PII: 0898-1221(89)90257-5 Computers Math. Applic. Vol. 17, No. 4-6, pp. 709-713, 1989 0097-4943/89 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1989 Pergamon Press plc M O D E R N S Y M M E T R Y A. HILL Department o f Mathematics, University College London, Gower Street, L o n d o n W C I E 6BT, England Abstract--In previous studies [1, 2], the a u t h o r - - w r i t i n g primarily as a geometric abstract a r t i s t - - h a s attempted an approach to symmetry/asymmetry from a phenomenological point of view taking mathematics to be regarded as " . . . the theoretical phenomenology of structure" [3]. T h e main ideas o f what is accepted as the m a t h e m a t i c a l t r e a t m e n t o f s y m m e t r y have an extremely long history. It is not even clear that the study originates with the ancient Greeks; suffice to say it can be loosely regarded as a p a r t o f g e o m e t r y an d as such is t h erefo re one o f the earliest forms o f " s c i e n c e " we can find. W h a t we can say is th at so far the earliest example o f a " s y s t e m " remains the b o o k s o f Euclid, a n d it is generally accepted t h at the priority here was the m e t h o d o l o g y r a t h e r t h a n the individual theorems. H o w e v e r , a l t h o u g h research m a y change this, " E u c l i d " is a c o r n e r s t o n e in Western mathematics, a n d the " e l e m e n t s " include topics n o t restricted to q u an - titative a n d " m e t r i c a l " geometry. As a system or b r a n c h o f m a t h e m a t i c s the p a r t we call plane g e o m e t r y is closed, there are n o m o r e t h e o r e m s to be discovered therein. H o w e v e r , this is n o t true o f m a n y topics we can regard as initiated by " E u c l i d " such as projective g e o m e t r y an d p o l y t o p e theory. I f o n e is interested in symmetry, the implications o f the "classical c o n c e p t " , we find t h at m at h em at i ci an s an d also physical s c i e n t i s t s - - p a r t i c u l a r l y c h e m i s t s - - w e r e initiating a m o r e ab st ract c o n c e p t o f s y m m e t r y which nevertheless could be seen as the result o f c o n t e m p l a t i n g f u n d a m e n t a l features o f very simple " s t r u c t u r e s " . It is tempting to call this c o n c e p t topological symmetry, b u t the t erm has n o t gained a n y c u r r e n c y despite the fact that the argum en t , i f naive, is neither illogical n o r a solecism. T h e n o t i o n will suggest that what is called s y m m e t r y a n d a s y m m e t r y can exist as features o f a co n n ect ed structure which remain invariant u n d e r certain simple d e f o r m a t i o n s such t h at the feature t h en can be regarded as strictly qualitative and i n d e p e n d e n t o f q u an t i t at i v e considerations, thus belonging to the elastic g e o m e t r y - - " e l a s t i c lines" as in " r u b b e r sheet" geometry, as t o p o l o g y is o ft en described in the literature o f popularizing (science a n d mathematics). W h a t are k n o w n as the five Platonic solids were generally conceived as literal "so l i d s", the fo rm s k n o w n as the t e t r a h e d r o n , cube, o c t a h e d r o n , d o d e c a h e d r o n , icosahedron, m o s t generally conceived as crystallographic o r volumic " s c u l p t u r a l " forms. Essentially o f course they can be seen as a n o t h e r set o f forms, and p e r h a p s the earliest well-known examples are the drawings o f L e o n a r d o d a Vinci in which they a p p e a r as " s k e l e t a l " forms. These representations were the step which led to the g e o m e t r y o f the solids being represented " s c h e m a t i c a l l y " so t h at the prism-shaped " l i m b s " ( " e d g e s " ) o f L e o n a r d o ' s "closed lattices" cou l d be replaced b y the lines ( " w i r e s " in the case o f a model) or the pencil lines o n p a p e r o f the linear models. T h e final stage in this d e v e l o p m e n t h ad to wait for the spirit o f the m o d e r n / a b s t r a c t way o f conceiving s t r u c t u r e s - - w e finally arrive at the Schlegel d i a g r a m in which the lengths o f the lines a n d the area o f the faces n o longer c a r r y o v er the " s y m m e t r y " o f the figures represented. Finally it becomes immaterial w h et h er the "l i n es" follow a n y regular feature, i.e. they need n o t be " s t r a i g h t " , an d in drawing them with " c u r v e s " these curved lines m a y in fact be uniquely different. W e have " j o i n e d u p " o r c o n n e c t e d a set o f points just as we please, a n d what is d r a w n can still be regarded as pertaining to some f o r m o f physical structure; the w a y w a r d paths o r " c o n n e c t i v e s " can be " s e e n " as elastic strings each o f which seems to have u n d e r g o n e a unique d e f o r m a t i o n as if the "el ast i ci t y " o f each line were intrinsically different or unique. It is often p o i n t e d out as strange that the ancient G reek s did n o t notice the f u n d a m e n t a l qualitative law which holds between the relation o f the dimensional elements o f a p o l y h e d r a l structure, n o w referred to as points (zero-d i m en si o n al )---"co rn ers"; lines ( o n e - d i m e n s i o n a l ) - - " e d g e s " ; and p o l y g o n s (two-dimensional) o r " f a c e s " . It seems Descartes was almost able to grasp CAMWA 17/4-6---P 7 0 9 710 A. HILL the n o t i o n but it was Euler by whose n a m e the fam o u s t h e o r e m is k n o w n , n o t surprisingly since Euler was a f o u n d i n g father o f t o p o l o g y a n d with his t h e o r e m a vast edifice o f t h eo rem s having to do with connectivity was initiated. I f we l o o k at the newer f o r m o f g e o m e t r y as the study o f structural features o f " a m o r p h o u s " o r informal linear structures, schematic diagrams o f degrees o f connectivity, what we t h en say o f the five Platonic structures is that each o f the respective sets o f elements is unidentifiable, interchangeable; we have a structure o f u t m o s t r e g u l a r i t y - - a n d r e d u n d a n c y - - b e l o n g i n g to the set o f regular coverings in two-dimensional space: plane tessellations either i n f i n i t e - - t h e three lattices m a d e respectively f r o m three-, four- a n d six-sided cells o r p o l y g o n s - - o r closed, as in the five closed systems which exhaust the possibilities f o r a closed system. W h e n we ask: w h at else is there? the answer is that we can exhibit structures all o f whose respective elements are distinguishable an d permit no interchanging. Such structures are described as asymmetric. Finally we show t h at a structure m a y have some o f its respective elements interchangeable while some remain identifiable; we d o n ' t call these " b o t h symmetric and a s y m m e t r i c " b u t say t h at they are symmetric b y virtue o f exhibiting some symmetries. It was only in the last c e n t u r y t h a t the d a u n t i n g task o f e n u m e r a t i n g all possible p o l y h e d r a l structures a t t r a c t e d the a t t e n t i o n o f mathematicians. While m u c h has been learnt between the first efforts a n d t o d a y , n o - o n e is very confident t h a t the p r o b l e m is going to suddenly b e c o m e easy an d in due course solved. F r o m the point o f view o f s y m m e t r y we discover t h a t when we ask a b o u t the possibilities the answer is that p o l y h e d r a with a n y s y m m e t r y at all fade o u t o f the " c a t a l o g u e " as the " s i z e " gets greater, i.e. as the n u m b e r o f vertices (points o r corners), edges (or lines) a n d faces increases. So, nearly all p o l y h e d r a are asymmetric! T h e r e are t h en three sets o f symmetric polyhedra: the f a m o u s Platonics, a mere five; the n o less f a m o u s Aristotelian solids exhibiting an almost equally high degree o f symmetry, o f which there are 13, a n o t h e r set revered, an d rightly so, by the ancient greeks; and lastly an infinite b u t diminishing set which struggle fo r existence, as one might p u t it, exhibiting various "degrees o f s y m m e t r y " . By c o n t r a s t we can c o n s t r u c t (or " e x h i b i t " ) a family o f linear structures which, conversely, are linear all symmetric. This family did n o t " e x i s t " until it was " i n v e n t e d " at the t u rn o f the century. T h e family or f o r m is k n o w n as a tree a n d we can confidently a d d t h at it was there all the time b u t had n o t been l o o k e d at mathematically. A tree, like the Schlegel diagram and maps, is a topologically linear structure, one-dimensional since it consists o f lines a n d points, b u t so c o n n e c t e d t h a t there are n o closed a r e a s - - l o o p s , polygons, circuits, f a c e s . . . T h e e n u m e r a t i o n o f classes o r families o f trees has been solved a l t h o u g h the overall p r o b l e m c o n t i n u e d to l o o k r a t h e r intractable until recently. A m o n g s t symmetric trees there is one special family or species which can be defined such that, h o w ev er large, they will always be symmetric. Such trees, while being quite simple structures, have as yet n o simple short n am e and are k n o w n as homeomorphically irreducible. T h e instruction goes like this: y o u r tree m u st n o t c o n t a i n points o f degree two, which means th at each p o i n t connects at least three lines o r only one line, the latter being called the terminal points o f the tree. F o r example, some capital letters o f the R o m a n a l p h a b e t are trees o f this k i n d - - T , Y, X, G , H , K - - w h i l e others are not, having points o f degree t w o - - A , E, F, L, M, N, V, W, Z. Let us l o o k at the following question o f " p a t t e r n m a k i n g " - - i t can be l o o k e d u p o n as purely m a t h e m a t i c a l or as belonging to gestalt t h e o r y o r even aesthetics: we wish to p a r t i t i o n circles with h o m e o m o r p h i c a l l y irreducible t r e e s - - w e will call t h em hitrees. I f we take each hitree in t u rn and Fig. 1 Modern symmetry Fig. 2 711 use it as a f o r m o f p a r t i t i o n we s o o n discover t h a t we can o b t ai n several distinguishable partititions f r o m the same hitree. In m a t h e m a t i c a l terms we say t h at a tree m a y take a distinct n u m b e r o f embeddings a n d this is a feature o f its symmetry. T h u s a tree m a y p erm i t o n l y o n e embedding, as shown in Fig. 1 (and here we keep to hitrees). While others will clearly allow m o r e t h a n one (see Fig. 2). I f we were to exhaust all such p a t t e r n s o r partitions, u p t o say hitrees with 20 lines, we would s o o n discover t h a t an increasing n u m b e r o f th em would be asymmetric despite the fact t h at all o f the hitrees e m p l o y e d a n d all t h a t we m a y ch o o se are symmetric. N o w it should c o m e as n o surprise t h a t each p a t t e r n (or partition) is in fact a rep resen t at i o n o f a p o l y h e d r o n , if we replace the circle by a p o l y g o n so t h at the terminal points o f the hitress are joined by straight lines o r edges we have ch an g ed n o t h i n g b u t the new " d i a g r a m " o r " f i g u r e " can be t a k e n as a Schlegel diagram. Thus, there exists a species o f p o l y h e d r o n generated in this manner: m a t h e m a t i c a l l y they would be described as having a homeomorphically irreducible spanning t r e e . R e t u r n i n g to o u r plane patterns o r partitions (or maps) we could ask t h a t the circle be replaced by a r e c t a n g l e - - f o r example a square; let us also specify t h a t the lines o f the tree p art i t i o n i n g o u r square are to be parallel with its sides. O u r p a t t e r n consists o f h o r i z o n t a l an d vertical lines only, thus the cells or areas o f the p a r t i t i o n will be "rect an g l es". In this special family o f partitions the recognizable (but n o t topologically) distinct partitions o f the circle with " o r t h o g o n a l l y e m b e d d e d " trees p r o d u c e s m a n y m o r e possibilities and these are o f course increased w h en the circle is replaced by the square (see Fig. 3). Whereas we can find o n l y two for one o f o u r trees when the s u r r o u n d is a circle, as so o n as the circle is replaced by the square it becomes obv i o u s t h at the n u m b e r is m o r e t h a n d o u b l e a n d the reader can see t h a t it is n o t h a r d to r e p r o d u c e the o t h e r four. Elsewhere I have discussed the fact that these o r t h o g o n a l partitions constitute the m o s t characteristic c o m p o s i t i o n a l schemes o f the ab st ract paintings o f Pier M o n d r i a n , certainly between the years 1918-44. T h e c o m p u t e r scientist F r i e d e r N a k e [4] a n d I were able t o p r o p o s e h o w to e n u m e r a t e all possible " M o n d r i a n s " in the rect an g u l ar fo rm at . This can be extended t o deal with the lozengical f o r m a t which M o n d r i a n frequently a d o p t e d , a n d this in t u rn comes u p with some surprising results. T h e example on the right-hand side is in fact the scheme chosen fo r w h at m u st be one o f M o n d r i a n ' s m o s t strikingly simple c o m p o s i t i o n s as the painting consists o f just the two intersecting black lines (or in the hitree) o n a white lozengical format. O f co u rse w h at the viewer is c o n f r o n t e d Fig. 3 712 A. HILL Fig. 4 with contains many other features, the lozenge appears not to be a perfect square, the lines are not o f exactly the same thickness, and when we examine the resulting polygons it is clear that even if the lines were of the same thickness the arrangement has no metrical symmetry, the two areas adjacent to the two sides of the triangular area are not of the same size, and it follows that the remaining area is not "symmetric". Bisecting the triangle and extending the "axis" helps one see this to be the case. One need hardly add that depending where the point o f intersection o f the lines (the four degree node o f the tree) is placed the artist could choose a great number o f possibilities by which he could in this manner partition the lozenge. Despite his importance for m o d e m art as one of the most respected "geometric" abstract painters and a founder o f abstract art, Mondrian took no interest or inspiration in any aspect of mathematics, not the time-honoured golden section nor any other formula; the idea o f calculation was inimical to him; his works, although extremely rigorous and perfectionist, resemble more the free toccata than the canonical fugue. It is often stressed in art historical exegesis that he belonged to the mystical and spiritual stream; whatever truth there may be in that his essential importance is that o f a radical plastician, perhaps the very last painter, a constructive painter no longer relating to La Belle Peinture and the continuous rhetoric of painting as it had been from Lascaux to Van Gogh. His ideas o f space, time, surface, structure came out o f cubism and took painting--as with other great innovators of the time--to the position o f the tableau object, the autonomous plastic art work. For the initiated modem artist there is no turning back from this arrival point: it is the watershed. To the artists who followed in the wake o f neo-plasticism (Mondrian and the group known as De Stijl)--pioneer constructivism, and the less messianic formalism inaugurated by Jakobson and his school, it is indeed the sciences and mathematics, although not exclusively, which provide a continuing inspiration and link with the scientific ethos as opposed to movements which seek for an identity in such areas a "automatism", the mystic, the unconscious, all somehow part of the modernist thrust along with the ubiquitous expressionism, not to mention the stereotypic "humanism" which is set to optimize the conventional image and icon which characterize the work of the work-a-day artist. Essentially modernism is not wholly identified with the formalist direction, it indeed recognizes its complement viz. the irrational, the subversive and anarchistic as best demonstrated by the iconoclastic dadaists and the montage pieces by the constructivists. It sees an end to the slothful conventions whereby art is to be equated with what is generally accepted as being such if only because it is done in artists' studios, old conventions jacked up by the inclusion of some modern terms o f plastic grammar and syntax, the attempt to project a modern art in the terms of the old--all of this can safely be abandoned. The modem artist has within his grasp modern science, modem mathematics, modem concepts due to various other disciplines, and by relating to these things--although this ensures no guarantee--he continues, paradoxically if you like, the tradition whereby the artist while being an individual (perhaps even a solipsist) works in a context a large part of which parallels and reflects the thinking of the new age. No longer is he the servant of the Church, nor need he replace Church by State (Marxism), but equally no longer is he trapped in the labyrinth of egotistic romanticism which leads to excesses such as self-expressionism, nor of course need he mock the artists o f the past by peddling an uncountable variety of "modernistic" Modern symmetry 713 f o r m u l a t i o n s o f the secular traditions in art; p o r t r a i t u r e , " l a n d s c a p e " , "still l i f e " - - a l l destined t o be a d e c a d e n t c h a r a d e which all t o o easily finds admirers. Some a b s t r a c t artists have f a v o u r e d s y m m e t r y an d m a d e great w o rk s which espouse the notion; one thinks o f Brancusi's E n d l e s s C o l u m n . Others, like M o n d r i a n , have strenuously av o i d ed symmetry. T h e y m a y n o t have k n o w n o f the words o f the celebrated F r e n c h biologist Claude Bernard which can be r e n d e r e d as: " I t is the asymmetric t h at creates life." T h e idea o f a s y m m e t r y has t o o often been relegated to the areas o f the n o n - i m p o r t a n t , the non-beautiful, even if it is u n d e r s t o o d t h a t in n a t u r e perfect s y m m e t r y is never to be f o u n d , only some f o r m o f a p p r o x i m a t i o n to the m a t h e m a t i c a l ideal. Clearly the artist is free to c h o o s e - - t h e r e is b o t h s y m m e t r y an d asymmetry. H o w e v e r , let us end by stating t h a t in a s y m m e t r y the idea o f an a p p r o x i m a t i o n is r a t h e r meaningless. T o p u t it technically, if a c o n n e c t e d structure has an a u t o m o r p h i s m g r o u p characterized as being the identity class this means t h a t it has the feature t h a t all its elements are distinguishable; this is a special an d m o s t f u n d a m e n t a l c o n d i t i o n - - c e r t a i n l y if regarded f r o m the p o i n t o f view o f i n f o r m a t i o n t h eo ry , it represents an " a b s o l u t e " as s o o n as this can be d e m o n s t r a t e d . W h a t is fascinating is t h a t n o algorithms exist for determining w h e t h e r a given structure is asymmetric o r not. In the case o f large structures it becomes necessary to painfully ap p l y the p ro ced u res to determine the a u t o m o r p h i s m number. Significantly it is in chemistry that such i n f o r m a t i o n m a y p r o v e crucial, an d m u c h effort has gone into the research, also u n d e r t a k e n by mathematicians. T o t h a t extent the p r o b l e m remains one o f a series t h a t have yet to be solved; the answer will have practical rewards a n d n o less in m a t h e m a t i c s a m o s t p r o f o u n d step will also have been taken. Which subsumes which, the d e m o n s t r a b l e metric c o n c e p t o f s y m m e t r y / a s y m m e t r y o r the " a b s t r a c t " c o n c e p t o f the a u t o m o r p h i s m grou p ? T h e latter is equally d e m o n s t r a b l e an d in o r d e r to furnish a p r o o f one is forced to state a piecemeal a c c o u n t o f the " n e i g h b o u r h o o d s i t u a t i o n " o f every p o i n t a n d this has to be d o n e by a sequence o f simple observations; it is a giant piece o f " m i c r o - c h e c k i n g " , as one might p u t it, a l t h o u g h o n e is n o t dealing in the real micro-world, just the zero-dimensional implications o f a strucutre o f at m o s t three dimensions. G e n e r a l i z a t i o n being one o f the key strategies in m a t h e m a t i c s - - " g e n e r a l i z e it u p to the next d i m e n s i o n " - - o n e can quickly grasp t h a t the c o n c e p t o f s y m m e t r y / a s y m m e t r y poses m a n y difficult questions. Some we m a y require for solving recondite problems, others remain m o r e like n i g h t m are chess problems. It is unlikely t h a t art can c o n t r i b u t e to this d a u n t i n g area, as it once did in the Renaissance, b u t there is no " l o g i c a l " reason why n o t since the n o t i o n s o f s y m m e t r y / a s y m m e t r y belong, in a sense, to b o t h science a n d art. R E F E R E N C E S 1. A. Hill, Art and Mathesis--Mondrian's structures. Leonardo 1, 3 (1968). 2. A. Hill, Program:Paragram:Structure DATA--Directions in Art, Theory and Aesthetics. Faber & Faber, London (1968). 3. P. Bernays, Comments on Wittgenstein's philosophy o f mathematics. Ratio II No. I 0959). 4. F. Nake, Asthetik als Informations-Ferarbeitung. 6.3. Mondrian StrukturelI-Topologisch. Springer, New York (1974).