characterizing the vertex neighbourhoods of semi-regular polyhedra, by t. r. s. walsh

8/9/2019 Characterizing the Vertex Neighbourhoods of Semi-Regular Polyhedra, by T. R. S. Walsh

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T . R . S. W A L S H *

C H A R A C T E R I Z I N G T H E V E R T E X N E I G H B O U R H O O D SO F S E M I - R E G U L A R P O L Y H E D R A

I . I N T R O D U C T I O NA po ly he dr on [4, p . 4 ] o r a p l a ne t e sse l l a t ion [4, p . 58] is c a l l ed semi-regular i f i t s fa c e s a r e r e g u l a r p o l y g o n s ( t h u s a l l i t s e d g es a r e e q u a l ) a n d i t sv e r t ic e s a r e a l l su r r o u n d e d a l ik e . T h i s i m p l i e s t h a t t h e c y c l i c s e q u e n c eS = (P l , P2 , . .. ,P~) r e p r e s e n t in g t h e d e g re e s o f t h e f a c e s s u r r o u n d i n g av e r t e x m u s t b e t h e s a m e f o r e a c h v e r t e x t o w i t h i n r o t a t i o n a n d r e f l e c t i o n .

I n t h i s p a p e r , w e s h o w t h a t the cyclic sequence S = ( p l , P2 , . . . ,Pq ) rep-resents the degrees o f the fac es surrounding each vertex o f a semi-regularcon vex polyhedron or tessellation o f the plan e i f and only i f:

(1) q>~ 3, and every member of S is at least 3;q

(2 ) . > /~ - 1 , with equa lity in the case o f a plan e tessellation,"i = l

(3 ) and fo r every odd number p in S, S contains a segment b, p , b .C o n d i t i o n ( 1 ) is n e c e s sa r y b e c a u se i f q~ < 2, t h e f i g u r e c o n s i s t s o f a s i n g le

p o l y g o n ; w h i l e a n y p o l y g o n w i t h s t r a ig h t s id e s m u s t h a v e a t l e a s t 3 o f t h e m .C o n d i t i o n (2 ) is n e c e s s ar y b e c au s e t h e s u m o f t h e i n t e r i o r a n g le s a t a

v e r t e x m u s t b e e q u a l t o o n e r o t a t i o n i f t h e f i g u r e i s t o l i e i n a p l a n e , a n dm u s t b e l es s t h a n o n e r o t a t i o n i f t h e f ig u r e i s t o b e s t r ic t ly co n v e x ( c o n v e xw i t h n o t w o a d j a c e n t f a c es l y i n g i n t h e s a m e p l an e ) .T h e u s u a l w a y i n w h i c h a l l t h e s e m i - r e g u l a r c o n v e x p o l y h e d r a a n d p l a n et e s se l la t i o n s a re f o u n d i s t o e l i m i n a t e so l u t i o n s o f (1 ) a n d ( 2) u s i n g s e p a r a t ea r g u m e n t s f o r e a c h o f se v e r a l s e ts o f so l u t i o n s [6 , p p . 1 1 6 - 2 6 ; 2 , p p . 2 5 - 3 2 ;3 , p . 3 9 4 ; 7 , p p . 2 0 2 - 3 ] , a n d t h e n t o p r o v e t h a t t h e so l u t i o n s n o t e l i m i n a t e dr e p r e se n t s e m i - r e g u l a r t e s se ll a t io n s o f t h e p l a n e o r sp h e r e .

I n [ 5 ] , t h e r e a p p e a r : a s h o r t p r o o f o f th e e x i s te n c e o f th e s e f ig u r e s[pp . 405-8] , a t ab le [p. 434] an d p ic tu res [p. 439] o f a l l bu t on e o f t hese m i - r e g u l a r c o n v e x p o l y h e d r a ( t h e r e m a i n i n g o n e a p p e a r s i n [1 , p . 1 37 ])a s we l l a s t h e k n o wn n o n - c o n v e x o n e s , a n d a t a b l e [ p . 4 3 8 ] o f a l l t h e s e m i -* T h e a u t h o r w i s h e s t o t h a n k P r o f . H . S . M . C o x e t e r f o r h is n u m e r o u s h e l p f u l s u g g e s t io n sd u r i n g t h e p r e p a r a t i o n o f t h i s p a p e r , a n d M . B u r t f o r h i s a s s i s t a n c e i n p r e p a r i n g t h ep l a t e s .

Geometriae Dedicata 1 (1972 ) 117 -123 . All Rights ReservedCopyright 1972 by D. Reidel Publishing Company, Dordrecht-Holland


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118 T . R . S . W A L S Hregula r p lane tesse l la t ions . P ic tures of a l l b ut two of the p lane tesse l la t ionsappear in [7, pp. 199 and 204-7, figures 75, 84, 85, 86, 88, 89, 94].

In t h is pape r , w e in t roduce the co ncep t o f a semi-regular map. W e d e d u c ethe nece ss i t y o f cond i t i ons (2 ) and (3 ) by show ing tha t t hey ho ld fo r a l lsemi-regula r maps on p lane- l ike or sphere- l ike surfaces provided tha tevery face is o f degree a t l eas t 3 . W e th en f ind a l l so lu t ions o f (1), (2), (3)and iden t ify each w i th a kno w n semi - regu la r convex po lyhed ron o r p l anetesse l la t ion . The proper t ies es tabl i shed in the course of th i s proof may beuse fu l fo r enum era t ing semi - regu la r map s on o the r t ypes o f su r face s.

I I . SEMI-REG ULAR MAPSA m a p [c f . 4 , p . 6] i s a par t i t ion of a connec ted , un bo un de d two-d imen siona lsu r face i n to s imply -connec ted po lygo na l r eg ions ( face s) by mean s o f pa i r -wise disjoint s imple curves (edges) join ing pa irs o f po ints (vertices). Ev erypo lyhedron i s a map , w h i l e a map w hose face s a re p l ane po lygons i s apo lyh edron i f t he num ber o f edges is f in ite , an d a d egene ra t e po lyh edron(like, fo r exam ple, a plane tessel la t ion) otherwise.

A m ap i s ca l led semi - regu lar i f t he cyc l ic o rde r o f t he degrees o f t he face ssu r round ing each ve r t ex is t he same to w i th in ro t a t i on and re fl e ct ion . Th i scyc l ic ord er de te rm ines a cyc l ic sequence S = (P l , P2 . . . . . p~) , w hich w e ca l lth e cycle o f t h e m a p .

Ev ery ver tex i s inc ident wi th mp p-gons , wh ere mp is the mu l t ip l ic i ty o fp in S . E ve ryp -gu n is inc ident wi th p ver t ices . So i f v ( the num be r of ver tices)i s f in i te and fp i s t he num ber o fp - gon s ,

(4 ) f p = ( ~ ) v .Eve ry ve r t ex i s i nc iden t w i th q edges (w he re q i s t he l eng th o f S ) and

every edge i s inc ident wi th 2 ver t ices . So i f e is th e n um be r o f edges,

(5) e = ( 2 ) v .I f f is t h e t o ta l n u m b e r o f f ac es ,

(6) f - - f = v .peS i=1

A convex bod y i s sphere- l ike , and i ts su rface is , like the p lane , s imply-


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S E M I - R E G U L A R P O L Y H E D R A 1 19

c o n n e c t e d . H e n c e w e m a y r e s t ri c t o u r a t t e n t i o n t o m a p s o n a s i m p l y -c o n n e c t e d su r fa c e ( a n o r i e n t a b l e s u r fa c e o f g e n u s 0 ) a n d u s e E u l e r 's f o r m u l a[4 , p . 9 ] to f ind v : q(7 ) f - e + v = v 1 - 2 + = 2 .

i = lq28 ) L e t t i n g D = - - ,P ii = lw e h a v e e i t h e r D > 0 a n d

2 q 2 ( m p / p )( 9 ) v = ~ , e = ~ , f P = Da s i n t h e c a se o f t h e c o n v e x p o l y h e d r a , o r e ls e D = 0 a n d o, e, fp a r e a l li n f i n i te , a s i n t h e c a s e o f t h e p l a n e t e s s e l l a t io n s .

T h i s p r o v e s t h e n e c e s s i t y o f c o n d i t i o n ( 2) .

I I I . R ESTR IC TIO N S O N TH E C Y C LE OF A SEMI- R EG U LA R MA PG i v e n a n y s e m i - r e g u l a r m a p ( n o t n e c e s s a r i l y o n e w h i c h o b e y s E u l e r ' sf o r m u l a ) w i t h c y c le S , c o n s i d e r a n y f a c e ( o f d e g r e e p , s a y ), a n d l e t b 1 . . . , bp b et h e d e g r e e s o f t h e f a c e s w h i c h s h a r e c o n s e c u t i v e e d g e s w i t h t h e p - g o n .T h e b ~ -g on , t h e p - g o n , a n d t h e b ~ + x -g o n a r e c o n s e c u t i v e f a c e s s u r r o u n d i n gt h e i r c o m m o n v e r t e x ( se e F ig u r e 1 );

\ p /0 ,

Fig. 1.

s o S c o n t a i n s a s e g m e n t bi , p, bi+ 1. T h i s i s t r u e f o r 1


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120 T . R . S . W A L S H

L E M M A 1 : I f S is the cycle fo r a semi-regular map, then fo r any p in S, Sm ust contain the seg me nts bx, p, b2; b2, p, ba ;.. . ; bp_ 1, P, bp; bp, p, b 1 o r som ebl , b2 , . . . , bp in S.

T h e se p s e g m e n t s a r e n o t n e c e s sa r i l y d i s t i n c t. I n f a c t , i f p i s e v e n , i t n e e do n l y b e t h e m i d d l e te r m o f a s i n g le s e g m e n t bl , p , b2; l e tt in g b 3 = b s . . . . .= b p -1 = b l a n d b 4 = b 6 . . . . . b p = b z w i ll s a ti s fy L e m m a 1. I f S c o n t a i n s as e g m e n t b l , p , b ~, t h e n l e t ti n g b 2 = b a . . . . . b p = b l w i ll s a ti s fy L e m m a 1w h e t h e r p i s ev e n o r o d d . B u t t h e s e a r e t h e o n l y c as e s w h e r e a s i n g le s e g m e n tsuffices.

I f p i s th e m i d d l e t e r m o f e x a c t ly t w o d i s t in c t s e g m e n t s b~,p , bz a n dbz, p , ba, t h e n b 4 = b 6 . . . . . b p = b 2 ; s o p m u s t b e e v en . T h e r e fo r e , i f p iso d d a n d n o s e g m e n t b , p , b e x is ts i n S , p m u s t b e t h e m i d d l e t e r m o f a tl e a s t 3 d i s t in c t s e g m e n t s i n S , w h i c h m e a n s t h a t t h e m u l t ip l i c it y o f p i n Si s a t l e as t 3 [ c f a l so 8 , p . 1 6 1 ] a n d S m u s t c o n t a i n a t l e a s t t w o o t h e r n u m b e r sd i st in c t f r o m e a c h o t h e r a n d f r o m p .

F o r e x a m p l e , t h e c y c l e o f t h e s e m i - r e g u l a r m a p i n F i g u r e 2 i s (3 , 4 , 3 , 2 , 3 ),i n w h i c h n o s e g m e n t b , 3 , b e x i st s .

F ig . 2 .

H o w e v e r , i f w e i n s is t t h a t e v e r y f a c e m u s t h a v e a t l e a s t 3 s i de s ( w h i c h isn e c e s s a ry i f t h e s id e s a r e t o b e s t ra i g h t, a n d w h i c h im p l i e s t h a t e v e r y m e m b e ro f S i s a t l e a s t 3 ), t h e n D i s m a x i m i z e d b y l e t t i n g p = 3 o c c u r t h r e e t i m e s a n dthe o the r two e l ement s o f S be 4 and 5 . The sequence (3 , 4 , 3 , 5 , 3 ) sa t i s f i e sL e m m a 1 ; h o w ev e r, D = - ~ o . S o w e h a v eL E M M A 2 : l f S is the cycle o f a semi-regular m aps without fac es o f degree 1or 2 on a simply-connected surface, then fo r every odd number p in S, S m ustcontain a segment b, p, b.

T h i s p r o v e s t h e n e c e s s i ty o f c o n d i t i o n ( 3) .


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S E M I - R E G U L A R P O L Y H E D R A 121IV. THE SOLUTIONS OF ( 1 ) , ( 2 ) , ( 3 )

W e n o w f i n d a ll t h e s o l u t i o n s o f (1 ), (2 ), ( 3) , a n d s h o w t h a t e a c h s o l u t i o n i st h e c y c le o f a k n o w n r e g u l a r o r s e m i - re g u l a r c o n v e x p o l y h e d r o n o r p l a n et e s s e ll a t io n . I n t h e p r o c e s s , w e r e c o n s t r u c t p a r t o f t h e t a b l e s i n [ 5 , p p . 4 3 4 ,4 38 ] g i vi n g f o r e a c h S t h e c o m m o n n a m e ( f o r t h e c o n v e x p o l y h e d r a - n oc o m m o n l y - a c c e p t e d n a m e s e x is t f o r t h e p l a n e t e ss e ll af io n s ) a n d t h e c o m m o ns y m b o l . T h e v a l u e s o f v, e, a n d f p c a n b e c o m p u t e d f r o m (8 ) a n d (9 ). P i c tu r e so f th e s e f ig u r e s a p p e a r i n t h e p l a t e i n t h e s a m e o r d e r a s t h e y a r e i n t r o d u c e db e l o w .

I f q = 3 , ( 2 ) b e c o m e si I 1 1( lO ) - - + - - + - - t> -

P l P 2 P a 2a n d (3 ) b e c o m e s : i f S c o n t a in s a n o d d n u m b e r p , t h e o t h e r t w o e l e m e n t s o fS m u s t b e e q u a l, a n d m u s t b e e i t h e r e q u a l t o p o r e v e n .

I f 3 ~ S , S = ( 3 , x , x ) , w h e r e b y (1 0 ), x~ < 1 2 ; s o w e h a v et e t r a h e d r o n { 3, 3 }3 , 3 , 3 )

(3 , 4 , 4)( 3 , 6 , 6 )( 3 , 8 , 8 )( 3 , 1 0 , 1 0 )( 3 , 1 2 , 1 2 )I f 3 S a n d 4(4, 4 , n)

t r i a n g u l a r p r i s m t { 2 , 3 }t r u n c a t e d t e t r a h e d r o n t {3 , 3 }t r u n c a t e d c u b e t {4 , 3 }t r u n c a t e d d o d e c a h e d r o n t {5 , 3}( p l a n e t e s s e l l a t i o n ) t { 6 , 3 } .

o c c u r s a t l e a s t tw i c e , w e h a v en - g o n a l p r i s m t { 2, n }

w h i c h s a ti sf ie s (3 ) a n d ( 1 0 ) f o r e v e r y n , a n d r e d u c e s w h e n n = 4 t o(4 , 4 , 4 ) cu be {4 , 3} .I f 4 o c c u r s e x a c t ly o n c e, t h e o t h e r t w o n u m b e r s m u s t b e e v e n . I f 6 ~ S ,

t h e n b y ( 10 ), t h e t h i r d n u m b e r ~< 1 2, a n d w e h a v e( 4 , 6 , 6 ) t r u n c a t e d o c t a h e d r o n t { 3, 4 }( 4, 6 , 8 ) t r u n c a t e d c u b o c t a h e d r o n t {~}( 4, 6 , 1 0) t r u n c a t e d i c o s i d o d e c a h e d r o n t {~ }(4, 6 , 12) (p lan e tess e l la t ion ) t{6a}.I f 6 S , t h e s e c o n d l a r g e s t t e r m m u s t b e a t l e a st 8. B u t b y ( 10 ), t h e r e m a i n -

i n g t e r m m u s t t h e n b e a t m o s t 8 ; s o w e h a v e o n l y(4 , 8 , 8 ) (p l a ne t e s se l l a t i on ) t {4 , 4} ,

w h i c h e x h a u s t s t h e s o l u t i o n s o f ( 10 ) w h i c h c o n t a i n 3 o r 4 .I f 3 S a n d 4 ~ S b u t 5 s S , S = ( 5 , x , x ) , x = 5 o r a n e v e n n u m b e r i> 6 .

B u t b y ( 1 0 ), x ~ < 6 ~ ; s o w e h a v e o n l y( 5, 5 , 5 ) d o d e c a h e d r o n { 5, 3 }( 5, 6 , 6 ) t r u n c a t e d i c o s a h e d r o n t { 3, 5 }


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1 2 2 T . R . S . WALSHF i n a l l y , i f 5 S , t h e s m a l l e s t n u m b e r i n S is a t l e a s t 6 . B u t t h e n b y ( 1 0 ),

t h e l a rg e s t n u m b e r i n S m u s t b e a t m o s t 6 ; so th e o n l y r e m a i n i n g s o l u t i o no f ( 1 0 ) i s

( 6 , 6 , 6 ) ( p l a n e t e s s e l l a t i o n ) { 6 , 3 }I f q = 4 , (2 ) b e c o m e s

1 1 1 1( 1 1 ) - - + - - + - - + - - I> 1 .Pl P:z P3 P4I f 3 o c c u r s i n S a t l e a s t 3 t i m e s , w e h a v e( 3 , 3 , 3 , n ) n - g o n a l a n t i p r i s m s { 2}

w h i c h s a ti sf ie s ( 3) a n d ( 1 1) f o r a l l n , a n d r e d u c e s w h e n n = 3 t o( 3 , 3 , 3 , 3 ) o c t a h e d r o n { 3 , 4 } .I f 3 o c c u r s in S e x a c t ly t w i c e, th e t w o o c c u r r e n c e s c a n n o t b e a d j a c e n t -

o t h e r w i s e , n e i t h e r 3 sa ti sf ie s c o n d i t i o n ( 3 ) - a n d t h e o t h e r t w o n u m b e r sm u s t b e e q u a l . S o S = ( 3 , x , 3 , x ) , w h e r e x ~ > 4 . B y ( 1 1) , x ~ < 6 ; s o w e h a v e

( 3, 4 , 3 , 4 ) c u b o c t a h e d r o n { 4}( 3 , 5 , 3 , 5 ) i c o s i d o d e c a h e d r o n { a}( 3 , 6 , 3 , 6 ) ( p l a n e t e s s e l l a t i o n ) { a} .I f 3 o c c u r s i n S e x a c t l y o n c e , t h e t w o n u m b e r s a d j a c e n t t o i t m u s t b e

e q u a l ( a n d e v e n : o t h e r w i se t h e r e m a i n i n g n u m b e r m u s t b e a 3 ). S oS = ( 3 , y , x , y ) , w h e r e y i s e v e n . I f y ~ >6 , t h e n b y ( 11 ), x ~ 3 ; s o y = 4 a n dS = ( 3 , 4 , x , 4 ) . B u t t h e n b y (1 1 ), x~ < 6 a n d w e h a v e

( 3, 4 , 4 , 4 ) r h o m b i c u b o c t a h e d r o n r { a}a n d p s e u d o - r h o m b i c u b o c t a h e d r o n [1 , p . 13 7]

( 3 , 4 , 5 , 4 ) r h o m b i c o s i d o d e c a h e d r o n r{as}( 3 , 4 , 6 , 4 ) ( p l a n e t e s s e l l a t i o n ) r { a } .I f 3 S , t h e s m a l l e s t n u m b e r in S m u s t b e i >4 . B u t t h e n b y (1 1 ), th e

r e m a i n i n g n u m b e r s m u s t b e ~


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SE MI - R E GUL A R POL YHE DR A 1 23l e a s t 4 , a n d b y (1 2), t h e r e m a i n i n g n u m b e r m u s t b e a t m o s t 4 ; a n d 3 c a n n o to c c u r le s s t h a n 3 t i m e s . So t h e o n l y r e m a i n i n g so l u t i o n s o f (1 2) a r e

(3, 3 , 3 , 4 , 4 ) (p l an e t e sse l l a t ion) no sym bo l(3 , 3 , 4 , 3 , 4) (pl an e tess e l la t io n) s {~}I f q > / 6, t h e n s i n ce e v e r y e l e m e n t i n S i s a t l e a s t 3 , b y ( 2) e v e r y e l e m e n t i n

S i s a t m o s t 3 a n d t h e r e a r e e x a c t l y 6 o f t h e m . T h u s t h e o n l y r e m a i n i n gso l u t i o n o f (1 ) a n d ( 2) i s

(3 , 3 , 3 , 3 , 3 , 3) (pla ne tess e l la t io n) {3, 6},w hich a l so sa t is f i e s (3) .

T h i s e x h a u s t s t h e so l u t i o n s o f ( 1) , (2 ), ( 3 ) a n d , t o g e t h e r w i t h L e m m a 2a n d t h e e x i s t e n c e p r o o f i n [5 , p p . 4 05 -8 1, p r o v e s t h e r e su l t q u o t e d i n t h es e c o n d p a r a g r a p h .

N o w s u p p o s e t h e r e s t r i c t io n t h a t e v e r y m e m b e r o f S i s a t l e as t 3 i s l if te d .T h e c o n c l u s io n o f L e m m a 2 is n o l o n g e r v a li d (s ee F i g u r e 2 ), b u t L e m m a 1s ti ll h o l d s . S o m e s o l u t i o n s o f (2 ) w h i c h s a t is f y L e m m a 1 c a n n o t p o s s i b l yb e t h e c y c l e o f a s e m i - r e g u l a r m a p b e c a u se f r o m ( 8 ) a n d ( 9) , a t l e a s t o n e o fv , e , a n d f p t u r n s o u t n o t t o b e a n i n t e g e r. H o w e v e r , o f a ll t h e s o l u t io n s o f(2 ) s o f a r in v e s t i g a te d w h i c h s a ti s f y L e m m a 1 a n d m a k e v , e, a n d f p i n t e g er s ,e v e r y o n e r e p r e se n t s a t l e a s t o n e s e m i - r e g u l a r m a p . I s t h i s g e n e r a l l y t r u e ;a n d i f n o t , i s t h e r e a r e a s o n a b l y s i m p l e c h a r a c t e r i z a ti o n o f t h e v e r t e xn e i g h b o u r h o o d s o f a n y s e m i- re g u la r p l a n a r m a p ?

B I B L I O G R A P H Y[1] B all, W . W. R ., Mathematical Recreations and Essays (revised by H . S. M . C oxeter),M acM illan, New York 1956.[2] C atalan, E ; 'M6m oire sur la Th6orie des Poly~dres',J. l'l~cole Polytechnique (Paris)41 (1865), 1-71.[3] Coxeter, H . S. M ., 'Regu lar an d Sem i-Regular Polytopes I ' , Math. Z. 46 (1940),380-407.[4] Co xeter, H . S. M., Regular Polytopes (2n d edition), M acmillan, New York 1963.[5] Cox eter, H. S. M., Lon guet-Higgins,M . S., and M iller, J. C . P., 'Uniform P olyhedra',Proc. Roy. Soc. London 246 (1954), 401-50.[6] Kepler, J., 'Harmonice Mtmdi', Opera Omnia, V ol. 5, Frankfurt 1864, pp. 75-334.[7] Kra'itchik, M ., Mathematical Recreations, Dov er Publications Inc., New Y ork 1942,pp 199-207.[8] Lines, L., Solid Geometry, Macm illan, London 1935.

Address of the author:T . R . S . W a l sh ,F a c u l t y o f M a t h e m a t i cs ,U n i v e r s it y o f W a t e r lo o ,W a t e r l o o , O n t a r i o , C a n a d a

(Received July 25, 1971)

characterizing the vertex neighbourhoods of semi-regular polyhedra, by t. r. s. walsh

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