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 >AMS AMERICAN MATHEMATICA L SOCIET Y Freehand® i s a registere d trademar k o f Adob e System s Incorporate d in th e Unite d State s and/o r othe r countries . Mathematica® i s a registere d trademar k o f Wolfra m Research , Inc . 2000 Mathematics Subject Classification. Primar y 51-01 , 20-01 . For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/mbk-47 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 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisition s Department, America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 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 .