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Continuou s Symmetr y Fro m Eucli d to Klei n
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Continuou s Symmetr y Fro m Eucli d to Klei n
Willia m Barke r Roge r How e
>AMS AMERICAN MATHEMATICA L SOCIET Y
http://dx.doi.org/10.1090/mbk/047
Freehand® i s a registere d t r ademar k o f Adob e System s Incorpora te d in th e Unite d Sta te s a n d / o r othe r countries .
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2000 Mathematics Subject Classification. P r imar y 51-01 , 20-01 .
For addit iona l informatio n an d upda te s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / m b k - 4 7
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 . Grou p theory . 3 . Symmetr y groups . I . Howe , Roger , 1945 -
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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 th e Euclidea n Plan e 11 9
II. Isometrie s i n th e Plane : Product s o f Reflection s ILL Transformation s i n th e Plan e 12 1 11.2. Isometrie s i n th e 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 th e 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 Transformation s 19 9 111.6. Th e Secon d Structur e Theore m 20 6 111.7. Rotatio n Angle s 21 1
Vll l Contents
IV. Similarit ie s i n t h e P l a n e IV. 1. Elementar y Propertie s o f Similaritie s 21 7 IV.2. Dilation s a s Similaritie s 22 4 IV.3. Th e Structur e o f Similaritie s 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 Geometr i 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. Appl icat ion s t o P l a n e G e o m e t r y VI. 1. Symmetr y i n Earl y Geometr y 28 7 VI.2. Th e Classica l Coincidence s 29 2 VI.3. Dilatio n b y Minu s Tw o 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 Apolloniu s 34 0
VII. Symmetr i c Figure s i n t h e P l a n 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 Wal lpape 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 Scal in g IX. 1. Lengt h o f Curve s 45 9 IX.2. Are a o f Polygona l Regions : Basi c Propertie s 46 7 IX.3. Are a an d Equidecomposabilit y 48 2 IX.4. Are a b y Approximatio n 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 tex t 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 an d a t the same time takes Feli x Klein' s Erlange r Program m seriously . Th e pri -mary focu s i s on transformation s o f th e 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 detai l th e concret e propertie s o f transformations as geometric objects; on the othe r hand , w e try t o sho w 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 trie d t o sho w th e intimat e relationshi p betwee n Fag-nano 's Problem (inscrib e in a given triangle a triangle o f minimal perimeter ) and th e proble m o f computing th e produc t o f three reflections . (Thi s latte r problem i s 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 th e concurrency o f the altitude s o f a triangle usin g only reflections , no t similar -ities a s does the traditiona l proof . A s a consequence, on e can late r conclud e that th e concurrenc y o f altitudes hold s equally wel l in elliptic geometr y 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 use d i n showin g tha t ever y stric t similarit y transformation ha s a fixed point an d th e computation o f the produc t o f two rotations i s interpreted i n terms o f traditiona l geometry .
The Erlanger Programm i s often treate d a s a component o f the large r 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 i t constantl y throughou t th e subsequen t material ; i n particular , i t play s a centra l rol e i n Chapter s VI I an d VIII . Bu t 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 wit h grou p theor y i n a subsequen t abstrac t algebr a cours e easier an d mor e meaningful .
ix
X Instructor Prefac e
We hav e expende d considerabl e effor t t o giv e a self-containe d developmen t and t o mak e explanation s a s clea r a s possible . Ou r student s hav e found th e text t o b e quit e readable ; w e hope th e sam e wil l b e tru e 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, discussin g different geometrie s — affine an d non-Euclidean (hy -perbolic an d spherical/elliptical ) — in a projective setting , an d endin g wit h an interpretatio n o f Einstein' s Special Theory of Relativity a s a n analo g i n higher dimension s of hyperbolic plane geometry. Unti l the completion o f the second volume we hope this text wil l 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 natura l plac e i n geometry .
This boo k ca n suppor t severa l differen t type s o f courses . Chapte r I gives a fairly complet e axiomati c 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 somethin g 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 transformationa l viewpoin t wit h geometr y 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 t o V . Differen t choice s o f materia l fro m thes e chapter s wil l resul t 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 transformationa l approac h t o establis h 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 th e rol e of the media l triangl e (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 simila r t o th e origina l triangl e b y a similarit y tha t stretche s b y 2 an d 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 t o th e nine-poin t circle . W e attemp t t o revea l the dept h o f thi s remarkabl e configuration . Severa l othe r topic s whic h ca n be approache d 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 hav e attempte d t o
Instructor Prefac e XI
give a carefu l treatment . Especiall y i n the cas e o f the wallpape r groups , we hope th e ingredient s tha t g o int o 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 spli t extensio n i s the basi c tool , bu t w e trea t i t in a concrete fashion . I n particular , thi s concep t help s t o organiz e bot h th e classification an d it s justification .
Chapter I X studie s scalin g an d dimensio n 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 exoti c shapes . Th e treatmen t i s relatively abbrevi -ated an d provide s man y opportunities fo r student s t o fill in the detail s (wit h generous hints) .
Throughout th e boo k th e approac h i s concrete, an d w e try t o give complet e explanations. Th e nee d fo r justification an d proo f i s taken fo r granted , an d there ar e many opportunitie s fo r student s to construc t 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 t o mathematica l reasoning .
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Student Prefac e
This boo k i s about Euclidea n geometry , th e sam e subjec t yo u studie d i n hig h school. However , th e viewpoin t i s probabl y ver y different . Th e goa l o f thi s text i s to presen t geometr y i n a wa y tha t honor s th e idea s o f transformatio n and symmetr y tha t hav e s o profoundl y 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 thumbnai l sketc h o f its historica l roots .
Geometry a s a deductiv e scienc e wa s invente d b y th e Greeks . Thei r wor k o n the subject , th e wor k o f man y thinker s ove r severa l centuries , wa s collecte d and wove n togethe r brilliantl y b y Eucli d o f Alexandri a aroun d 30 0 B.C . Fo r nearly tw o thousan d years , Euclid' s Elements , expoundin g Gree k geometry , was th e hig h poin t o f huma n 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 th e sam e sid e les s tha n tw o righ t angles , th e tw o straigh t lines, i f produced indefinitely , mee t o n tha t sid e o n whic h th e angle s are les s tha n 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 th e 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 th e angle s ar e clos e t o 90° , th e 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) .
xii i
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 littl e 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 intersectio n point , w e can neve r rul e ou t tha t suc h a poin t exist s "jus t a little further " alon g th e lines .
So people who were inclined t o worry abou t suc h logica l an d 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 i t b y turning i t int o a theorem, i.e. , they tried t o show it followed a s a logica l necessit y fro m th e first fou r postulates . However , n o one succeede d 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 tha t do , in fact , follo w fro m denyin g th e 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 th e 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 traditiona l meanin g of geometr y coul d no t encompas s th e ne w wealt h o f phenomena . A revise d understanding o f the natur e o f geometry wa s urgentl y needed .
The basis for thi s new understanding came from a subject invente d a t nearl y the sam e tim e a s non-Euclidea n geometr y — th e 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 attracte d muc h attentio n — th e proble m of givin g formula s fo r th e solution s o f polynomia l 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' idea s wer e hard fo r hi s contemporar y 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 relevanc e o f group theor y t o geometr y bega n t o grow , wit h group s arisin g from symmetries — transformation s o f a syste m whic h preserv e th e essentia l characteristics o f th e system .
The semina l connectio n betwee n geometr y an d grou p theor y 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 P r o g r a m m . 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 Euclidea n 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 tw o 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 th e geometry . (I n Euclid' s Elements, thi s ide a wa s 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 thos e 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 bu t admirabl y distinguished .
3 The "r " o n th e en d o f "Erlanger " i s no t a misprint . It' s ho w Germa n gramma r works .
XVI Student Prefac e
(i) th e se t o f symmetrie s forme d a grou p i n th e sens e o f Galoi s an d
(ii) yo u ca n reconstruc t th e geometr y fro m it s symmetr y group. 4
In short , th e fundamenta l 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 it s 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 associate d geometrie s ar e stil l onl y partiall y understood .
Through th e en d o f th e nineteent h centur y thi s wa s al l pure mathematics , inspired b y nagging question s 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 Universit y o f Koenigsberg an d wa s 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 th e 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 a n exaggeratio n t o sa y tha t symmetr y an d group s hav e bee n a dominan t theme i n modern physics . I n particular , th e structur e 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 theor y an d symmetry , whic h wer e first in -troduced i n response to question s abou t 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 dramati c 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 — the y wer e no t eve n goin g t o produc e ne w theorems , onl y tid y u p th e syste m
Student Prefac e xvn
Our book takes Klein's Erlanger Programm seriously , while still retaining th e flavor of a classica l stud y o f geometry i n which triangles , circles , quadrilat -erals, an d othe r simpl e shapes ar e the primar y object s o f investigation. Th e study of transformations i s integrated wit h serious attention t o the beautifu l results o f syntheti c geometr y W e do thi s i n tw o 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 th e 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 thi s 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 a n abstract , orga -nizational viewpoint , bu t a s a practical approac h tha t enhance s you r under -standing an d highlight s th e beaut y o f thi s timeles s subject .
Above all , w e hope yo u enjo y th e journey yo u ar e abou t 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 item s with thos e label s 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 reverberation s i n mathematics , science , an d philosoph y tha t hav e shape d an d continu e t o shape th e natur e o f ou r thinking .
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Acknowledgments
When a book has been in preparation fo r eleven years, there are many peopl e who have contributed helpfu l advic e an d wh o have supported th e author s i n numerous importan t ways . Thoug h w e can lis t onl y a small numbe r o f them — students, colleagues , friends , an d famil y — they hav e al l 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 th e fal l o f 199 6 with fundin g fro m th e Nationa l Scienc e Foun -dation. W e thank th e NS F fo r th e encouragemen t an d suppor t the y gav e t o the projec t durin g that formativ e initia l perio d (NS F Gran t DUE-9555134) .
Jim Rei d an d Rober t Rosenbau m o f Wesleya n Universit y wer e highl y in -volved th e firs t year , attendin g th e class an d 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 th e year s an d ha s contribute d man y valuabl e suggestion s tha t have entered ou r expositio n i n significant ways ; hi s unwavering suppor t an d advice hav e bee n deepl y appreciated .
We als o than k Thoma s Berge r o f Colb y Colleg e an d Anit a Sale m o f Rock -hurst College , bot h o f whom taugh t course s fro m earl y version s o f our man -uscript o n multipl e occasion s — it wa s reassuring t o hav e confirmation tha t the manuscrip t worke d outsid e o f th e 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 of the Universit y o f Washington fo r sharin g thei r insight s 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 Duk e 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 th e 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 taugh t Continuous Symmetry ove r th e pas t te n year s a t bot h Yal e and Bowdoi n — we gratefully than k al l our man y student s fo r th e insightfu l comments the y gave , fo r th e carefu l proofreadin g the y provided , an d fo r th e
xix
X X Acknowledgments
patience they displaye d i n dealing with a sometimes roug h work-in-progress . Their enthusias m fo r the materia l wa s a great suppor t durin g the long hour s spent preparin g th e manuscript . W e cannot lis t the m all , bu t a few 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 th e 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 tha t le d t o improvement s i n th e text . W e ar e grateful fo r thei r insight , help , an d support .
We als o wis h t o than k Ree d Hastings , Bowdoin , 1983 , whose generou s 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 th e 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 — the y wer e generous with thei r tim e an d accurat e wit h thei r advice . Arlen e O'Sean , th e AMS Cop y Editor , wa s insightfu l an d meticulou s i n he r excellen t editin g o f our manuscript. W e are particularly thankfu l t o AMS Editors Ed Dunne an d Ina Mett e fo r thei r unfailin g encouragemen t an d suppor t fo r thi s project , their gentl e reminder s abou t deadlines , thei r 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 th e productio n o f thi s 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 a t Bowdoi n spen t man y hours these past year s helping the author s cope with al l the administrativ e issue s involved wit h thi s writing project , al l with efficiency , skill , an d goo d humor .
Finally, the greates t thank s 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 t o 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 use d t o dra w al l th e othe r figures .
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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. , an d 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 America n 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
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Index
AAA (Angle-Angle-Angle ) Thales's statemen t of , 28 9
AAA an d SS S criteri a fo r similarity , 270
altitude, 106 , 296 angle
acute, 3 7 central, 80 , 465 complementary, 3 7 congruence, 3 6 corresponding, 7 8 definition of , 2 4 directed, definitio n of , 24-2 5 equal modul o 360 , 39 exterior, o f triangle , 56 , 80 initial ra y of , 3 9 inscribed i n a circle , 80-8 1 intercepted ar c o f a n inscribed , 8 0 interior, 3 0 linear pair , 3 5 measure, 3 5 obtuse, 3 7 of incidence , 31 2 of reflection, 31 2 opposite a side o f a triangle , 3 3 remote interior , o f triangle , 5 6 right, 3 7 straight, 3 9 sum o f angle s o f triangle , 7 1 supplementary, 3 5 terminal ra y of , 3 9 trivial, 3 9 vertical pair , 3 6
Angle Addition , 3 5 angle bisecto r
as locus o f points equidistan t fro m sides, 6 1
definition of , 3 6 existence an d uniquenes s of , 3 7
Angle Bisecto r Theorem , 34 0
Angle Construction , 3 5 angle measure , 3 5
directed, 38-4 5 Angle Measur e Axioms , 34-4 5
statement of , 35 , 120 antipodal point , 1 2 Appolonius o f Perga , 34 0 Archimedean Order , 2 area, 45 9
computation vi a Jorda n measure , 490-505
Euclid's approach , 475-47 6 formulas fo r familie s o f simila r
figures, 507 , 515-51 6 of a disk , 488-49 0 of a parallelogram , formul a for ,
473 of a rectangle , formul a for , 47 3 of a secto r o f a disk , 49 0 of a square , formul a for , 471-47 2 of a trapezoid , 47 7 of a triangl e
in term s o f ASA, 47 8 in term s o f base an d height , 47 4 in term s o f SAS , 478 in term s o f SSS , 477
of polygonal regions , 467-48 7 scales unde r dilatio n wit h th e
square o f the dilatio n factor , 489, 50 6
area formulas , 45 9 area functio n o n polygona l region s
axioms fo r an , 47 0 constructed fro m Jorda n measure ,
503 existence an d uniquenes s of , 47 0 uniqueness vi a dissection , 47 1
ASA (Angle-Side-Angle ) statement of , 4 8 Thales's statemen t of , 28 8
533
534 Index
axiom syste m axioms o r assumption s in , 6 model for , 7 undefined objec t in , 6
axiomatic method , 6 Axioms fo r Euclidea n Geometry ,
collected, 119-12 0
Basic Similarit y Principle , 9 9 basis fo r discret e plana r translatio n
group, 417-42 2 definition of , 41 8 existence of , 418 , 43 8
betweenness, 2 2 for thre e point s o n a line , 2 2
billiards, 32 2 Bisector/Fixed Poin t Relation , 15 6 Bolzano's Theorem , 115-11 9
for Euclidea n plane , 11 8 boundary
of a tr iangula r region , 46 8 bounded figure, 35 4
Cantor set , 52 6 and bas e b expansions, 52 9 generalized, 52 8 unsymmetric, 52 9
center o f a group , 41 5 centers o f similitude , 230 , 239-240 ,
344 centralizer
of a transformation , 204 , 22 9 centrally symmetri c figure, 511-51 2 centroid, 292 , 294-29 6
a triangl e an d it s media l triangl e have th e same , 30 0
aka cente r o f gravity , 29 5 dilation b y — 2 around the ,
298-309 divides eac h media n i n a rati o o f
1:2, 29 5 lies o n th e Eule r line , 30 1
change o f scal e change o f lengt h unde r a , 46 2
chirality, 30 9 circle, 8 0
arc o f a , 45 9 center of , 56 , 11 2 central angl e of , 8 0
definition of , 55 , 11 2 diameter of , 56 , 11 2 inscribed angl e of , 8 0 intercepted ar c of , 8 0 intersection criterion , 11 4 intersection wit h a line , 11 4 length o f a n ar c o f a , 464-46 5 line tangen t to , 11 4 Line-Circle Theorem , 11 4 of Appolonius , 340-34 6 orthogonal, 34 4 point o f contac t o f tangen t line ,
114 possible intersectio n wit h another ,
112 radius of , 56 , 11 2 radius segmen t of , 11 4 Theorem o f Thales , 8 0 Two Circl e Theorem , 11 2
circumcenter, 292-29 3 lies o n th e Eule r line , 30 1 of a triangl e i s th e orthocente r o f
the media l triangle , 30 1 circumcircle, 29 3 circumference
arbitrarily larg e relativ e t o diameter, 51 3
of a circle , 459-46 4 of a se t i n th e plane , 51 0
classical coincidences , 292-29 8 Classification o f Plan e Isometries ,
188 classification procedur e fo r
symmetry groups , 393 , 40 0 Classification Theore m fo r
Similarities, 24 2 closed region , 46 8 closed se t i n th e plane , 50 7 coincidence i n a triangl e
of al t i tude s (orthocenter) ,
300-301, 31 8 of angl e bisector s (incenter) ,
293-294 of median s (centroid) , 294-29 5 of perpendicula r bisector s
(circumcenter), 292-29 3 collinear, 18 , 2 2
Index 535
commutativity
and fixed points , 15 4 examples, 15 5
compatibility
of poin t grou p an d translatio n orbit, 386-38 7
of poin t group s an d translatio n subgroup, 38 1
completeness of Euclidea n plane , 11 7 of rea l numbers , 2- 3
composition
definition of , 14 6
inversion of , 15 3
is a n algebrai c operation , 14 8
is associative , 14 9
is no t commutative , 14 8 juxtaposit ion notatio n for , 15 0
of a rotatio n an d a reflection , 17 8
of a rotat io n an d a translation , 201, 26 6
of a n eve n numbe r o f reflection s i s orientation-preserving, 19 4
of a n od d numbe r o f reflection s i s orientation-reversing, 19 4
of fou r reflection s i s a compositio n of tw o reflections , 19 2
of orientation-preservin g isometries i s orientation-preserving, 19 5
of thre e reflections , singula r case , 188
of tw o mappings , 14 6 of tw o reflections , 165-18 0 of tw o rotations , 17 8 of tw o translation s i s a translation ,
176, 17 8 of unifor m dilations , 226 , 23 9 preserves isometries , 14 7 preserves one-to-oneness , 14 7 preserves ontoness , 14 7 preserves transformations , 14 7
congruence
and geometri c equivalence , 251-256
as equivalenc e relation , 25 4 criteria for , 254-25 5
equivalence o f transformationa l and forma l definitions , fo r triangles, 25 2
for lin e segments , 2 5 for triangle s i s provide d b y
isometries, 16 1 of angles , 3 6 of triangles , 4 6 oriented, 26 7 transformational definition , 25 1
conjugacy as geometri c equivalenc e fo r
transformations, 256-26 6 classes o f isometries , 263 , 27 5 classes o f similarities , 273 , 27 5 criteria for , 259-26 5 is a n equivalenc e relation , 25 8 of rotation s an d orientation , 26 5 of subgroups , 35 2 of two-fol d product s take n i n bo t h
orders, 26 5 conjugation
and commutativity , 17 9 by similarities , 273 , 27 5 definition of , 25 8 examples of , 179-18 0 of a glid e reflectio n b y a similarity ,
187, 27 3 of a reflectio n b y a similarity , 179 ,
260, 27 3 of a rotat io n b y a similarity ,
179-180, 265 , 27 3 of a stric t similarit y b y a
translation, 27 5 of a translatio n b y a similarity ,
179, 27 3 of a unifor m dilatio n b y a
similarity, 235 , 27 3 of a n isometr y b y a similarit y
gives a n isometry , 23 9 of on e transformatio n b y another ,
definition of , 179 , 25 8 takes inverse s t o inverses , 26 1 takes product s t o products , 26 1
Contraction Mappin g Theorem , 25 0 convex quadrilateral , 3 0 convex set , 3 2
definition of , 2 7
536 Index
coordinate system s definition of , 1 7 equivalence of , 1 9
coordinates of a poin t i n th e Euclidea n plane ,
109 of a poin t o n a line , 1 7
Crossbar Theorem , 30 , 3 4 crystallographic notation , standard ,
455 crystallographic restriction , 44 1 cyclic group , 35 4
definition, 35 0 of infinit e order , 35 4 of orientation-preservin g
isometries, 36 5
Desargues' Littl e Theorem , 77 , 8 3 diameter
of a se t i n th e plane , 50 8 of a square , 50 9 of a triangle , 51 7 of a n equilatera l triangle , 50 9 properties of , 517-51 8 relationship wit h radius , 511-51 2
dihedral group , 35 0 a finite non-orientation-preservin g
symmetry grou p i s a , 36 8 is a reflectio n group , 373-37 4
dilatation classification, 24 8 definition of , 14 4 isometric dilatations ,
classification, 20 5 must b e a similarity , 24 8 translation i s a , 14 4 uniform dilatio n i s a , 22 6
dilation facto r negative value , 22 8 of a composition , 21 8 of a similarity , 135 , 21 7
dilation, uniform , see unifor m dilation
dimension-exponent relationship , 52 2 directed angl e measure , 84-9 4
is translat io n invariant , 91 , 14 5 transformation b y similarities , 23 8
directed lin e definition of , 2 4
discrete grou p o f isometries , 375 , 414, 45 6
dissection, 467 , 48 2 transitive o n polygona l regions ,
484 distance
between point s o n a line , 1 8 from poin t t o line , 6 1 properties of , 1 8 triangle inequalit y for , 18-1 9
ellipse, 31 4 equality modul o a number , 3 9 equidecomposable
a rectangl e an d a squar e o f th e same are a are , 482-48 4
a triangl e an d a parallelogra m with th e sam e bas e an d hal f th e height are , 47 5
definition of , 48 2 equivalence relatio n o n polygona l
regions, 48 4 two parallelogram s wit h th e sam e
base an d heigh t are , 47 4 two polygona l region s o f th e sam e
area are , 482-48 7 equivalence classes , 25 6 equivalence relatio n
congruence i s an , 25 4 definition of , 25 , 25 3 examples of , 25 6 relation o f ke y propertie s t o grou p
properties, 258-25 9 Erlanger Programm , 25 1
for Euclidea n geometry , 276-28 5 Euclidean geometr y
is implici t i n th e structur e o f th e Euclidean group , 276-28 5
Euclidean grou p matr ix realizatio n of , 280-28 3
Euler line , 301-30 2 contains th e circumcenter ,
orthocenter, centroi d an d nine-point center , 30 1
Euler points , 303 , 30 6 Euler, Leonhar d
discovered th e Eule r line , 30 2 Exterior Angl e Theorem , 5 6
Index 537
Fagnano's Problem , 313 , 315-32 2 Fermat point , 322 , 32 9 Fermat Problem , 322-33 9
solution of , 32 8 statement of , 32 2
Fermat 's Principl e
implies th e La w o f Reflection , 31 1
statement of , 31 0 Fermat, Pierre , 310 , 32 2 Feuerbach points , 30 4 Feuerbach's Theorem , 30 5 Feuerbach, Kar l Wilhelm , 30 5 finite grou p
of isometrie s fixes a point , 36 4
of rotation s i s cyclic , 36 5 First Structur e Theorem , 156-160 ,
165 statement of , 15 9
fixed line s behavior unde r conjugation , 25 9 for isometries , 18 9
for similarities , 24 6 fixed point s
behavior unde r conjugation , 25 9
definition of , 154 , 15 6 existence fo r contractio n
mappings, 25 0 for isometries , 18 9 for similarities , 240-24 1 for symmetr y group s o f bounde d
figures, 36 3 strict similaritie s hav e unique ,
240-250 Fixed Point s an d Fixe d Lines , 18 9 fractal, 459 , 523-52 9
dimension o f a , 52 6 generator fo r a , 52 3 initiator fo r a , 52 3
is a self-simila r figure, 52 5 frieze group , 375 , 39 9 41 5
central reflectio n o f a , 40 0 classification: seve n conjugac y
classes, 41 0 conjugacy classe s o f non-spli t
groups: onl y two , 408 , 41 3 conjugacy classe s o f spli t groups :
only five, 40 4
contained i n a spli t friez e group , 407
definition of , 40 0 flow char t fo r classification , 41 2 generators fo r a , 410-41 1 isomorphism classes : onl y four ,
415 midpoint restricte d subgrou p of ,
408-410 point groups : onl y four , 40 3 sample friez e pat tern s for , 41 1 split i f poin t grou p doe s no t
contain centra l reflection , 40 6 frieze pat tern s
complete set : seve n conjugac y classes, 41 1
definition of , 35 0 example, 35 0 from Anasaz i pottery , 41 3 non-split groups : tw o conjugac y
classes, 40 8 split groups : five conjugac y
classes, 40 5 fundamental domain , 331 , 37 3
for a translatio n group , 42 3
G-related, 36 1 geometric equivalence , 25 1
and similarities , 266-27 5 with respec t t o a group , definitio n
of, 26 7 geometric optics , 31 0 geometric propertie s
for Euclidea n geometry , 25 5 of isometries , 263-26 4 of similarities , 27 4
geometry early histor y of , 28 7 etymology of , 28 7
glide axi s and th e orthi c triangle , 31 6
glide lengt h and th e orthi c triangle , 316-31 7
glide reflection , 181-18 8 a generi c threefol d produc t o f
reflections i s a , 18 4 and th e orthi c triangle , 18 7 as produc t o f a reflectio n an d a
point inversion , 18 6
538 Index
axis of , 18 3 definition of , 18 3 equality of , 18 4 glide axi s i s the onl y fixed lin e o f a
non-trivial, 18 6 has squar e equa l t o a translation ,
187 non-trivial, 18 3 (oriented) glid e length , 18 3 specification b y data , 184 , 18 6 standard form , 18 2
glide reflectio n propert y fo r a n inscribed triangle , 31 9
global rotationa l direction , 84-8 6 great circle , 1 3 group
abelian, 20 0 center of , 41 5 cyclic, 350 , 354 , 36 5 cyclic o f orde r m , definition , 35 0 dihedral, 35 0 isomorphism, 27 9 normal subgroup , 204 , 23 8 of symmetries , 20 4 of transformations , 19 9 subgroup of , 20 0
group homomorphis m definition of , 205 , 37 7
group isomorphism , 279 , 353 , 439 is a n equivalenc e relation , 35 5
group o f transformations , 165 , 199-206
as symmetr y group , 20 4 centralizer o f a n elemen t o f a , 20 4 commutative, 19 9 definition, 19 9 dilatation group , 20 5 examples, 20 3 intersection o f two , 20 3 normal subgrou p of , 20 4 of the line , 20 5 subgroup of , 19 9 translations for m a commutative ,
199
half plan e definition of , 2 8 edge of , 2 8
Heron o f Alexandria , 47 7
Heron's Formula , 47 7 Hinge Theorem , 6 1 homogeneous isotropi c medium , 31 0 hyperbolic geometry , 63 , 78
identity mapping , 12 2 is an isometry , 13 5
image o f a se t b y a mappin g definition of , 13 0 of a union , 13 4 of a n intersection , 13 4
incenter, 292-29 3 Incidence Axiom s
examples, 8-1 7 minimal example , 1 5 statement of , 7 , 12 0
incircle, 29 3 Inscribed Angl e Theore m
and reflection s i n perpendicula r bisectors, 29 8
statement of , 8 0 intercepted arc , 46 5
measure of , 8 0 of a n angl e inscribe d i n a circle , 8 0
interior of a triangula r region , 46 8 of a n angle , 3 0 of the domai n V o f standar d
lattices, 43 0 invariant propertie s
for subset s o f plane , 25 5 for transformation s o f plane , 26 3
invariant se t fo r a group , 356-357 , 361
inverse compute examples , 15 4 definition of , 15 0 of a composit e mapping , 152-15 3 of a similarit y i s a similarity , 21 8 of a n isometr y i s an isometry , 15 2
invertible equivalence wit h one-to-on e an d
onto, 15 1 involution, 24 7 isometric mappin g i s a
transformation, 14 5 isometries
form a grou p unde r composition , 199
Index 539
form a norma l subgrou p o f th e similarities, 23 8
represented a s a matri x group , 28 0 isometry
as distance-preservin g mapping , 145
as product s o f reflection , 15 9 classification of , 18 8 definition of , 13 5 examples, 135-13 9 First Structur e Theorem , 15 9 fixed poin t classification , 15 7 fixed point s of , 15 7 groups o f th e line , 20 5 inverse i s agai n a n isometry , 15 2 of th e line , 19 0 orthogonal extensio n of , 19 0 preserved b y composition , 14 7 preserves geometri c objects , 139 ,
144 preserves hal f planes , 14 4 preserves parallelism , 14 4 preserves perpendicularity , 14 4 properties of , 139 , 14 4 type i s specifie d b y fixed point s
and fixed lines , 18 9 isomorphism o f groups , 27 9 isoperimetric theorem , 52 0 isosceles triangl e
equivalent condition s for , 292 , 29 7 symmetry of , 29 1
Jordan measurabl e set s all disk s an d sector s o f disk s are ,
496 all polygona l region s are , 49 6 closed unde r union , intersection ,
difference, 49 3 from inne r an d oute r Jorda n
measure, 49 3 independent o f choic e o f squar e
lattice, 50 1 with respec t t o a fixed squar e
lattice, 49 2 Jordan measure , 490-50 5
behavior unde r similarities , 506-507
independent o f choic e o f squar e lattice, 50 1
is define d fo r polygona l regions , 496
is invarian t unde r isometry , 50 2 of a lin e segmen t i s zero , 496-49 8 properties of , 493-49 5 satisfies th e axiom s fo r a n are a
function, 50 3 with respec t t o a lattic e o f
squares, 49 2
kaleidoscope, 369-37 2 Klein, Felix , 25 1 Koch curve , quadratic , 52 7 Koch snowflake , 523-52 6
similarity dimensio n of , 52 6
lattice, 38 2 a genera l lattic e i s simila r t o a
unique standard , 43 5 classification o f plana r types , 439 ,
440 (fat) rhombic , 433 , 43 9 for discret e plana r translat io n
group, definitio n of , 41 7 for discret e plana r translat io n
groups, 416-43 8 hexagonal, 431 , 433 , 43 9 invariant unde r a give n poin t
group, 44 2 invariant unde r a reflection ,
425-429 invariant unde r a reflectio n i s
rectangular o r rhombic , 42 6 (long) rhombic , 433 , 43 9 oblique, 433 , 43 9 rectangular, 425 , 43 1 rhombic, 425 , 43 1 rhombic extensio n o f a
rectangular, 428-429 , 43 7 square, 431 , 433 , 43 9 s tandard, 422-42 8 strict rectangular , 433 , 43 9 symmetries of , 429-43 8 symmetry classe s of , 433 , 43 9 with symmetr y o n th e boundar y
of V, 43 2 lattice o f squares , 49 1 lattice poin t group , 42 9 length, 45 9
540 Index
Leonardo's Theorem , 36 9 line
intersection wit h a circle , 11 4 isometries of , 19 0
line segmen t definition of , 2 3 directed, definitio n of , 2 4 has equa l directio n wit h another ,
124 perpendicular bisecto r of ,
definition an d characterization , 55
line segment s congruence for , 2 5
Line-Circle Theorem , 11 4 linear pair , 3 5
Mandelbrot, Benoit , 52 3 mapping
composition, 14 6 definition of , 12 2 equality of , 12 2 fixed point s of , 15 4 identity, 12 2 inverse mapping , 15 0 one-to-one, definitio n of , 13 0 onto, definitio n of , 13 1
measures o f shape , 511-513 , 51 9 medial triangle , 295 , 298-299 , 30 6
a triangl e i s simila r t o its , 29 9 median, 29 3 midpoint, 2 3 minimal t ranslat io n displacement ,
399 Mirror Principle , 31 3 Mobius band , 84-8 6 Moulton plane , 1 7
and Rule r Axiom , 2 1 incidence in , 1 3
multisquare regio n definition of , 49 0 inner an d oute r approximatio n by ,
490-492
Napoleon symmetr y group , 335 , 36 2 Napoleon tesselation , 330-337 , 362 ,
456 fundamental domai n of , 33 2 special, 33 8
Napoleon translatio n group , 334 , 36 2 Napoleon triangl e
inner, 33 7 Napoleon's Theorem , 216 , 33 0 natural numbers , 1 nine poin t cente r
is th e midpoin t o f th e Eule r line , 301
nine poin t circle , 29 5 is ak a th e Feuerbach circle, 30 5 is tangen t t o th e incircl e an d th e
excircles, 30 4 is th e circumcircl e o f th e media l
triangle, 30 2 is th e circumcircl e o f th e orthi c
triangle, 30 3 three-dimensional interpretation ,
308-309 non-Euclidean geometry , 7 8
one-to-one correspondence , 1 7 one-to-one mappin g
definition of , 13 0 preserved b y composition , 14 7
onto mappin g definition of , 13 1 preserved b y composition , 14 7
orbit for a group , 36 1 for a subgroup , 36 2 of p i s th e smalles t G-invarian t se t
containing p , 35 8 of a point , fo r a group , 357-35 9 properties of , 35 8
orientation, see also parity , 191-19 9 and composition , 195 , 23 6 of a similarity , 23 2 of a n isometry , 19 4
orientation-preserving subgroup o f a symmetr y group ,
359-360 origin, choic e of , 1 ornamental group , 37 5 orthic triangl e
and Fagnano' s Problem , 315-32 2 angles o f the , 32 1 characterizations o f the , 31 9 definition of , 187 , 297 , 31 5 solves Fagnano' s Problem , 31 8
Index 541
orthocenter, 292 , 296 is the incente r o f the orthi c
triangle, 31 8 lies on th e Eule r line , 30 1
orthogonal extensio n of isometrie s o f the line , 19 0
orthogonal projectio n to a line , definitio n of , 129-13 0
overlap, 46 8
p-component of a plan e isometry , 37 6 of a n isometry , definitio n of , 21 0
p-factorization defines a grou p homomorphism ,
377 of a plane isometry , 37 6
pairwise adjustmen t o f reflections , 174
parallel line segments , definitio n of , 6 2 line throug h a point , existenc e of ,
63 lines mus t b e perpendicula r t o th e
same lines , 73 , 78 lines perpendicula r t o sam e lin e
are, 6 2 lines, definition , 6 2 lines, distanc e between , 7 9 lines, transitivity , 7 2
Parallel Postulate , 64 , 66 , 69-8 4 and Rectangl e Hypothesis , 7 9 and Theore m o f Thales , 8 0 and Triangl e Su m Hypothesis , 6 8 implies constanc y o f interio r angl e
sum fo r triangles , 7 1 implies equa l alternativ e angles , 7 1 implies transitivit y o f parallelism ,
67 independence of , 7 7 statement of , 70 , 12 0
parallelism, 62-7 0 parallelogram
analog o f Pythagorea n Theore m for diagonals , 9 8
as conve x quadrilateral , 8 1 definition, 7 3 Desargues' Littl e Theorem , 7 7 diagonals bisect , 7 4
diagonals, definitio n of , 7 3 equality o f opposit e sides , 82 have diagonal s tha t bisec t eac h
other, 8 1 is convex, 8 1 opposite side s o f ar e congruent , 7 4
Parallelogram Constructio n Theorem, 7 6
Parallelogram Existenc e Theorem , 75
Parallelogram Uniquenes s Theorem , 74
parity, see also orientatio n and conjugatio n o f a rotation , 18 0 and directe d angl e measure , 18 0 behavior unde r compositio n o f
isometries, 19 5 of a similarity , 23 2 of a n isometry , 180 , 194 , 19 7 of product s o f similarities , 23 6
perpendicular bisector , 5 5 perpendicular lin e
as shortes t distanc e fro m poin t t o line, 6 0
definition, 3 7 existence an d uniqueness , 4 9
planar lattice , 37 5 plane geometr y axio m syste m
undefined object s for , 7 Plane Separatio n Axiom , 27-3 4
statement of , 28 , 12 0 Poincare disk , 15 , 63
falsity o f Paralle l Postulat e in , 7 7 incidence in , 1 0
point group , 375-37 9 at differen t point s ar e conjugate ,
379 compatibility wit h translatio n
orbit, 38 6 compatibility wit h translatio n
subgroup, 38 1 definition of , 37 7 examples of , 37 8 of a friez e grou p i s Ci , C2 , Di , o r
D2, 40 3 of a wallpape r group , 440-44 3 representative o f a n elemen t of ,
377
542 Index
point inversio n definition of , 12 5 equality of , 12 5 equals a half-turn , 12 7 group theoreti c characterizatio n
of, 279 , 28 3 is a n isometry , 13 7
polygonal regio n area of , 47 0 convex, 48 0 definition of , 256 , 46 7 is trangulable , 48 1 non-overlapping, 46 8 preserved b y union s an d
intersections, 47 0 Pons Asinorum, 48 , 53 , 28 8
and isoscele s triangles , 29 0 products o f tw o reflections ,
redundancy of , 17 4 Pythagoras, 28 7 Pythagorean Theore m
analog fo r diagonal s o f a parallelogram, 98 , 10 7
converse of , 10 7 Euclid's proo f o f the , 47 9 proof b y similarity , 9 7 proof usin g are a o f squares , 48 0 proof usin g similarit y scalin g o f
area, 513-51 5
quadrilateral convex, 3 0 cyclic, 323-32 5 definition of , 3 0 diagonals of , 3 0 Saccheri, 66-68 , 7 3 Varignon's Theorem , 10 7 with opposit e angl e supplementar y
is cyclic , 32 3
radius of a se t i n th e plane , 50 8 of a square , 50 9 of a n equilatera l triangle , 50 9 relationship wit h diameter ,
511-512 rational numbers , 1
incompleteness of , 5
ray definition of , 2 3 opposite t o give n ray , 2 3 translates, 8 6 translates fro m translations , 14 2
Ray Separatio n Theorem , 8 3 real Cartesia n 3-spac e
incidence in , 1 4 real Cartesia n plane , 8
and Rule r Axiom , 2 0 distance on , 1 9 t ru th o f Paralle l Postulat e in , 7 7
real Cartesia n space , 1 7 real number s
and relate d numbe r systems , 1 Archimedean orderin g principle , 2 axioms for , 6 Bolzano's Theorem , 3-4 , 11 5 bounded sequence s of , 3 , 11 5 completeness of , 2- 3 convergent sequence s of , 3 , 11 5 coordinatizing a lin e with , 1 , 1 7 decimal expansion s of , 6 open an d close d interval s in , 2 separation b y rationa l numbers , 4 ,
103 standard algebrai c propertie s of , 2
real projectiv e plane , 1 7 incidence in , 1 2
rectangle and Saccher i quadrilaterals , 7 3 as parallelogram , 8 2 definition of , 6 6
Rectangle Hypothesis , equivalenc e t o Parallel Postulate , 7 9
reflection of light , 309-31 5
reflection group , 369 , 415 , 45 8 definition of , 37 4
reflection i n a lin e definition of , 12 3 equality of , 12 3 group theoreti c characterizatio n
of, 279 , 28 3 is a n isometry , 135-137 , 14 3
reflection propert y fo r a n inscribe d triangle, 31 6
refraction, 31 1
Index 543
rhombic extensio n o f a rectangula r lattice, 428-42 9
rhombus, 8 2 fat, 43 2 long, 43 2
rosette group , 37 5 rotation
around a point , throug h a directe d angle, 12 6
characterization a s produc t o f reflections, 17 2
equality of , 12 7 is a n isometry , 13 8 non-trivial, 12 6 pairwise adjustmen t o f reflections ,
172, 17 4 rotation angle , 165 , 20 8
additivity unde r composition , 212 , 238
characterization i n term s o f a lin e and it s image , 214-215 , 23 9
characterization o f translation s i n terms of , 21 2
effect o f conjugation , 26 4 for a n eve n isometry , 211 , 214-21 5 for a n eve n similarity , 234 , 237 ,
239 not possibl e fo r od d isometry , 21 5 tells whe n a produc t o f reflection s
is a translation , 21 3 rotat ional direction , 8 4 Ruler Axiom , s ta temen t of , 17 , 12 0 Ruler Placemen t Theorem , 1 9
SAA (Side-Angle-Angle) , 6 0 Saccheri quadrilateral , 66-68 , 7 9
definition of , 6 6 Saccheri quadrilateral s
are rectangle s unde r Paralle l Postulate , 7 3
SAS (Side-Angle-Side ) Axio m independence o f previou s axioms ,
54 s ta tement of , 47 , 12 0
SAS Similarit y Theorem , 10 7 Second Structur e Theorem , 206-21 1
s ta tement of , 20 7 Segment Construct io n Theorem , 25 ,
27
semidirect product , 377 , 39 8 sets
equality of , 2 4 shear paralle l t o a line , definitio n of ,
128-129 shortest translatio n i n a translatio n
group, 42 4 Sierpinski's carpet , 52 8 similarities
classification of , 24 2 fixing a poin t an d a line , 22 2 fixing a poin t for m a group , 23 4 form a group , 21 8 groups of , 24 7 mapping a give n poin t t o a give n
point, 24 6 mapping a segmen t t o a segment ,
238 of a line , 234 , 24 8 the composit e o f tw o i s another ,
218 similarity, 217-25 3
as a distanc e rati o preservin g mapping, 23 0
as a transformatio n preservin g lines an d preservin g circles , 22 3
as a n equa l distanc e preservin g transformation, 22 2
as geometri c equivalenc e unde r similarities, 27 1
as produc t o f a unifor m dilatio n and a n isometry , 23 1
criteria for , 27 2 definition of , 21 7 dilation factor , 135 , 21 7 equivalence o f transformationa l
and forma l definitions , fo r triangles, 26 9
fixed point s o f a , 240-25 0 inverse o f on e i s another , 21 8 is a transformation , 21 7 is a n equivalenc e relation , 27 1 is determine d b y it s actio n o n
three points , 23 3 of triangles , 9 4 pari ty of , 23 2
preserves geometri c shapes , 219 , 222
544 Index
preserves parallelism , 22 2 preserves perpendicularity , 22 2 properties of , 219 , 222 rotation angl e of , 233-23 7 strict, definitio n of , 24 1 strict, structur e o f a , 24 2 structure o f a , 231-23 5 Structure Theore m fo r
Similarities, 23 2 Structure Theore m fo r Stric t
Similarities, 24 1 transformational definitio n of , 26 8 with dilatio n factor , definitio n of ,
135 Similarity an d Are a Principle , 50 6 similarity dimension , 459 , 52 6 similarity extensio n
from lin e t o plane , 23 5 Similarity Fixe d Poin t Theore m
analytic proo f of , 25 0 proof of , 242-246 , 24 9 statement of , 24 1
Similarity Principle , Basic , 9 9 Similarity Theore m
proof, 98-10 6 statement of , 95 , 224
Similarity/Dilation Theorem , 23 1 sine
addition formul a for , 47 8 split a t p , 38 8 split group , 387-39 3
a wallpape r grou p wit h a rhombi c lattice i s a , 44 8
classification, 390-39 3 conjugacy, 392 , 39 6 construction, 39 0 decomposition int o poin t grou p
and translatio n subgroup , 38 7 example, 38 8 recognition, 38 9
split i n G , 38 8 splitting point , 39 3 splitting set , 39 4 square roo t
of a translation , 43 6 SSA (Side-Side-Angle ) conjecture , 6 0 SSS (Side-Side-Side) , 49 , 5 3 SSS Similarit y Theorem , 10 7
strain orthogona l t o a line , definitio n of, 12 8
stretch reflection , definitio n of , 24 1 stretch rotation , definitio n of , 24 1 Structure Theore m fo r Similarities ,
232-233 Structure Theore m fo r Stric t
Similarities, 24 1 superposition
and symmetry , 28 9 symmetry, 347-45 8
of a figure, definition , 34 8 symmetry grou p
equivalence, 35 2 finite; Leonardo' s Theorem , 36 9 of a bounde d figure, 363-37 4 of a bounde d figure fixes a point ,
363 of a figure, definition , 204 , 34 9
symmetry type , 354-35 5 classification of , 351-35 3 of conve x quadrilaterals , 37 2
Taylor circle , 33 9 tesselation
Napoleon, 330-337 , 362 , 45 6 of plane b y parallelograms , 42 3 regular plane , 33 7
Thales of Miletus , 28 7 Theorem of , 80 , 28 8 theorems attribute d to , 28 7
Theorem o f Thales , 80 , 28 8 equivalence t o Paralle l Postulate ,
80 Three Circl e Theorem , 30 7
and th e nin e poin t circle , 30 8 Three Reflection s Theorem , see Firs t
Structure Theore m transformation
criteria fo r invertibility , 15 1 definition of , 13 1 Euclidean geometri c propertie s of ,
263 examples, 121-13 5 means a n invertibl e mapping , 15 1
transformational descriptio n o f geometric conditions , 29 3
Index 545
transitivity of parallelism , 7 2 of ra y translation , 88 , 93
translation characterization a s produc t o f
point inversions , 17 5 characterization a s produc t o f
reflections, 16 7 collection form s abelia n group , 20 0 composition o f two yield s a
translation, 176 , 17 8 criteria fo r equality , 124 , 14 4 definition of , 12 4 displacement of , 166 , 18 3 equivalence o f directed lin e
segments under , 14 2 equivalence o f rays under , 14 2 is a produc t o f uniform dilation s
with reciproca l dilatio n factors , 230
is an isometry , 137 , 14 4 is the squar e o f a glide reflection ,
187 non-trivial, 12 4 of directe d angl e measure , 91-9 3 of directe d angles , 89-9 0 of rays , 86-8 8 pairwise adjustmen t o f reflections ,
167, 17 4 parallel o r perpendicula r t o a line ,
436 parallel t o a segment , definitio n of ,
123 square roo t o f a , 43 6 transitivity o f ray , 88
translation displacemen t minimal, fo r a pattern , 39 9
translation grou p basis for , 417-42 2 definition o f discret e planar , 41 7
translation orbit , 382-38 7 compatibility wit h poin t group ,
386 definition of , 38 3 properties of , 38 3
translation subgroup , 375 , 380-38 2 compatibility wit h poin t group ,
381
definition of , 38 0 is in bijectio n wit h an y translatio n
orbit, 38 3 of a friez e grou p i s infinite cyclic ,
401 there i s only on e conjugacy clas s
of friez e group , 40 2 transversal lin e
alternate interio r angle s for , 6 4 definition of , 6 4 equal alternat e interio r angle s fo r
guarantees parallelism , 6 5 interior angle s for , 6 4
triangle AA Similarit y Condition , 9 6 AAA Similarit y Condition , 9 6 "all ar e isosceles " (?!?) , 10 8 altitude of , 10 6 ASA Congruenc e Condition , 4 8 congruence vi a isometries , 161 , 252 congruence wit h another , 4 6 constancy o f interior angl e sum , 7 1 definition of , 2 4 equilateral, 48 , 53 equilateral, transformationa l
characterization, 215-21 6 exterior angl e of , 5 6 filled, 46 7 inscribed i n a triangle , 31 9 interior of , 3 2 is convex , 3 3 isosceles, 48 , 29 1 mirror propert y fo r a n inscribed ,
313, 31 6 Pythagorean Theorem , 9 7 remote interio r angle s of , 5 6 SAA Congruenc e Criterion , 6 0 SAS Congruenc e Condition , 4 7 SAS Similarit y Condition , 10 7 similarity, definitio n of , 9 4 similarity, vi a similarit y
transformations, 26 9 SSA Congruenc e Criterio n fo r
obtuse, 6 0 SSS Congruenc e Condition , 49 , 5 3 SSS Similarit y Condition , 10 7 with give n sides , 11 0
Triangle Inequality , 5 9
546 Index
triangle inequalit y independence o f Incidenc e an d
Ruler Axioms , 2 1 strict, 5 8
Triangle Isometr y Theorem , 16 1 triangle similarity , 21 7 Triangle Su m Hypothesis , 6 8
implies th e Paralle l Postulate , 6 9 Triangle Theorem , 11 0 triangular region , 46 7
a unio n o f tw o ca n b e triangulated, 48 0
an intersectio n o f tw o ca n b e triangulated, 46 9
boundary, 46 8 degenerate, 46 7 interior, 46 8
triangulation, 46 8 refinement of , 46 8
trigonometric functions , 98 , 10 8 trigonometry wit h directe d angles ,
108-110 Two Circl e Theorem , 11 2
uniform dilation , 21 7 commutes wit h reflectio n i n lin e
containing fixed point , 23 0 composition o f tw o unifor m
dilations, 226 , 23 9 conjugation b y a n isometr y give s a
uniform dilation , 23 5 definition of , 12 7 equality of , 12 8 is a dilatation , 22 6 is a similarity , 22 4 properties of , 226 , 22 9 signed, 22 8 terminology, 22 9
unit circle , 10 9 unit o f length , 1
Varignon's Theorem , 10 7 vertex sum , 32 5 Vertical Angl e Theorem , 36 , 28 8 vertical pai r o f angles , 3 6 virtual parallelopipe d o f th e nin e
point circle , 30 9 volume, 521-52 2
wallpaper grou p automatic splittin g classes , 44 8 classification o f plana r lattic e
types, 439-44 0 classification: seventee n classes ,
454-455 contained i n a spli t wallpape r
group, 44 9 non-split groups : fou r classes ,
450-451 ornamental groups , 37 5 point group s an d lattic e pairs , 44 2 point groups : onl y ten , 44 1 seventeen isomorphis m classes , 45 8 split groups : thir tee n classes , 44 4
wallpaper pat tern s definition of , 41 6 example, 35 0 non-split groups : fou r classes ,
450-451 split groups : thir tee n classes ,
444-445
