some problems from the visual arts

Some Problems from the Visual Arts

Anthony H i l l

Chelsea School of Art, London, England

1. The number of n e t s f o r a polytope

We pose t h e following problem:

n e t s a r e t h e r e f o r a simply connected polyhedron? k! This problem

would appear t o be very hard, added t o which t h e author i s unaware

i f e i t h e r t h e above problem - or t h e general problem - has received

a t ten t ion .

how many topologica l ly d i s t i n c t face-

A t t h e time of wri t ing, t h e goal of f inding a general expression f o r

t h e number of face n e t s seems remote. Currently study has been

l i m i t e d t o examining i s o l a t e d r e s u l t s and observations with a view t o

wr i t ing a computer programme.

A polygonal ne t pN* i s an edge connected plane map with p faces .

approach t o our problem i s t o arrive at t h e condi t ions by which any

pN* i s a polyhedral ( face) ne t PN*.

One

For any pN* t o be a PN* t h e

following, conditions must be s a t i s f i e d : faces f a c e

(a) I n any pN* t h e r e must be a t l e a s t 4 / and every /'must contain at least 3 edges.

(b) pN* must be ' t r e e shaped' ( i . e . every node of pN* must l i e

on t h e boundary c i r c u i t 1. ( c )

(d)

The boundary c i r c u i t must cons is t of 2m edges.

There must be i n pN* at l e a s t one 'edge i d e n t i f i c a t i o n

scheme' I, which i s t o be defined as fallows:

spanning t r e e Tf ;

I n pN* d r a w t h e face-

l a b e l edges of t h e boundary c i r c u i t of pNw

# According t o Panofsky, (1) DUrer, i n his t r e a t i s e on s o l i d s was DUrer's schemes were face- t h e first t o develop t h e idea of t h e net .

ne ts , but t h e idea of a net can be general ized so as t o include

appropriately defined schemes of K-dimensional elements from K

Kn-l. daughter of George Boole) proposing K

1 to Thus at t h e beginning of t h e century we f i n d Al ic ia S t o t t ( t h e

ne ts f o r t h e hyper-solids. (2) 3

208

Hill: Some Problems from the Visual Arts 209

a,b,a ,b , . . . successively and match a edges with b edges, with l i n e s

i n t e r s e c t i n g t h e edges t o connect t o poin ts of Tf .

we s h a l l c a l l G(I ) , cons is t s of t h e nodes and edges of T f , p lus t h e

isomorphic to t h e graph of some polyhedron PI, then pN* = PN*.

shor t , it w i l l ' f o l d up' t o make a polyhedron. For a pN* = PN*, P ' w i l l be t h e dual of P.

one G ( I ) , such t h a t G ( 1 ' ) = P ' ;

r e s u l t i n some P , I must conform t o t h e following:

The graph, which

m new edges t h a t comprise I. I f pN* contains a G ( 1 ) which i s

I n

But pN* may have more than

For a l l I of PN* t o G ( 1 " ) = P", e tc .

(1). Clearly no two edges of a face i n pN* with t h e same l a b e l

a r e t o be matched.

( 2 ) .

sharing an edge.

(3).

Adjacent faces of pN* w i l l not be matched, i . e . faces

No more than one edge o f a face i n pN* w i l l be matched t o

edges of some other face.

f ( 4 ) . No edge of T may belong t o two t r i a n g u l a r faces i n G ( 1 )

if a l l p i n PN* a r e t r i a n g u l a r ,

These conditions seem s u f f i c i e n t t o ensure t h a t G ( 1 ) is isomorphic t o

some P, where P i s both simply and t r i p l y connected.

except when P i s t h e te t rahedron

I n f i g . 1 we show t h e construct ion of I on t h e smallest PN* which i s

not s e l f dual.

/ I I I I \ \ \

f i g . 1

To obtain PN* from P we d r a w t h e map of P ( i t s Schlegel diagram) and

l o c a t e Tf on P.

point spanning t r e e Tn of P, t h e s e then def ine a separa t ion scheme s

on P, t h e r e s u l t of which i s a PN* of P, as shown i n f i g . 2.

The edges i n P not crossed by those of Tf form a

210 Annals New York Academy of Sciences

9 10

8 v7 9 10

f i g . 2

Carrying out t h e respec t ive operation on t h e map of P' we obtain a PN*

of P I , i n t h i s case employing t h e Tn of P' f o r t h e seperat ion as shown

i n f i g . 3. implicat ions of t h i s we discuss later.

The r e s u l t i n g ne t i s t o be taken as a dual n e t , t h e

1

5

I 7 : x /

0

f ig , 3

I n any P t h e r e are mTf and m+n Tn (Tf of P = Tn of PI , and v i c e verse) .

For a start we need t o know t h e values of m and n i n any P.

Since every unique PN* of P i s t h e r e s u l t of a seperat ion employing a

Tn of P, t h e number of PN* of P is known when we have found t h e number

of T (and / o r T f ) i n P

t h e map of P.

t h e number of waJrsin which they can span

( i n a self dual s o l i d it is s u f f i c i e n t t o count only

Hill: Some Problems from the Visual Arts

one kind of spanning t r e e s ince T = T f ) .

21 1

t h e conditions t h a t must hold f o r a pN* t o be a PN*, w e

Eb/2 + Ei = E of P , when Eb are t h e boundary edges of pN*

E, + 1 = number of faces i n P

Eb/2 + 1 = number of nodes i n P

t h e poin ts i n pN* with t h e l a r g e s t valency v' must have

t h e i n t e r i o r edges.

1 ) i n P

The Platonic s o l i d s and dual ne ts

The s o l i d s , i n t h r e e dimensions, i n which each s e t of K elements i s

respect ively ind is t inguishable , i .e . s o l i d s i n which t h e network i s a

completely regular symmetric planar graph, make up t h e s m a l l family of

t h e f i v e Platonic s o l i d s .

obtained t h a t answered t h e problem of t h e number of PN* when P = a

Pla tonic so l id . But t o d a t e we have only r e s u l t s f o r t h e f i r s t t h r e e ;

t h a t t h e te t rahedron has j u s t two f a c e n e t s and t h e cube, eleven, has

of course been observed and commented upon, it is b r i e f l y c i t e d i n

Steinhaus (3) .

To date we have been ab le t o add only one o ther r e s u l t , it is t h a t

t h e octahedron has ekeven ne ts , t h e same number as i t s dual , t h e cube.

From t h i s f a c t it seems reasonable t o conjecture t h a t t h e remaining

two w i l l a l s o have t h e same number of n e t s , s ince each i s t h e dual of

t h e other . (We r e f e r of course t o t h e dodecahedron and icosahedron).

A f u r t h e r conjecture i s t h a t s ince Tf + Tn = t h e number of ne ts i n

t h e case of t h e t h r e e w e have s tudied, t h i s should hold f o r t h e

remaining two.

It w a s hoped t h a t r e s u l t s would have been

It remains t o be shown under what condi t ions PN* = PIN*, such t h a t i n

each P/P'N* t h e r e e x i s t s j u s t one I tak ing P i n t o P ' . This r e l a t i o n -

sh ip i s probably a s p e c i a l case, although not necessar i ly l i m i t e d t o

completely symmetric and / or three-dimensional so l ids .

Another s e r i e s of r e s u l t s of i n t e r e s t a r e those f o r t h e family of

self-dual n-gon based pyramids, when n = 3, 4 , 5 , ... . Enumeration

of t h e PN* of t h i s c l a s s of P could be taken as t h e answer t o t h e

problem what i s t h e number of T ?


F i r s t we have t o f i n d t h e number o f topological ly d i s t i n c t Tn, and

then t h e number of ways ('non-congruent') i n which they can belong t o

P as a subgraph.

The counting s e r i e s so far gives:

P3 = 2, P4 = 8, P5 = 15, P6 = 33. I n f i g 4 we show t h e 3 topologica l ly d i s t i n c t T f o r P and t h e non- 4

which they can appear congruent ways i n

f i g . 4

Face-nets from t r i a n g u l a r animals One approach i n t h e search fo r pN* t h a t are PN* l i e s i n t h e inspect ion

of tree-shaped animals. We s h a l l denote by A: an animal with n c e l l s ,

each c e l l of m edges.

Clear ly only a l i m i t e d number of PN* can be discovered amongst t h e

animals on t h e square and t r i a n g u l a r l a t t i c e s ;

11 PN* f o r t h e cube.

number, and by relaxing t h e approach so as t o become topological i n

essence, we are not l imi ted t o del tahedra, f o r , by considering t h e

general case of p o s s i b i l i t i e s i n A

accomadate d i s t o r t i o n s and / o r t h e PN* t o be made of rubber.

4 Am giving us only t h e

But t h e t r i a n g u l a r l a t t i c e y i e l d s a s i g n i f i c a n t

3 we assume t h e l a t t i c e t o

When m = 2 , of t h e 3pN* two are immediately recognizable as PN* o f t h e

te t rahedron (unique i n t h a t it has j u s t two PN*). When m = 3, we f i n d


t h a t amongst t h e 11 tree-shaped pN* t h e r e are j u s t 6 t h a t a r e PN*,

where P i s t h e dua l of t h e t r i a n g u l a r pr ism, we s h a l l c a l l t h i s

f i g u r e t h e ( t r i a n g u l a r ) hexahedron. In spec t ion of t h e PN* f o r t h e

hexahedron found t h a t one o f t h e s e gave r i s e t o two G ( I ) , and t h i s

became t h e f i r s t ca se where P # P I , i . e . t h e prism had one more PN*

than i t s dual .

We cons ide r next t h e r e s u l t s ob ta ined when m = 4. Of t h e 66 animals , 62 are t r e e shaped and appear t o supply a l l t h e nets

f o r bo th t h e r e g u l a r octahedron and an a l t e r n a t i v e octahedron wi th

node values 5 .5 .4 .4 .3.3.

r e spec t ive ly .

We f i n d t h a t j u s t t h r e e a r e PN* of 0 a l o n e , so t h a t 0 ' has a t l ea s t

59 PN*, as a g a i n s t 11 f o r 0.

In spec t ion o f t h e number of G(1) i n t h e PN* for 0' r e v e a l s as many as

69 m a l l t h a t t a k e 0 ' i n t o i t s dua l . This i s t h e second example

where PN* # PIN*.

It would seem from t h e p rev ious example (p r i sm & hexahedron) t h a t with

dual P, whichever has t h e g r e a t e r number of nodes (and less f a c e s

t h e s e being more than one k i n d ) w i l l have t h e g r e a t e r number of PN*,

when and i f PN* # PIN*.

We s h a l l r e f e r t o t h e s e as 0 and 0'

Thus from t h e 62 PN*, f o r m = 4 , we can d e r i v e 100 d i s t i n c t G(1) and

observing t h a t some PN* have up t o 4 t h i s r a i s e s t h e gene ra l ques t ion :

what i s t h e number o f a l l G ( 1 ) i n any PN*?

3 With r ega rd t o pe r su ing t h e sea rch for PN* amongst AZm, as m i nc reases ,

t h e number of P with v >, 5 diminishes . Of course i f we could i n s p e c t

f u r t h e r t ree-shaped animals for va lues of m up t o 10 , w e cou ld

d i scove r PN* for a number of P, i nc lud ing t h o s e of t h e icosahedron!

Examination of t h e t o p o l o g i c a l l y d i s t i n c t Tf t h a t appear i n 0 and 0 '

shows t h a t t h e r e are 11, 5 of which appear only i n 0 while a l l of them

appear i n 0 ' . Of t h e 11, t h e one t h a t p rov ides t h e g r e a t e s t number of

t o p o l o g i c a l l y d i s t i n c t PN* is t h e Tf w i th no branch p o i n t s , of which

t h e r e are two kinds - t h e Hamiltonian and t h e non-Hamiltonian.

One s p e c i e s of PN* whose Tf i s of t h e k ind we have j u s t mentioned i s


t h e r ibbon; t h i s i s a PN* and / o r animal whose f a c e s are con ta ined

between two p a r a l l e l l i n e s .

The cube cannot have a r ibbon amongst i t s PNX bu t when we t a k e t h e

c a s e of A2m, f o r a l l va lues of m t h e r e would seem t o be a t l e a s t one

PN* t h a t i s a r ibbon. So i t fol lows t h a t any r ibbon of 3 n t r i a n g l e s

i s a PN* for some P.

3

One such s e r i e s of P permit ing a r ibbon PN* a r e contained i n t h e family

of t h o s e p f o r 2m ( t r i a n g u l a r ) f a c e s which have t h e node va lues :

3 . 3 . 3 . 3

4 .4 .4.3.3.

5 .5 .4 .4 .3 .3 .

6.6.4.4.4.3.3.

2. On t h e number of Mondrians

The i d e a of a p r e c i s e a n a l y s i s of a l l t h e a b s t r a c t (and p u r e l y

o r thogona l ) compositions o f Mondrian, of enumerating f u r t h e r

p o s s i b l e Mondrians presupposes an a p p r o p r i a t e methodology.

Amongst t h e var ious approaches open, t h e one p resen ted h e r e i s based

upon t h e fol lowing cons ide ra t ions .

Without embarking on a comprehensive account of t h e d i f f e r e n t k inds of

t h e s e composi t ions, w e can s t a t e that o f t h e 160 or so p a i n t i n g s

Mondrian made between 1917 and 1944 ( t h e y e a r of h i s d e a t h ) almost a l l

of them con ta in a 'network' of b l ack ' l i n e s ' of v i r t u a l l y uniform

th i ckness . Thus i f w e i gnore a l l o t h e r c c n s i d e r a t i o n s ( e . g . t h e

d i s t r i b u t i o n of coloured areas;

network as t h e l i n e a r ' i n f r a - s t r u c t u r e ' o f t h e compositions. I n t h i s

approach we t h e r e f o r e a r e concerned s o l e l y with t h e topology of a

network and / o r o f a d e f i n a b l e system of c o n s t r u c t i n g d i s t i n c t sub-

s e t s of a l a t t i c e a r r a y .

p ropor t ions e t c . ) w e can conceive t h e

The f i r s t a t tempt by t h e au tho r t o s e t up such a s tudy , w a s a pape r

of 1968 ( b ) , coupled with some f u r t h e r obse rva t ions i n ano the r

p u b l i c a t i o n of t h e same y e a r ( 5 ) .


A s a r e s u l t o f pub l i sh ing t h e above,F. Nake, of t h e Un ive r s i ty of

S t u t t g a r t , wrote t o say t h a t he had planned a s t a t i s t i c a l a n a l y s i s o f

t h e same compositions, u s ing a computer.

I t was agreed t h a t i f one could deduce something l i k e a se t of

'Mondrian axioms' t h i s would imply t h e p o s s i b i l i t y o f c o n s t r u c t i n g

more 'Mondrians' ( a f i r s t a t tempt a t deducing t h e Mondrian axioms w a s

made i n (4)). An agreement t o c o l l a b o r a t e on t h e s e proposals has l e d t o t h e au tho r s

p re sen t ing t h e problem t h a t i s d i scussed below.

However it would seem u s e f u l t o first d e s c r i b e some gene ra l i s s u e s

which supply t h e con tex t t o t h i s problem.

I n gene ra l we a r e concerned wi th a l i n e s t r u c t u r e which can be con-

ceived as a graph G. I t s most important (a l though obvious) f e a t u r e i s

t h a t it must permit a l a t t i c e r e p r e s e n t a t i o n . By t h i s we mean t h a t

t h e a b s t r a c t graph G i s isomorphic t o a p l a n a r geometr ic graph G ' =

(V' , E ' ) such t h a t a l l edges of G ' a r e p a r a l l e l t o e i t h e r o f t h e two

d i r e c t i o n s of a r e c t a n g u l a r l a t t i c e and such t h a t t h e s i d e s of t h e

surrounding r e c t a n g l e of G' a r e a l l contained i n E ' .

With r ega rd t o t h e compositions under examination, we f i n d t h a t

t h e r e a r e some 125 or sa# where t h e l i n e a r i n f r a - s t r u c t u r e comprises

a connected graph. Furthermore we conceive of t h e s e graphs as be ing

homeomorphically i r r e d u c i b l e , f o r i f we t a k e t h e c e l l s i n such a

graph t o be s u b s e t s o f t h e i n f i n i t e l a t t i c e , a l l of them are convex.

By f a r t h e l a r g e s t p ropor t ion of t h e i n f r a - s t r u c t u r e s employed by

Mondrian t u r n out t o be c l o s e d graphs, and wi th t h e except ion of

j u s t t h r e e of t h e s e , which are l e s s t han t r i p l y connected, t hey a r e

a l l isomorphic t o t h e graphs of polyhedra.

Concerning po lyhedra l graphs i n g e n e r a l , amongst t h o s e t h a t permit a

l a t t i c e r e p r e s e n t a t i o n we f i n d t h a t i nc luded a r e t h r e e of t h e f i v e

#

square but i n which t h e square i s t u r n e d 45 degrees .

compositions ( a l l o r thogona l , i n t h e same sense as t h e o t h e r s ) a r e

known as Mondrian's ' l o z e n g i c a l ' compositions.

A s p e c i a l ca se i s made f o r t h e a d d i t i o n a l 17 composi t ions which a r e

These


completely regular graphs. We show i n f i g . 5 these three 'cubic '

graphs :

f i g . 5

When we ask what i s t h e smallest polyhedral graph, p e d t i n g a l a t t i c e

representat ion, which i s asymmetric (has group order 1) we f i n d t h a t

t h i s e x i s t s i n j u s t one form. The graph has 6 i n t e r i o r c e l l s and i s

shown i n f i g . 6 with t h e 5 d i f f e r e n t l a t t i c e representat ions it can

take:

f i g . 6

It i s s i g n i f i c a n t t h a t Mondrian, who always avoided Euclidean symmetry,

(while c l e a r l y having no conception of abs t rac t symmetry) never

employed completely symmetric graphs ( t h e 3 ' c u b i c s ' ) . However he

did employ asymmetric graphs, t h e example shown i n f i g . 6 w a s employed

i n at l e a s t 5 compositions.

We have ye t t o discover t h e proportion of h i s schemes t h a t are

asymmetric, and we hope t o enumerate asymmetric graphs permiting a

l a t t i c e representat ion.

One problem t h a t t o us seems t o be of i n t e r e s t , o n t h e way t o enumerate


a l l p o s s i b l e Mondrians can be s t a t e d i n t h e fol lowing way.

We c a l l a p l a n a r connected und i rec t ed graph without loops and w i t h a l l

p o i n t s ( v e r t i c e s ) having degrees 3 or 4 an M-graph i f f it has a

l a t t i c e r e p r e s e n t a t i o n . L e t C be t h e c l a s s of a l l non-isomorphic

M-graphs wi th r i n t e r i o r c e l l s .

f o r r = 1, 2, ... . Then t h e problem i s t o enumerate C

The t a b l e below g ives t h e r e s u l t s achieved so f a r f o r r = 1 (1) 6.

These f i g u r e s were ob ta ined by a g e n e r a t i v e p rocess which w e a r e going

t o exp la in .

for each c l a s s of isomorphic graphs of C must be chosen, perhaps i n

some s t anda rd form).

fol lowing t y p e s of c o n s t r u c t i o n t o g e t a graph w i t h r+l c e l l s :

a. add one c e l l t o t h e l e f t , t o p , r i g h t , bottom r e s p . of G ;

b.

Let us assume we have C for a given r (one r e p r e s e n t a t i v e

We t a k e each graph Ge C and apply t h e

d i v i d e an i n t e r i o r c e l l "along t h e l a t t i c e " whenever t h i s i s

p o s s i b l e (it i s p o s s i b l e for a l l c e l l s g r e a t e r t han one l a t t i c e

un i t ) ;

c. d i v i d e an i n t e r i o r c e l l "ho r i zon ta l ly" and " v e r t i c a l l y " r e s p . i n

between success ive l i n e s of t h e l a t t i c e , and redraw t h e ob ta ined

graph on t h e l a t t i c e .

D 7----7 'hi m r::: :::m

; _ _ _ _ - ; ' J -

-TIcm ~.---..~ ~- - - - - - - -

[ T i pl-1 7 - - [ T ] L..

C - - - _ - _ J L.. ' I

f i g . 7


(The process i s out l ined i n Fig. 7 ) . graphs from each member of C . check i f it i s isomorphic t o one derived already.

case we keep t h e graph.

Thus we get a t l e a s t 4+2r new

Whenever we generate a new graph we

I f t h i s i s not t h e

Denote t h e s e t of a l l non-isomorphic graphs

so generated by Br+l . Then c l e a r l y B r+l= cr+l' For , by

construct ion, each G e Br+l i s an M-graph and has r+l i n t e r i o r c e l l s .

We d idn ' t prove t h e inclusion B r+13 Cr+l ye t .

t h i s , we would be i n a pos i t ion t o wr i te a computer program t h a t

constructs and therefore enumerates C . Par t of it must be an

algorithm t o check f o r isomorphism of two M-graphs.

In f i g . 8 we show a way of represent ing t h e r e s u l t s f o r r = 1 (1) 5.

I f we were able t o get

f i g . 8

While f o r every M-graph t h e r e e x i s t representat ions on t h e l a t t i c e

with a minimal a r ray , we have chosen t o draw them here with t h e

surrounding rectangle as a square.


f i g . 9

I n f i g . 9 we show t h a t t h e po lyhedra l M-graphs f o r r = 6 , i n f i g . 8 po lyhedra l graphs a r e shown wi th a t h i c k e r l i n e .

3. I d e n t i t y Rraphs de r ived from Trees

Various p l a n a r i d e n t i t y graphs G , i n p a r t i c u l a r t h e smallest cubic

i d e n t i t y g raphs , can be ob ta ined by j o i n i n g up success ive end-points

of p l ane t rees T ( f i g . 10 ) . One might t h i n k t h a t i f T i s an

1- - - - - - -

* - - - .c-

'r; ?--I- f i g . 10


i d e n t i t y t r e e then G i s necessar i ly an i d e n t i t y graph, but counter-

examples are e a s i l y found ( f i g . 11). In f a c t a l l counter-examples a r e

f i g . 11

of t h i s kind, i . e . "hammocks" ( f i g . 1 2 ) . This has already been proved

( 6 )

f i g . 1 2

We pose t h e question:

'asymmetrically embedded' i n t h e plane? --- t h a t i s , i f T does not

(even a f t e r a d i s t o r t i o n of t h e p lane) admit a Euclidean symmetry:

r o t a t i o n about a point ( o r i e n t a t i o n preserving) o r r e f l e c t i o n i n a

l i n e (or ien ta t ion revers ing) --- see f i g . 13. O f course, we have t o

exclude t h e "hammocks";

i s G necessar i ly an i d e n t i t y graph i f T i s

f i g . 13

Hill: Some Problems from the Visual Arts 22 1

but i n any c a s e we a r e r e a l l y i n t e r e s t e d only i n homeomorphically

i r r e d u c i b l e t r e e s , i . e . t h o s e without p o i n t s o f degree 2 , and t h e only

"hammock" graphs a r i s i n g from such t r e e s a r e as i n f i g . 14 and t h e

t r e e s he re are

f i g . 14

symmetr ical ly embedded.

Nor it i s easy t o see t h a t i f G admits a non- iden t i ca l automorphism a

which t a k e s t h e o u t s i d e c y c l e c on to i t s e l f then ( a l lowing a p re l imi -

nary d i s t o r t i o n , of cour se ) we can r e a l i z e u by a Eucl idean symmetry,

and s o T is symmetr ical ly embedded. For example i f , i n f i g . 15,

f i g . 1 5

e t c t h e n a w

p o i n t ( s ) , r e a l i z i n g a by a r o t a t i o n .

t h e c e n t r e ;

an i n v o l u t i o n (a2 = t h e i d e n t i t y ) and can be r e a l i z e d by a r o t a t i o n

through 180' or as a r e f l e c t i o n i n a l i n e . )

= w2 e t c , and we can work our way i n t o t h e c e n t r a l

i n f a c t t h e r e w i l l be on ly one c e n t r a l p o i n t un le s s a i s

(There w i l l b e no t r o u b l e a t

We can be s u r e t h a t c goes onto i tsel f under every automorphism o f

G i f we can i d e n t i f y c " a b s t r a c t l y " , t h a t i s , i n t h e a b s t r a c t graph G .

Now R . O . Davies has shown ( 7 ) t h a t , i f G i s not a "hammock", c i s t h e


unique c y c l e o f maximal l e n g t h among t h e cyc le s o f G t h e removal of

t h e edges of which from G l e a v e s a tree. Thus c is c h a r a c t e r i z e d

a b s t r a c t l y , and t h e ques t ion i s answered.

The r e s u l t can make it easy t o check t h a t a given p l a n a r graph G i s

an i d e n t i t y graph: i f w e can f i n d a spanning s u b t r e e T f o r G , and

draw T and G i n t h e p l ane i n t h e i n d i c a t e d way, t h e n w e need on ly

look f o r Euclidean symmetries. For example: i f w e t a k e t h e t ree i n

f i g . 16, w e s e e t h a t it has two d i s t i n c t embeddings; one i s

symmetric ( a ) , t h e o t h e r i s asTJmmetric ( b ) . If we choose t h e

asymmetric embedding t h e r e s u l t is t h e i d e n t i t y graph ( c ) - l i k e t h e

graph i n f i g . 10, it i s a minimal cubic i d e n t i t y graph ( t h e r e a r e

j u s t two such g raphs ) .

( a )

The graph shown i n f i g 6 - t h e minimal asymmetric M-graph was

a r r i v e d a t by t h e same c o n s t r u c t i o n ;

asymmetric ernbedding bu t no symmetric embedding.

i n t h i s ca se t h e t r ee has an

References :

1. PANOFSKY, I. 1955. Albrecht D U r e r . Oxford Un ive r s i ty P r e s s .

(The t r ea t i s e of W r e r r e f e r r e d t o i s t h e r e v i s e d e d i t i o n of

1938) .

2 . COXETER, H.S.M. 1948. Methuen & Co. (The paper o f Mrs. Boole

r e f e r r e d t o d a t e s from 1900) .

3. STEINHAUS, H. 1964. 100 Problems i n Elementary Mathematics.

Pergamon P r e s s .


4. HILL, A . 1968. A r t and Mathesis : Mondrian's S t r u c t u r e s .

Leonard0 Vol. 1 no. 3.

5. HILL, A. 1968. Programme. Paragram. S t r u c t u r e . DATA

Di rec t ions i n A r t , Theory & A e s t h e t i c s , ( e d i t e d by Anthony H i l l )

Faber & Faber. New York Graphic S o c i e t y Ltd.

6. BALABAN, A . T . , DAVIES, R.O. , HARARY, F . , HILL, A. & WESTWICK, R .

1970.

J. A u s t r a l i a n Math. SOC. (Forthcoming or j u s t pub l i shed)

Cubic i d e n t i t y graphs and p l a n a r graphs d e r i v e d from trees.

7. DAVIES, R.O., & HILL, A. 1970. Unpublished MS.

some problems from the visual arts

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