Symmetry In The World Of Man And Nature;mayy2001

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Symmetry in the World of Man and Nature 1. Classification of Isometries Shailesh A Shirali

1 . In tr o d u c tio n

Shailesh Shirali has been at the Rishi Valley School (Krishnamurti Foundation of India), Rishi Valley, Andhra Pradesh, for more than ten years and is currently the Principal. He has been involved in the Mathematical Olympiad Programme since 1988. He has a deep interest in talking and writing about mathematics, particularly about its historical aspects. He is also interested in problem solving (particularly in the fields of elementary number theory, geometry and combinatorics).

S y m m etry a s a n id ea h a s a n a sp ect o f u n iv ersa lity to it. In v irtu a lly ev ery fa cet o f h u m a n en d eav o u r a n d n a tu ra l p h en o m en a , cu ttin g a cro ss th e b o u n d a ries o f tim e a n d sp a ce, w e ¯ n d m a n ifesta tio n s o f sy m m etry. A s H erm a n n W ey l w rites in h is w o n d erfu l b o o k [1 ], a b o o k th a t is certa in ly essen tia l rea d in g fo r a n y o n e w ith a n in terest in th e su b ject, \ S y m m etry, a s w id e o r a s n a rrow a s y o u m ay d e¯ n e its m ea n in g , is o n e id ea b y w h ich m a n th ro u g h th e a g es h a s tried to co m p reh en d a n d crea te o rd er, b ea u ty a n d p erfectio n " . T h e a ll-em b ra cin g n a tu re o f th e co n cep t o f sy m m etry is sta g g erin g : w ith in its fo ld lie su b jects a s fa r rem ov ed fro m o n e a n o th er a s p a rticle p h y sics, rela tiv ity, cry sta llo g ra p h y, ra n g o li p a ttern s a n d Isla m ic a rt. S cien tists a re fa r fro m b ein g th e o n ly o n es to p o n d er a b o u t th e co n cep t, a n d W illia m B la k e's im m o rta l p eo m s re° ect m a n 's a g e-o ld fa scin a tio n w ith sy m m etry : T yger! T yger! bu rn in g bright In the forests of the n ight, W hat im m ortal han d or eye D are fram e thy fearfu l sym m etry? T h o se k een o n th e a n a ly sis o f p o etry w ill n o te th a t B la k e h im self ch o o ses to b rea k th e sy m m etry in th e en d , w ith h is u se o f th e w o rd `sy m m etry '. H ere is W o rd sw o rth : T o see a W orld in a G rain of S an d A n d a H eaven in a W ild F low er H old In ¯ n ity in the palm of you r han d A n d E tern ity in an hou r. (W h a t sy m m etry is b ein g re° ected in th ese lin es?)

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Some of the finest examples of symmetry come from nature itself: the striking axial symmetry of a butterfly’s wings, the rotational symmetry of a flower (“Nature’s gentlest children”, in Weyl’s words), the astonishing 3dimensional symmetry of certain radiolaria and pollen grains, the spiral arrangement of seeds in a sunflower, the logarithmic spiral on a snail shell, the spiralling double helix of the DNA molecule, etc.

It h a s b een sa id th a t 'sy m m etry is d ea th '. T h e p u n is su g g estiv e { `sy m m etry ' a n d `cem etery '(!); th e p u rp o rt p resu m a b ly is th a t sy m m etry ca rries a n a sso cia tio n o f sta sis a n d la ck o f ch a n g e, w h erea s life is ev er ch a n g in g , ev er m ov in g . B u t in fa ct so m e o f th e ¯ n est ex a m p les o f sy m m etry co m e fro m n a tu re itself: th e strik in g a x ia l sy m m etry o f a b u tter° y 's w in g s, th e ro ta tio n a l sy m m etry o f a ° ow er (\ N a tu re's g en tlest ch ild ren " , in W ey l's w o rd s), th e a sto n ish in g 3 -d im en sio n a l sy m m etry o f certa in ra d io la ria a n d p o llen g ra in s, th e sp ira l a rra n g em en t o f seed s in a su n ° ow er, th e lo g a rith m ic sp ira l o n a sn a il sh ell, th e sp ira llin g d o u b le h elix o f th e D N A m o lecu le, etc. It is o n e o f th e stra n g e fa cts a b o u t th e w o rld th a t life h a s th e u rg e a s w ell a s th e ca p a city to crea te sy m m etric fo rm s. A n o th er cu rio u s th in g is th e p h en o m en o n o f left-rig h t a sy m m etry in so m e o rg a n ic m o lecu les (fo r ex a m p le, su g a rs) a n d th e p referen ce o f liv in g fo rm s fo r o n e ty p e o f o rien ta tio n . W h y th is sh o u ld b e so , a n d w h eth er sy m m etry is in ev ita b le in a n y fo rm o f life, is so m eth in g th e rea d er co u ld re° ect u p o n . In m a th em a tics to o , th e co n cep t o f sy m m etry h a s p lay ed a stro n g ly u n ify in g ro le. \ T h e in v estig a tio n o f th e sy m m etries o f a g iv en m a th em a tica l stru ctu re h a s a lw ay s y ield ed th e m o st p ow erfu l resu lts..." w ro te E m il A rtin . T h e ex a m p le th a t co m es m o st im m ed ia tely to m in d is F elix K lein 's w o rk o n th e u n i¯ ca tio n o f g eo m etry (th e so -ca lled E rlan gen program m e { see P a rt 2 in th e fo rth co m in g issu e). In th is tw o -p a rt ex p o sito ry a rticle, w e m a k e a b rief su rv ey o f th e b a sic p rin cip les o f sy m m etry. W e d iscu ss th e d i® eren t k in d s o f sy m m etry th a t a n o b ject ca n h av e, a n d n o te h ow th e sy m m etries o f a n o b ject fo rm a g ro u p in a v ery n a tu ra l m a n n er. F o llo w in g th is, w e cla ssify th e ¯ n ite 2 -d im en sio n a l sy m m etry g ro u p s. T h is is fo llow ed b y a n ex cu rsio n in to th e stu d y o f rep ea tin g p a ttern s: strip p a ttern s (a lso k n ow n a s frieze p a ttern s o r b o rd er p a ttern s), w a ll-p a p er p a ttern s a n d cry sta ls. T h e ex p o -

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sitio n is la rg ely self-co n ta in ed , th o u g h w e p resu p p o se so m e k n ow led g e o f elem en tary g ro u p th eo ry a n d lin ea r a lg eb ra . T h e su b ject is a rich a n d fa scin a tin g o n e a n d o n ly \ ex trava g a n t in co m p eten ce o n th e a u th o r's p a rt" (a s H a rd y m ig h t h av e p u t it, in h is in im ita b le la n g u a g e) w ill fa il to b rin g o u t its b ea u ty. 2 . T h e C o n c e p t o f S y m m e tr y W h a t ex a ctly is sy m m etry ? E v ery o n e w o u ld a g ree th a t a sq u a re is sy m m etric in th e lin e jo in in g th e m id p o in ts o f a p a ir o f o p p o site sid es an d in ea ch o f its d ia g o n a ls; lik ew ise, th a t a circle is sy m m etric in a n y o f its d ia m eters (see F igu re 1 ). A circle w h en re° ected in a n y o f its lin es o f sy m m etry fa lls b a ck u p on itself, a s d o es a sq u a re. T h ese co n sid era tio n s m o tiva te th e m o d ern a p p ro a ch to sy m m etry. (M u ch o f w h a t is sta ted b elow h a s b een d o n e k eep in g in m in d 2 -d im en sion a l sp a ce. T h is is o n ly fo r th e sa k e o f sim p licity ; th e sa m e trea tm en t w o rk s fo r 3 d im en sio n a l sp a ce.) L et th e u n d erly in g 2 -d im en sio n a l sp a ce in w h ich th e o b jects u n d er co n sid era tio n a re em b ed d ed b e d en o ted b y R 2 , a n d let th e d ista n ce b etw een p o in ts x a n d y in R 2 b e d en o ted b y d (x ;y ). C o n sid er a m a p p in g f o f R 2 in to itself; d e¯ n e f to b e a n isom etry if it leav es a ll d ista n ces u n a ltered , th a t is, if d (f (x );f (y )) = d (x ;y ) fo r a ll x a n d y (`iso ' m ea n s `th e sa m e', an d `m etric' h a s th e co n n o ta tio n o f d ista n ce). F u rth er, d e¯ n e th e iso m etry to b e direct if it p reserv es o rien ta tio n , a n d in direct if it ca u ses a rev ersa l o f o rien ta tio n . B y th is w e m ea n th e fo llow in g : let a n iso m etry f a ct o n ¢ A B C ; a n d let it ta k e A to A 0;B to B 0 a n d C to C 0. T h en f is direct if th e d irectio n o f th e cy cle A 0 ! B 0 ! C 0 ! A 0 is th e sa m e a s th a t o f th e cy cle A ! B ! C ! A ; if n o t, it is in direct. (A ltern a tiv es to `d irect' a n d `in d irect' a re even a n d odd.)

The square has essentially eight symmetries: reflections in the dotted lines shown, plus four rotations about its centre; these include the identity operation which corresponds to rotation by 0o. The circle has infinitely many symmetries: reflections in all its diagonals, and rotations about its centre by any angle whatever.

Figure 1. Symmetries of a square and a circle.

Iso m etries a re fa m ilia r o b jects; ex a m p les a re re° ectio n in a lin e a n d ro ta tio n a b o u t a p o in t, a s a lso th e tra n sla tio n T a g iv en b y T a (x ) = x + a , w h ere a is a n y ¯ x ed ele-

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Groups were first invented to analyze symmetries of certain algebraic structures called field extensions, and as symmetry is a common phenomenon in all sciences, it is still one of the main ways in which group theory is applied.

m en t o f R 2 . N o te th a t re° ectio n is a n in d irect iso m etry, w h erea s ro ta tio n s a n d tra n sla tio n s a re d irect iso m etries. S y m m e try G ro u p o f a n O b je c t N o ta tio n s: W e sh a ll d en o te p o in ts b y u p p erca se letters a n d lin es b y low erca se letters. R o ta tio n s w ill b e d en o ted b y `½ ' (w ith a su b scrip t to sh ow th e cen tre o f ro ta tio n ) a n d so m etim es b y `R o t'; re° ectio n s w ill b e d en o ted b y `¾ ' (w ith a su b scrip t to sh ow th e a x is o f re° ectio n ). H a lftu rn s to o w ill b e d en o ted b y `¾ '. S o ¾ P d en o tes a h a lftu rn a b o u t th e p o in t P (i.e., a ro ta tio n th ro u g h 1 8 0 d eg rees), a n d ¾ l d en o tes re° ectio n in th e lin e l. L et I d en o te th e set o f a ll iso m etries in R 2 . It is triv ia l to v erify th e fo llow in g : ² T h e id en tity m a p p in g ¶, g iv en b y ¶(x ) = x fo r a ll x , b elo n g to I . ² If f ;g 2 I , th en th e co m p o sitio n f ± g 2 I . ² If f 2 I th en f p o ssesses a n in v erse f ¡ 1 2 I . In d eed , I fo rm s a g ro u p u n d er fu n ctio n a l co m p o sitio n , th e grou p of rigid m otion s o f R 2 , d en o ted (b y a b u se o f n o ta tio n ) b y th e sa m e sy m b o l I . (N o te th a t a re° ectio n is its ow n in v erse; i.e., its o rd er is 2 .) E x e rc ise s 1 . S h ow th a t th e set o f a ll p o ssib le tra n sla tio n s in R is a su b g ro u p o f I .

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2 . S h ow th a t th e set o f a ll p o ssib le ro ta tio n s (a b o u t a ll p o ssib le p o in ts) d o es n o t fo rm a su b g ro u p o f I , b u t th a t th e set o f a ll p o ssib le ro ta tio n s to g eth er w ith th e set o f a ll p o ssib le tra n sla tio n s d o es fo rm a su b g ro u p . L et a n o b ject X b e g iv en , a n d co n sid er th e iso m etries f

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in I th a t leav e X ¯ x ed ; th a t is, f (X ) = X (th e p o in ts o f X ex ch a n g e p la ces a m o n g st th em selv es). If X w ere a sq u a re, th en re° ectio n in eith er d ia g o n a l w o u ld q u a lify fo r su ch a m a p p in g , a s w o u ld ro ta tio n a b o u t its cen ter b y 9 0 d eg rees. T h e set o f a ll su ch iso m etries is ca lled th e grou p of sym m etries o f X a n d is d en o ted b y th e sy m b o l S (X ); its elem en ts a re th e sym m etries o f X . E x a m p les a re ea sy to list: a b u tter° y h a s tw o sy m m etries, th e id en tity m a p a n d a re° ectio n ; a 3 -p eta lled ° ow er h a s th ree sy m m etries, th e id en tity m a p a n d ro ta tio n s th ro u g h 1 2 0 d eg rees a n d 2 4 0 d eg rees a b o u t its cen ter; a n d so o n . T able 1 d isp lay s th e n u m b er o f sy m m etries co rresp o n d in g to a few fa m ilia r o b jects.

X

|S (X)|

Isoceles triangle 2 Parallelogram 2 Rectangle 4 Equilaterial triangle 6 Square 8 Circle ∞ Table 1. Orders of a few symmetry groups.

(N o te th a t `iso sceles' m ea n s `iso sceles n o n -eq u ila tera l', `recta n g le' m ea n s `n o n -sq u a re recta n g le', a n d `p a ra llelo g ra m ' m ea n s `n o n -recta n g u la r `p a ra llelo g ra m '.) E x e r c ise s 3 . L ist th e eig h t sy m m etries o f a sq u a re. 4 . S h ow th a t a reg u la r n sid ed p o ly g o n p o ssesses 2 n sy m m etries. (Its g ro u p o f sy m m etries is th e dihedral grou p of order 2 n , d en o ted b y th e sy m b o l D n . It co n ta in s n ro ta tio n s a n d n re° ectio n s.) 5 . F in d a fa m ilia r m a n -m a d e o b ject in d a ily u se w h o se sy m m etry g ro u p is iso m o rp h ic to Z 3 . (W h y is it Z 3 a n d n o t D 3 ?) It is in terestin g to a sk w h eth er, g iv en a n a rb itra ry g ro u p G , w e ca n ¯ n d a n o b ject X fo r w h ich S (X ) is iso m o rp h ic to G ; th is is eq u iva len t to a sk in g fo r a cla ssi¯ ca tio n o r en u m era tio n o f a ll th e su b g ro u p s o f I . F o r in sta n ce, is th ere a n o b ject X fo r w h ich S (X ) »= Z 4 (th e cy clic g ro u p o f o rd er 4 )? Y es in d eed ! A sw a stika (eith er o n e w ill d o , th e N a zi o r th e H in d u sw a stika ) is su ch a n o b ject (see F igu re 2 ). T h e co rresp o n d in g 3 -a rm ed sy m b o l h a s

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a sy m m etry g ro u p iso m o rp h ic to Z 3 (th e cy clic g ro u p o f o rd er 3 ). O b v io u sly, th e g ro u p s Z n a re a ll `rea liza b le' in th e sen se w e h av e in m in d .

Figure 2. Nazi and Hindu swastikas.

In three dimensions, reflection in a plane cannot be physically realized. Thus, a left shoe cannot be transformed into a right shoe, no matter how you move it about — the only way to do so would be to make a quick dash to 4-dimensional space!

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M ov in g to th ree d im en sio n s, w e ¯ n d a sm a ll co m p lica tio n . In tw o d im en sio n s, a ll sy m m etries a re rea liza b le b y g en u in e p h y sica l m ov em en ts. F o r in sta n ce, re° ectio n in a lin e ca n b e p h y sica lly a ch iev ed b y ro ta tio n a b o u t th e lin e b y 1 8 0 d eg rees; th is m irro rs th e a ctu a l situ a tio n ex a ctly, in clu d in g th e o rien ta tio n rev ersa l ca u sed b y re° ectio n . (In fa ct th is is th e o n ly w ay o f d o in g it. N o a m o u n t o f slid in g a b o u t o n a p la n e w ill ev er a ch iev e a rev ersa l o f o rien ta tio n .) H ow ev er in th ree d im en sio n s, re° ectio n in a p la n e ca n n o t b e p h y sica lly rea lized . T h u s, a left sh o e ca n n o t b e tra n sfo rm ed in to a rig h t sh o e, n o m a tter h ow y o u m ov e it a b o u t { th e o n ly w ay to d o so w o u ld b e to m a k e a q u ick d a sh to 4 -d im en sio n a l sp a ce! (T h is b rin g s a tten tion to a n u n ex p ected h a za rd o f jo u rn ey in g to h ig h er d im en sio n a l sp a ces: a ca relessly d ro p p ed sh o e m ay rev erse its `p a rity ', a n d w e m ay b e left w ith tw o left sh o es o r tw o rig h t sh o es. A stro n a u ts, b e fo rew a rn ed !) F ro m th is p o in t o n , w h en w e refer to 3 -d im en sio n a l sp a ce w e sh a ll ex clu d e fro m co n sid era tio n a ll o rien ta tio n -rev ersin g iso m etries. E x e rc ise s 6 . H ow m a n y sy m m etries d o es a reg u la r tetra h ed ro n h av e? A reg u la r o cta h ed ro n ? 7 . S h ow th a t a cu b e h a s 2 4 sy m m etries. H ow m a n y o f th ese h av e o rd er 2? O rd er 3 ? O rd er 4 ? O rd er 6 ? W h a t p h y sica l m ov em en t co rresp o n d s to a sy m m etry o f o rd er 3 ? 8 . C o n sid er a 2 -co lo u red fo o tb a ll w h o se su rfa ce is a sy m m etrica l m o sa ic o f reg u la r p en ta g o n s a n d h ex a g o n s, w ith th e p en ta g o n s o f o n e co lo u r a n d th e h ex a g o n s o f a n o th er co lo u r. H ow m a n y sy m m etries d o es th e fo o tb a ll h av e?

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9 . W h a t sy m m etries d o es a n in ¯ n ite h elix h av e? (A sp rin g o ® ers a co n v en ien t m o d el; see F igu re 3 .) 1 0 . T h e toy, `R u b ik cu b e', m a rk eted in th e ea rly 1 9 8 0 's b y th e H u n g a ria n a rch itect E rn o R u b ik is a 3 £ 3 £ 3 cu b e d iv id ed in to th ree lay ers p erp en d icu la r to ea ch o f its th ree p rin cip a l a x es (see F igu re 4 ). T h e in tern a l stru ctu re o f th e cu b e is su ch th a t ea ch lay er ca n b e ro ta ted ab o u t its cen tre. E a ch o f th e o u ter lay ers h a s a d i® eren t co lo u r, so a fter a few su ch ro ta tio n s th e colo u rs g et h o p elessly scra m b led . T h e ch a llen g e is to resto re it to its o rig in a l p ristin e sta te. (T h e task is d ecid ed ly n o n -triv ia l!) T h e p ro b lem w e p o se h ere is to ¯ n d th e o rd er o f th e g ro u p g en era ted b y th e six b a sic m ov em en ts. W arn in g: T h e n u m b er is ex trem ely la rg e, a n d th e g ro u p h a s a v ery in trica te stru ctu re! P erh a p s th is m ay a cco u n t fo r th e n o to rio u s d i± cu lty o f th e p u zzle. 3 . Iso m e trie s in T w o D im e n sio n s W e n ow p ro ceed to cla ssify th e iso m etries in tw o d im en sio n s. A s m en tio n ed ea rlier, th e iso m etries in R 2 in clu d e th e tra n sla tio n s, ro ta tio n s a n d re° ectio n s. T h ere is a fo u rth ty p e o f iso m etry, th e glide re° ection , w h ich w e d e¯ n e a s fo llow s. L et ` b e a lin e in th e p la n e, let ¾ ` d en o te re° ectio n in `, let a b e a n o n -zero v ecto r p a ra llel to ` a n d let T a d en o te th e tra n sla tio n m a p g iv en b y T a (x ) = x + a . T h en th e p ro d u ct g = ¾ ` ± T a is referred to a s a g lid e re° ectio n . N o te th a t ¾ ` ± T a = T a ± ¾ `. A g lid e h a s in ¯ n ite o rd er, a n d it rev erses o rien ta tio n . It is ea sy to ex h ib it a p h y sical o b ject th a t p o ssesses g lid e sy m m etry : fo o tp rin ts o n a b ea ch ! (S ee F igu re 5 .)

Figure 3. A helix.

Figure 4. The Rubik cube. Problem: How many configurations are possible for the Rubik cube? That is, what is the order of the group generated by the six basic motions of the cube (the rotations through 90o of each of the six faces)?

Figure 5. Schematic picture of footprints on a beach.

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T h e o r e m 1 . A n isom etry that ¯ xes tw o distin ct poin ts ¯ xes the en tire lin e passin g throu gh them . A n isom etry that ¯ xes three n on -collin ear poin ts is the iden tity m ap. P r o o f. L et f b e a n iso m etry th a t ¯ x es tw o p o in ts A ;B . T h en , a n y p o in t P o n th e lin e A B is u n iq u ely sp eci¯ ed b y th e tw o d ista n ces A P ;B P , a n d a s d ista n ces a re left u n ch a n g ed , P to o m u st b e ¯ x ed b y f . T h e seco n d a ssertio n is p rov ed sim ila rly. T h e o r e m 2 . A n isom etry that ¯ xes tw o distin ct poin ts is either a re° ection or the iden tity m ap. P r o o f. L et f b e a n iso m etry th a t ¯ x es tw o p o in ts A ;B ; th en , it ¯ x es lin e A B , p o in tw ise. L et C b e a p o in t n o t o n A B . S in ce th e d ista n ces A C ;B C a re ¯ x ed , f (C ) ca n b e in o n e o f ju st tw o p o ssib le lo ca tio n s. If f (C ) = C th en f is th e id en tity. If n o t, let g b e th e re° ectio n in th e lin e A B . T h en , g ± f ¯ x es A ;B ;C a n d h en ce is th e id en tity. T h u s, f is th e in v erse o f g a n d so , f = g . T h e o r e m 3 . A n isom etry that ¯ xes exactly on e poin t is a produ ct of tw o re° ection s. A n isom etry that ¯ xes a poin t is a produ ct of at m ost tw o re° ection s. P r o o f. L et f b e a n iso m etry th a t ¯ x es ex a ctly o n e p o in t A . L et B b e a n o th er p o in t, a n d let B 0 = f (B ); th en B ;B 0 a re eq u id ista n t fro m A , so A lies o n lin e m w h ich b isects B B 0 a t rig h t a n g les. L et g d en o te re° ectio n in m . T h en g ± f ¯ x es A ;B , so g ± f is eith er a re° ectio n o r th e id en tity m a p . T h e la tter lea d s to f = g , w h ich ca n n o t b e, a s f ¯ x es ex a ctly o n e p o in t. T h erefo re g ± f is so m e re° ectio n h . T h u s, f = g ± h . T h e seco n d a ssertio n fo llow s a s a co ro lla ry o f th e p ro o f. 2 T h e o r e m 4 . A n y isom etry in 2-dim en sion al space can be expressed as a produ ct of n o m ore than three re° ection s. P r o o f W e n eed o n ly co n sid er th e ca se w h en th e iso m etry f h a s n o ¯ x ed p o in ts. L et P ;Q b e p o in ts w ith Q = f (P ).

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L et m b e th e lin e b isectin g P Q a t rig h t a n g les, a n d let g d en o te re° ectio n in m . T h en g ± f ¯ x es P a n d so is a p ro d u ct o f a t m o st tw o re° ectio n s. T h erefo re f is a p ro d u ct o f a t m o st th ree re° ectio n s. 2 T h e o r e m 5 . G iven tw o coplan ar trian gles con gru en t to on e an other, there exists a u n iqu e isom etry m appin g on e trian gle on to the other. T h e p ro o f is left to th e rea d er (E x ercise 1 4 ). T h e o re m 6 . (a ) T he produ ct of a tran slation an d a re° ection (in either order) is either a re° ection or a glide. (b ) T he sam e con clu sion holds for the produ ct of a rotation an d a re° ection (in either order). P r o o f. T h e sim p lest a p p ro a ch is v ia co o rd in a tiza tio n . L et th e a x is o f re° ectio n b e ch o sen to b e th e x -a x is. D en o te th e re° ectio n m a p b y ¾ , a n d let th e tra n sla tio n T b e g iv en b y T (x ;y ) = (x + a ;y + b); let f = ¾ ± T . T h en f ta k es (x ;y ) to (x + a ;¡ y ¡ b): T

¾

(x ;y ) ! (x + a ;y + b) ! (x + a ;¡ y ¡ b): D e¯ n e a u x ilia ry m a p s ® a n d ¯ a s fo llow s: ® (x ;y ) = (x + a ;y );¯ (x ;y ) = (x ;¡ b ¡ y ): T h en ® is a tra n sla tio n a lo n g th e x -a x is, w h ile ¯ is a re° ectio n in th e lin e y = ¡ b= 2 . B y co m p u ta tio n , w e see th a t ¾ ± T = ® ± ¯ . It fo llow s th a t ¾ ± T is a g lid e o r a re° ectio n (th e la tter ca se co rresp o n d s to a = 0 ; w h en a 6= 0 , th e a x es o f th e m a p s T a n d ¾ a re p a ra llel, so ¾ ± T is a g lid e b y d e¯ n itio n ). T h e p ro o f th a t T ± ¾ is a re° ectio n o r a g lid e is h a n d led sim ila rly, a n d w e leav e th e p ro o f o f p a rt (b ) to th e rea d er. 2 T h e o r e m 7 . A n y isom etry in I is the iden tity, a tran slation , a rotation , a re° ection or a glide re° ection . P r o o f. A s ea rlier, w e n eed o n ly co n sid er th e ca se w h en th e iso m etry f h a s n o ¯ x ed p o in ts; th en b y T h eo rem 4 , f ca n b e ex p ressed a s a p ro d u ct o f th ree re° ectio n s. S in ce th e p ro d u ct o f tw o re° ectio n s is eith er th e id en tity, a

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tra n sla tio n o r a ro ta tio n (see E x ercise 1 3 ), w e ca n in v o k e T h eo rem 6 . T h e resu lt fo llow s. 2 E x e rc ise s 1 1 . L et ¾ P ;¾ Q ;¾ R d en o te h a lf-tu rn s in th ree d istin ct p o in ts P ;Q ;R . S h ow th a t ¾P ± ¾Q ± ¾R = ¾R ± ¾Q ± ¾P : 1 2 . L et m b e a n y lin e, let ¾ m d en o te re° ectio n in m , a n d let f b e a n y iso m etry. S h ow th a t th e iso m etry f ± ¾ m ± f ¡1 is th e sa m e a s re° ectio n in th e lin e f (m ). 1 3 . L et m ;n b e g iv en lin es. S h ow th a t th e co m p o site m a p ¾ m ± ¾ n is eith er th e id en tity m a p (if m a n d n a re th e sa m e), a tra n sla tio n (if m k n ), o r a ro ta tio n (if m a n d n a re n o n -p a ra llel; in th is ca se, th e cen tre o f ro ta tio n is th e p o in t m \ n ). 1 4 . P rov e T h eo rem 6 . Address for Correspondence

Suggested Reading

Shailesh A Shirali Rishi Valley School Chittoor District Rishi Valley 517 352 Andhra Pradesh, India.

[1] Hermann Weyl, Symmetry, Princeton University Press, 1952. [2] George E Martin Transformation Geometry: An Introduction to Symmetry, Springer Verlag. [3] L Tarasov, This Amazingly Symmetrical World , Mir Publishers, 1986. [4] D’Arcy Thomson, Growth and Form, Cambridge University Press, 1917.

Absence of evidence is not evidence of absence.

Carl Sagan The Dragons of Eden

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