sam e a finite non-orientation-preservin g area are , 482-48 7 symmetry grou p i s a , 36 8 equivalence classes , 25 6 is a reflectio n group , 373-37 4 equivalence relatio n dilatation congruence i s an , 25 4 classification, 24 8 definition of , 25 , 25 3 definition of , 14 4 examples of , 25 6 isometric dilatations , relation o f ke y propertie s t o grou p classification, 20 5 properties, 258-25 9 must b e a similarity , 24 8 Erlanger Programm , 25 1 translation i s a , 14 4 for Euclidea n geometry , 276-28 5 uniform dilatio n i s a , 22 6 Euclidean geometr y dilation facto r is implicit i n th e structur e o f th e negative value , 22 8 Euclidean group , 276-28 5 of a composition , 21 8 Euclidean grou p of a similarity , 135 , 21 7 matrix realizatio n of , 280-28 3 dilation, uniform , see unifor m Euler line , 301-30 2 dilation contains th e circumcenter , dimension-exponent relationship , 52 2 orthocenter, centroi d an d directed angl e measure , 84-9 4 nine-point center , 30 1 is translation invariant , 91 , 14 5 Euler points , 303 , 30 6 transformation b y similarities , 23 8 Euler, Leonhar d directed lin e discovered th e Eule r line , 30 2 definition of , 2 4 Exterior Angl e Theorem , 5 6 Index 537

Fagnano's Problem , 313 , 315-32 2 contained i n a spli t friez e group , Fermat point , 322 , 32 9 407 Fermat Problem , 322-33 9 definition of , 40 0 solution of , 32 8 flow char t fo r classification , 41 2 statement of , 32 2 generators fo r a , 410-41 1 Fermat's Principl e isomorphism classes : onl y four , implies th e La w o f Reflection , 31 1 415 statement of , 31 0 midpoint restricte d subgrou p of , Fermat, Pierre , 310 , 32 2 408-410 point groups : onl y four , 40 3 Feuerbach points , 30 4 sample friez e pattern s for , 41 1 Feuerbach's Theorem , 30 5 split i f point grou p doe s no t Feuerbach, Kar l Wilhelm , 30 5 contain centra l reflection , 40 6 finite grou p frieze pattern s of isometrie s fixes a point , 36 4 complete set : seve n conjugac y of rotation s i s cyclic , 36 5 classes, 41 1 First Structur e Theorem , 156-160 , definition of , 35 0 165 example, 35 0 statement of , 15 9 from Anasaz i pottery , 41 3 fixed line s non-split groups : tw o conjugac y behavior unde r conjugation , 25 9 classes, 40 8 for isometries , 18 9 split groups : five conjugac y for similarities , 24 6 classes, 40 5 fixed point s fundamental domain , 331 , 37 3 behavior unde r conjugation , 25 9 for a translation group , 42 3 definition of , 154 , 15 6 existence fo r contractio n G-related, 36 1 mappings, 25 0 geometric equivalence , 25 1 and similarities , 266-27 5 for isometries , 18 9 with respec t t o a group , definitio n for similarities , 240-24 1 of, 26 7 for symmetr y group s o f bounde d geometric optics , 31 0 figures, 36 3 geometric propertie s strict similaritie s hav e unique , for Euclidea n geometry , 25 5 240-250 of isometries , 263-26 4 Fixed Point s an d Fixe d Lines , 18 9 of similarities, 27 4 fractal, 459 , 523-52 9 geometry dimension o f a , 52 6 early histor y of , 28 7 generator fo r a , 52 3 etymology of , 28 7 initiator fo r a , 52 3 glide axi s is a self-simila r figure, 52 5 and th e orthi c triangle , 31 6 frieze group , 375 , 39 9 41 5 glide lengt h central reflectio n o f a , 40 0 and th e orthi c triangle , 316-31 7 classification: seve n conjugac y glide reflection , 181-18 8 classes, 41 0 a generi c threefol d produc t o f conjugacy classe s o f non-spli t reflections i s a , 18 4 groups: onl y two , 408 , 41 3 and th e orthi c triangle , 18 7 conjugacy classe s o f spli t groups : as product o f a reflectio n an d a only five, 40 4 point inversion , 18 6 538 Index

axis of , 18 3 Heron's Formula , 47 7 definition of , 18 3 Hinge Theorem, 6 1 equality of , 18 4 homogeneous isotropi c medium , 31 0 glide axi s i s the onl y fixed lin e o f a hyperbolic geometry , 63 , 78 non-trivial, 18 6 has square equa l to a translation , identity mapping , 12 2 187 is an isometry , 13 5 non-trivial, 18 3 image o f a se t b y a mappin g (oriented) glid e length , 18 3 definition of , 13 0 specification b y data, 184 , 18 6 of a union, 13 4 standard form , 18 2 of an intersection , 13 4 glide reflectio n propert y fo r a n incenter, 292-29 3 inscribed triangle , 31 9 Incidence Axiom s global rotationa l direction , 84-8 6 examples, 8-1 7 great circle , 1 3 minimal example , 1 5 group statement of , 7 , 12 0 abelian, 20 0 incircle, 29 3 center of , 41 5 Inscribed Angl e Theore m cyclic, 350 , 354 , 36 5 and reflection s i n perpendicula r cyclic o f order m , definition , 35 0 bisectors, 29 8 dihedral, 35 0 statement of , 8 0 isomorphism, 27 9 intercepted arc , 46 5 normal subgroup , 204 , 23 8 measure of , 8 0 of symmetries, 20 4 of an angl e inscribe d i n a circle, 8 0 of transformations, 19 9 interior subgroup of , 20 0 of a triangular region , 46 8 group homomorphis m of an angle , 3 0 definition of , 205 , 37 7 of the domai n V o f standar d group isomorphism , 279 , 353, 439 lattices, 43 0 is an equivalenc e relation , 35 5 invariant propertie s group o f transformations, 165 , for subset s o f plane, 25 5 199-206 for transformation s o f plane, 26 3 as symmetry group , 20 4 invariant se t fo r a group, 356-357 , centralizer o f an elemen t o f a , 20 4 361 commutative, 19 9 inverse definition, 19 9 compute examples , 15 4 dilatation group , 20 5 definition of , 15 0 examples, 20 3 of a composite mapping , 152-15 3 intersection o f two, 20 3 of a similarity i s a similarity, 21 8 normal subgrou p of , 20 4 of an isometr y i s an isometry , 15 2 of the line , 20 5 invertible subgroup of , 19 9 equivalence wit h one-to-on e an d translations for m a commutative , onto, 15 1 199 involution, 24 7 isometric mappin g i s a half plan e transformation, 14 5 definition of , 2 8 isometries edge of , 2 8 form a group unde r composition , Heron o f Alexandria, 47 7 199 Index 539

form a normal subgrou p o f th e is define d fo r polygona l regions , similarities, 23 8 496 represented a s a matrix group , 28 0 is invariant unde r isometry , 50 2 isometry of a lin e segmen t i s zero , 496-49 8 as distance-preservin g mapping , properties of , 493-49 5 145 satisfies th e axiom s fo r a n are a as product s o f reflection , 15 9 function, 50 3 classification of , 18 8 with respec t t o a lattic e o f definition of , 13 5 squares, 49 2 examples, 135-13 9 First Structur e Theorem , 15 9 kaleidoscope, 369-37 2 fixed poin t classification , 15 7 Klein, Felix , 25 1 fixed point s of , 15 7 Koch curve , quadratic , 52 7 groups o f the line , 20 5 Koch snowflake , 523-52 6 inverse i s again a n isometry , 15 2 similarity dimensio n of , 52 6 of the line , 19 0 orthogonal extensio n of , 19 0 lattice, 38 2 preserved b y composition , 14 7 a genera l lattic e i s simila r t o a preserves geometri c objects , 139 , unique standard , 43 5 144 classification o f planar types , 439 , preserves hal f planes , 14 4 440 preserves parallelism , 14 4 (fat) rhombic , 433 , 43 9 preserves perpendicularity , 14 4 for discret e plana r translatio n properties of , 139 , 14 4 group, definitio n of , 41 7 type i s specifie d b y fixed point s for discret e plana r translatio n and fixed lines , 18 9 groups, 416-43 8 isomorphism o f groups , 27 9 hexagonal, 431 , 433, 43 9 isoperimetric theorem , 52 0 invariant unde r a give n poin t isosceles triangl e group, 44 2 equivalent condition s for , 292 , 29 7 invariant unde r a reflection , symmetry of , 29 1 425-429 invariant unde r a reflectio n i s Jordan measurabl e set s rectangular o r rhombic , 42 6 all disk s an d sector s o f disk s are , (long) rhombic , 433 , 43 9 496 oblique, 433 , 43 9 all polygona l region s are , 49 6 rectangular, 425 , 43 1 closed unde r union , intersection , rhombic, 425 , 43 1 difference, 49 3 rhombic extensio n o f a from inne r an d oute r Jorda n rectangular, 428-429 , 43 7 measure, 49 3 square, 431 , 433, 43 9 independent o f choic e o f squar e standard, 422-42 8 lattice, 50 1 strict rectangular , 433 , 43 9 with respec t t o a fixed squar e symmetries of , 429-43 8 lattice, 49 2 symmetry classe s of , 433 , 43 9 Jordan measure , 490-50 5 with symmetr y o n th e boundar y behavior unde r similarities , of V, 43 2 506-507 lattice o f squares , 49 1 independent o f choic e o f squar e lattice poin t group , 42 9 lattice, 50 1 length, 45 9 540 Index

Leonardo's Theorem , 36 9 Napoleon translatio n group , 334 , 36 2 line Napoleon triangl e intersection wit h a circle , 11 4 inner, 33 7 isometries of , 19 0 Napoleon's Theorem , 216 , 33 0 line segmen t natural numbers , 1 definition of , 2 3 nine poin t cente r directed, definitio n of , 2 4 is the midpoin t o f the Eule r line , has equa l directio n wit h another , 301 124 nine poin t circle , 29 5 perpendicular bisecto r of , is ak a th e Feuerbach circle, 30 5 definition an d characterization , is tangent t o th e incircl e an d th e 55 excircles, 30 4 line segment s is the circumcircl e o f the media l congruence for , 2 5 triangle, 30 2 Line-Circle Theorem , 11 4 is the circumcircl e o f the orthi c linear pair , 3 5 triangle, 30 3 three-dimensional interpretation , Mandelbrot, Benoit , 52 3 308-309 mapping non-Euclidean geometry , 7 8 composition, 14 6 definition of , 12 2 one-to-one correspondence , 1 7 equality of , 12 2 one-to-one mappin g fixed point s of , 15 4 definition of , 13 0 identity, 12 2 preserved b y composition , 14 7 inverse mapping , 15 0 onto mappin g one-to-one, definitio n of , 13 0 definition of , 13 1 onto, definitio n of , 13 1 preserved b y composition , 14 7 measures o f shape , 511-513 , 51 9 orbit medial triangle , 295 , 298-299 , 30 6 for a group, 36 1 a triangl e i s simila r t o its , 29 9 for a subgroup , 36 2 median, 29 3 of p i s the smalles t G-invarian t se t midpoint, 2 3 containing p , 35 8 minimal translatio n displacement , of a point , fo r a group , 357-35 9 399 properties of , 35 8 Mirror Principle , 31 3 orientation, see also parity , 191-19 9 Mobius band , 84-8 6 and composition , 195 , 23 6 Moulton plane , 1 7 of a similarity , 23 2 and Rule r Axiom , 2 1 of a n isometry , 19 4 incidence in , 1 3 orientation-preserving multisquare regio n subgroup o f a symmetry group , definition of , 49 0 359-360 inner an d oute r approximatio n by , origin, choic e of , 1 490-492 ornamental group , 37 5 orthic triangl e Napoleon symmetr y group , 335 , 36 2 and Fagnano' s Problem , 315-32 2 Napoleon tesselation , 330-337 , 362 , angles o f the, 32 1 456 characterizations o f the , 31 9 fundamental domai n of , 33 2 definition of , 187 , 297 , 31 5 special, 33 8 solves Fagnano' s Problem , 31 8 Index 541 orthocenter, 292 , 29 6 diagonals, definitio n of , 7 3 is the incente r o f the orthi c equality o f opposite sides , 8 2 triangle, 31 8 have diagonal s that bisec t eac h lies on the Eule r line , 30 1 other, 8 1 orthogonal extensio n is convex, 8 1 of isometries o f the line , 19 0 opposite side s o f are congruent , 7 4 orthogonal projectio n Parallelogram Constructio n to a line , definition of , 129-13 0 Theorem, 7 6 overlap, 46 8 Parallelogram Existenc e Theorem , 75 p-component Parallelogram Uniquenes s Theorem , of a plane isometry , 37 6 74 of an isometry , definitio n of , 21 0 parity, see also orientatio n p-factorization and conjugatio n o f a rotation, 18 0 defines a group homomorphism , and directe d angl e measure, 18 0 377 behavior unde r compositio n o f of a plane isometry , 37 6 isometries, 19 5 pairwise adjustmen t o f reflections , of a similarity, 23 2 174 of an isometry , 180 , 194 , 19 7 parallel of products o f similarities, 23 6 line segments, definitio n of , 6 2 perpendicular bisector , 5 5 line through a point, existenc e of , perpendicular lin e 63 lines must b e perpendicular t o th e as shortest distanc e fro m poin t t o same lines , 73 , 78 line, 6 0 lines perpendicular t o sam e lin e definition, 3 7 are, 6 2 existence an d uniqueness , 4 9 lines, definition , 6 2 planar lattice , 37 5 lines, distance between , 7 9 plane geometr y axio m syste m lines, transitivity, 7 2 undefined object s for , 7 Parallel Postulate , 64 , 66, 69-8 4 Plane Separatio n Axiom , 27-3 4 and Rectangl e Hypothesis , 7 9 statement of , 28 , 12 0 and Theore m o f Thales, 8 0 Poincare disk , 15 , 63 and Triangl e Su m Hypothesis , 6 8 falsity o f Parallel Postulat e in , 7 7 implies constanc y o f interior angl e incidence in , 1 0 sum fo r triangles , 7 1 point group , 375-37 9 implies equa l alternative angles , 7 1 at differen t point s ar e conjugate , implies transitivity o f parallelism , 379 67 compatibility wit h translatio n independence of , 7 7 orbit, 38 6 statement of , 70 , 12 0 compatibility wit h translatio n parallelism, 62-7 0 subgroup, 38 1 parallelogram definition of , 37 7 analog o f Pythagorean Theore m examples of , 37 8 for diagonals , 9 8 of a friez e grou p i s Ci, C2 , Di, o r as conve x quadrilateral, 8 1 D2, 40 3 definition, 7 3 of a wallpaper group , 440-44 3 Desargues' Littl e Theorem , 7 7 representative o f an elemen t of , diagonals bisect , 7 4 377 542 Index point inversio n ray definition of , 12 5 definition of , 2 3 equality of , 12 5 opposite t o give n ray , 2 3 equals a half-turn , 12 7 translates, 8 6 group theoreti c characterizatio n translates fro m translations , 14 2 of, 279 , 28 3 Ray Separatio n Theorem , 8 3 is an isometry , 13 7 real Cartesia n 3-spac e polygonal regio n incidence in , 1 4 area of , 47 0 real Cartesia n plane , 8 convex, 48 0 and Rule r Axiom , 2 0 definition of , 256 , 46 7 distance on , 1 9 is trangulable, 48 1 truth o f Paralle l Postulat e in , 7 7 non-overlapping, 46 8 real Cartesia n space , 1 7 preserved b y union s an d real number s intersections, 47 0 and relate d numbe r systems , 1 Pons Asinorum, 48 , 53 , 28 8 Archimedean orderin g principle , 2 and isoscele s triangles , 29 0 axioms for , 6 products o f tw o reflections , Bolzano's Theorem , 3-4 , 11 5 redundancy of , 17 4 bounded sequence s of , 3 , 11 5 Pythagoras, 28 7 completeness of , 2- 3 Pythagorean Theore m convergent sequence s of , 3 , 11 5 analog fo r diagonal s o f a coordinatizing a lin e with , 1 , 1 7 parallelogram, 98 , 10 7 decimal expansion s of , 6 converse of , 10 7 open an d close d interval s in , 2 Euclid's proo f o f the, 47 9 separation b y rationa l numbers , 4 , proof b y similarity , 9 7 103 proof usin g are a o f squares , 48 0 standard algebrai c propertie s of , 2 proof usin g similarit y scalin g o f real projectiv e plane , 1 7 area, 513-51 5 incidence in , 1 2 rectangle quadrilateral and Saccher i quadrilaterals , 7 3 convex, 3 0 as parallelogram, 8 2 cyclic, 323-32 5 definition of , 6 6 definition of , 3 0 Rectangle Hypothesis , equivalenc e t o diagonals of , 3 0 Parallel Postulate , 7 9 Saccheri, 66-68 , 7 3 reflection Varignon's Theorem , 10 7 of light , 309-31 5 with opposit e angl e supplementar y reflection group , 369 , 415 , 45 8 is cyclic , 32 3 definition of , 37 4 reflection i n a lin e radius definition of , 12 3 of a se t i n th e plane , 50 8 equality of , 12 3 of a square , 50 9 group theoreti c characterizatio n of a n equilatera l triangle , 50 9 of, 279 , 28 3 relationship wit h diameter , is an isometry , 135-137 , 14 3 511-512 reflection propert y fo r a n inscribe d rational numbers , 1 triangle, 31 6 incompleteness of , 5 refraction, 31 1 Index 543 rhombic extensio n o f a rectangula r semidirect product , 377 , 39 8 lattice, 428-42 9 sets rhombus, 8 2 equality of , 2 4 fat, 43 2 shear paralle l t o a line , definitio n of , long, 43 2 128-129 rosette group , 37 5 shortest translatio n i n a translatio n rotation group, 42 4 around a point , throug h a directe d Sierpinski's carpet , 52 8 angle, 12 6 similarities characterization a s produc t o f classification of , 24 2 reflections, 17 2 fixing a poin t an d a line , 22 2 equality of , 12 7 fixing a poin t for m a group , 23 4 is a n isometry , 13 8 form a group , 21 8 non-trivial, 12 6 groups of , 24 7 pairwise adjustmen t o f reflections , mapping a give n poin t t o a give n 172, 17 4 point, 24 6 rotation angle , 165 , 20 8 mapping a segmen t t o a segment , additivity unde r composition , 212 , 238 238 of a line , 234 , 24 8 characterization i n term s o f a lin e the composit e o f tw o i s another , and it s image , 214-215 , 23 9 218 characterization o f translations i n similarity, 217-25 3 terms of , 21 2 as a distanc e rati o preservin g effect o f conjugation , 26 4 for a n eve n isometry , 211 , 214-21 5 mapping, 23 0 for a n eve n similarity , 234 , 237 , as a transformatio n preservin g lines an d preservin g circles , 22 3 239 not possibl e fo r od d isometry , 21 5 as a n equa l distanc e preservin g tells whe n a product o f reflection s transformation, 22 2 is a translation , 21 3 as geometri c equivalenc e unde r rotational direction , 8 4 similarities, 27 1 Ruler Axiom , statemen t of , 17 , 12 0 as product o f a unifor m dilatio n Ruler Placemen t Theorem , 1 9 and a n isometry , 23 1 criteria for , 27 2 SAA (Side-Angle-Angle) , 6 0 definition of , 21 7 Saccheri quadrilateral , 66-68 , 7 9 dilation factor , 135 , 21 7 definition of , 6 6 equivalence o f transformationa l Saccheri quadrilateral s and forma l definitions , fo r are rectangle s unde r Paralle l triangles, 26 9 Postulate, 7 3 fixed point s o f a , 240-25 0 SAS (Side-Angle-Side ) Axio m inverse o f on e i s another, 21 8 independence o f previou s axioms , is a transformation , 21 7 54 is a n equivalenc e relation , 27 1 statement of , 47 , 12 0 is determined b y it s actio n o n SAS Similarit y Theorem , 10 7 three points , 23 3 Second Structur e Theorem , 206-21 1 of triangles , 9 4 statement of , 20 7 parity of , 23 2 Segment Constructio n Theorem , 25 , preserves geometri c shapes , 219 , 27 222 544 Index

preserves parallelism , 22 2 strain orthogona l t o a line, definitio n preserves perpendicularity , 22 2 of, 12 8 properties of , 219 , 22 2 stretch reflection , definitio n of , 24 1 rotation angl e of , 233-23 7 stretch rotation , definitio n of , 24 1 strict, definitio n of , 24 1 Structure Theore m fo r Similarities , strict, structur e o f a , 24 2 232-233 structure o f a , 231-23 5 Structure Theore m fo r Stric t Structure Theore m fo r Similarities, 24 1 Similarities, 23 2 superposition Structure Theore m fo r Stric t and symmetry , 28 9 Similarities, 24 1 symmetry, 347-45 8 transformational definitio n of , 26 8 of a figure, definition , 34 8 with dilatio n factor , definitio n of , symmetry grou p 135 equivalence, 35 2 Similarity an d Are a Principle , 50 6 finite; Leonardo' s Theorem , 36 9 similarity dimension , 459 , 52 6 of a bounded figure, 363-37 4 similarity extensio n of a bounded figure fixes a point , from lin e to plane , 23 5 363 Similarity Fixe d Poin t Theore m of a figure, definition , 204 , 34 9 analytic proo f of , 25 0 symmetry type , 354-35 5 proof of , 242-246 , 24 9 classification of , 351-35 3 statement of , 24 1 of convex quadrilaterals, 37 2 Similarity Principle , Basic , 9 9 Similarity Theore m Taylor circle , 33 9 proof, 98-10 6 tesselation statement of , 95 , 224 Napoleon, 330-337 , 362 , 45 6 Similarity/Dilation Theorem , 23 1 of plane b y parallelograms , 42 3 sine regular plane , 33 7 addition formul a for , 47 8 Thales split a t p, 38 8 of Miletus, 28 7 split group , 387-39 3 Theorem of , 80 , 28 8 a wallpape r grou p with a rhombi c theorems attribute d to , 28 7 lattice i s a , 44 8 Theorem o f Thales, 80 , 28 8 classification, 390-39 3 equivalence t o Paralle l Postulate , conjugacy, 392 , 39 6 80 construction, 39 0 Three Circl e Theorem , 30 7 decomposition int o point grou p and th e nin e poin t circle , 30 8 and translatio n subgroup , 38 7 Three Reflection s Theorem , see Firs t example, 38 8 Structure Theore m recognition, 38 9 transformation split i n G , 38 8 criteria fo r invertibility , 15 1 splitting point , 39 3 definition of , 13 1 splitting set , 39 4 Euclidean geometri c propertie s of , square roo t 263 of a translation, 43 6 examples, 121-13 5 SSA (Side-Side-Angle ) conjecture , 6 0 means a n invertibl e mapping , 15 1 SSS (Side-Side-Side) , 49 , 5 3 transformational descriptio n o f SSS Similarity Theorem , 10 7 geometric conditions , 29 3 Index 545 transitivity definition of , 38 0 of parallelism, 7 2 is in bijectio n wit h an y translatio n of ray translation, 88 , 93 orbit, 38 3 translation of a friez e grou p i s infinite cyclic , characterization a s product o f 401 point inversions , 17 5 there i s only on e conjugacy clas s characterization a s product o f of friez e group , 40 2 reflections, 16 7 transversal lin e collection form s abelia n group , 20 0 alternate interio r angle s for , 6 4 composition o f two yield s a definition of , 6 4 translation, 176 , 17 8 equal alternat e interio r angle s fo r criteria fo r equality , 124 , 14 4 guarantees parallelism , 6 5 definition of , 12 4 interior angle s for , 6 4 displacement of , 166 , 18 3 triangle equivalence o f directed lin e AA Similarit y Condition , 9 6 segments under , 14 2 AAA Similarit y Condition , 9 6 equivalence o f rays under , 14 2 "all ar e isosceles " (?!?) , 10 8 is a product o f uniform dilation s altitude of , 10 6 with reciproca l dilatio n factors , ASA Congruenc e Condition , 4 8 230 congruence via isometries, 161 , 252 is an isometry , 137 , 14 4 congruence wit h another , 4 6 is the squar e o f a glide reflection , constancy o f interior angl e sum, 7 1 187 definition of , 2 4 non-trivial, 12 4 equilateral, 48 , 53 of directed angl e measure , 91-9 3 equilateral, transformationa l of directed angles , 89-9 0 characterization, 215-21 6 of rays, 86-8 8 exterior angl e of , 5 6 pairwise adjustmen t o f reflections , filled, 46 7 167, 17 4 inscribed i n a triangle, 31 9 parallel o r perpendicular t o a line, interior of , 3 2 436 is convex, 3 3 parallel t o a segment, definitio n of , isosceles, 48 , 29 1 123 mirror propert y fo r a n inscribed , square roo t o f a , 43 6 313, 31 6 transitivity o f ray, 88 Pythagorean Theorem , 9 7 translation displacemen t remote interio r angle s of , 5 6 minimal, fo r a pattern, 39 9 SAA Congruenc e Criterion , 6 0 translation grou p SAS Congruenc e Condition , 4 7 basis for , 417-42 2 SAS Similarit y Condition , 10 7 definition o f discrete planar , 41 7 similarity, definitio n of , 9 4 translation orbit , 382-38 7 similarity, vi a similarit y compatibility wit h poin t group , transformations, 26 9 386 SSA Congruenc e Criterio n fo r definition of , 38 3 obtuse, 6 0 properties of , 38 3 SSS Congruence Condition , 49 , 5 3 translation subgroup , 375 , 380-38 2 SSS Similarity Condition , 10 7 compatibility wit h poin t group , with give n sides , 11 0 381 Triangle Inequality , 5 9 546 Index triangle inequalit y unit circle , 10 9 independence o f Incidence an d unit o f length, 1 Ruler Axioms , 2 1 strict, 5 8 Varignon's Theorem , 10 7 Triangle Isometr y Theorem , 16 1 vertex sum , 32 5 triangle similarity , 21 7 Vertical Angl e Theorem , 36 , 28 8 Triangle Su m Hypothesis , 6 8 vertical pai r o f angles , 3 6 implies th e Paralle l Postulate , 6 9 virtual parallelopipe d o f the nin e Triangle Theorem , 11 0 point circle , 30 9 triangular region , 46 7 volume, 521-52 2 a unio n o f two ca n b e wallpaper grou p triangulated, 48 0 automatic splittin g classes , 44 8 an intersectio n o f tw o ca n b e classification o f planar lattic e triangulated, 46 9 types, 439-44 0 boundary, 46 8 classification: seventee n classes , degenerate, 46 7 454-455 interior, 46 8 contained i n a spli t wallpape r triangulation, 46 8 group, 44 9 refinement of , 46 8 non-split groups : fou r classes , trigonometric functions , 98 , 10 8 450-451 trigonometry wit h directe d angles , ornamental groups , 37 5 108-110 point group s an d lattic e pairs , 44 2 Two Circl e Theorem , 11 2 point groups : onl y ten , 44 1 uniform dilation , 21 7 seventeen isomorphis m classes , 45 8 commutes wit h reflectio n i n lin e split groups : thirtee n classes , 44 4 containing fixed point , 23 0 wallpaper pattern s composition o f two unifor m definition of , 41 6 dilations, 226 , 23 9 example, 35 0 conjugation b y a n isometr y give s a non-split groups : fou r classes , uniform dilation , 23 5 450-451 definition of , 12 7 split groups : thirtee n classes , equality of , 12 8 444-445 is a dilatation , 22 6 is a similarity , 22 4 properties of , 226 , 22 9 signed, 22 8 terminology, 22 9