PII: 0898-1221(86)90135-5 Comp. & Maths. with Appls. Vol. 12B. Nos. I/2, pp. 1-17, 1986 0886-9561/86 $3.00+ .00 Printed in Great Britain. © 1986 Pergamon Press Ltd. L I M I T S O F P E R F E C T I O N ISTVAN H A R G I T T A I t Institute o f Materials Science and Departments o f Chemistry and Physics, University of Connecticut, Storrs, CT 06268, U.S.A. A b s t r a c t - - T h i s personal narrative is an introduction to a collective effort by a number o f scientists and artists to examine the role and significance of symmetry in the most diverse domains of nature and human activity. Material symmetry, devoid o f the rigor o f geometrical symmetry, is viewed applicable to material objects as well as abstractions with limitless implications. To mark the 350th anniversary of Johannes Kepler's death, the Hungarian Post Office issued a beautiful memorial stamp (Fig. 1 ). Next to Kepler's portrait his famous model o f the planetary system is shown. This was fitting since of all o f Kepler's discoveries this is the best known to the general public, although it is viewed by some as his most spectacular failure[l]. A closer look at Kepler's activities, however, justifies the selection o f the Hungarian Post Office. Although he is most famous for his three laws of heavenly mechanics, there is another piece of work that was also a milestone in a different branch o f science, crystallography. If one is astonished by the depth of his understanding o f the physics o f the sky with the then available data, it is not less astonishing that Kepler could discuss the " a t o m i c " arrangement in crystals two hundred years before Dalton and three hundred years before X-ray crystallography began. In his new year's gift of the hexagonal snowflake Kepler[2] not only examines the hexagonal symmetry of the snow crystals but lays the foundation o f the principle o f densest packing in crystal structures. Densest packing is then, of course, the key to the symmetry o f crystal habit. The planetary model from the regular solids is also a densest packing model. Kepler's search for harmony was the bridge between his two lines o f activities. Although the snowflake paper seems to be almost an accident on the background o f his astronomy, the Hungarian stamp gave more credit to the complete Kepler than was probably envisioned by the planners of the stamp themselves. According to the regular solids model, taking the six planets, known to Kepler, in order, the greatest distance of one planet from the Sun stands in a fixed ratio to the least distance o f the next outer planet from the Sun. There are, of course, conveniently five ratios for the six planets. A regular solid can be interposed between two adjacent planets so that the inner planet, when at its greatest distance from the Sun, lies on the inscribed sphere o f the solid, while the outer planet, when at its least distance, lays on the circumscribed sphere. There are molecular structures which can be best described by polyhedra enveloping other polyhedra. The structure of [ C o 6 ( C 0 ) 1 4 ] 4 - is shown is Fig. 2: an omnicapped cube of carbonyl oxygens envelopes an octahedron formed by cobalt atoms[3]. One of today's most successful models in structural chemistry is based on extremely, simple considerations of space distribution. The valence shell electron pair repulsion (VSEPR) model[4] postulates that the geometry of the molecule is determined by the space requirements o f the electron pairs in the valence shell of its central atom. The bond configuration around atom A in an AX, molecule is such that the electron pairs in the valence shell be at maximum distances from each other. Thus the arrangement may be visualized so that the electron pairs occupy well-defined parts of the space about the central atom. In a different concept, these space segments are called localized molecular orbitals. It is easy to demonstrate the three-dimensional consequences of the VSEPR model. Only a few balloons have to be blown up and connected at their narrowing ends in groups o f two, three or four[5]. The linear, equilateral triangular, and tetrahedral arrangements o f these as- semblies are what the VSEPR model predicts for the electron pairs. Another beautiful analogy is found on walnut trees[6]. As two, three, or four walnuts grow sometimes together, the above tVisiting Professor (1983/85). Permanent address: Hungarian Academy o f Sciences, P.O. Box 117 Budapest, H-1431, Hungary. CAMWAI 2 : 1/2 (B) -B I ISTVAN HARGITTAI . . f - ' " ° " " \ ~ ~ .......... ],-.v ..... Fig. 1. Kepler memorial stamp. (Hungarian Post Office, 1980.) arrangements occur unfailingly (see Fig. 3). The soft balloons and hard walnuts may even be viewed as representing weaker and stronger interactions and thereby represent even more subtle analogies for molecular structure. Can molecular geometry be so simple? There is obvious oversimplification in the model. Of the many effects determining molecular structure, one is taken into account and all the others are ignored. The model is applicable where this particular effect, to wit the repulsions o f the electron pairs due to their space requirements, is dominant. The VSEPR model is successful because this effect is important enough in extensive classes o f compounds. In comparing the complexity of planetary motion and molecular structure, the real analogy is in the possibility of selecting a dominant effect and ignoring the others. This approach works much better for the planetary motion where the dominant effect is the gravitational attraction of the Sun while the others are perturbations. In the world of molecules, the dominant effect may change from one compound to another. In spite of the tremendous amount o f accumulated knowledge about molecular structure, its basic principles are still being clarified. One o f the characteristics of the forces keeping the molecule together is that they are very strong, whereas the gravitational forces are very weak. To discover the law of gravitational interactions, the observations had to be made on a large scale. The laboratory was the planetary system itself. As we compare the symmetries of molecules and crystals, a striking difference is that there are no limitations for molecules and there are well-defined limitations for crystals. A consequence is the finite number (32) of symmetry classes for crystal habit with no such limits for molecules. ~ O Fig. 2. The structure o f [Co6(CO)~41 +- , the omnicapped cube o f the carbonyl oxygens envelopes the cobalt octahedron. (After[3].) Limits of perfection 3 Fig. 3. Walnut clusters with two, three, and four walnuts with linear, equilateral triangular, and tetrahedral arrangements. (Photographs by the author following the idea of Niac and Florea[6].) 4 ISTVAN HARGITTAI T h e m o l e c u l e s are a m o r e fundamental building unit in the hierarchy o f structures than the crystals and m a n y crystals t h e m s e l v e s are built f r o m molecules. Unfortunately, m o l e c u l e s are usually not to be seen b y the n a k e d e y e whereas the crystals are. T h e y are so a p p e a l i n g l y s y m m e t r i c a l that they h a v e b e c o m e a sort o f idol for s y m m e t r y . For the BEla Bart6k centenary a couple o f years ago (1981), Victor Vasarely p r o d u c e d a limited edition o f ten serigraphs created for ten o f B a r t 6 k ' s musical pieces. Each serigraph was a c c o m p a n i e d b y a p o e m , each written b y a different living Hungarian poet. M o r e than once the w o r d crystal was related to B a r t r k ' s music. I f one considers their s y m m e t r y properties in a strictly technical sense, h o w e v e r , they could not be farther f r o m each other. B a r t r k ' s music is i n t e r w e a v e d throughout b y the Fibonacci n u m b e r s and the golden section. T h e s e s y m m e t r i e s characterize for e x a m p l e the scattered leaf arrangements o f m a n y plants and are o m n i p r e s e n t in other d o m a i n s o f the a n i m a t e world as well. On the other hand, " c r y s t a l l i z a t i o n is d e a t h " , as crystallographers t h e m s e l v e s like to point out. Incidentally, pentagonal s y m m e t r y is con- spicuously present in primitive o r g a n i s m s and crystailographer Nikolai Belov[7] suggested that it was their m e a n s o f self-defense against crystallization. T h e restrictions on crystal s y m m e t r y start with the fact that, strictly s p e a k i n g , it exists only theoretically• Its main characteristic is translational s y m m e t r y , i.e. infinite periodic rep- etition, and in reality, o f course, crystals a l w a y s end s o m e w h e r e ; but apart f r o m this, there is no five-fold s y m m e t r y axis, nor are there axes with higher order than six. Considering the m o r e easily visualized t w o - d i m e n s i o n a l surface o f the regular p o l y g o n s , only the equilateral triangle, the square and the regular h e x a g o n can be used to c o m p l e t e l y c o v e r an area without gaps, w h e r e a s the regular p e n t a g o n , h e p t a g o n , etc., can not. Similar limitations extend to the third dimension. In spite o f the limitations in crystal s y m m e t r y , the crystals have a unique appeal and are widely used also in analogies. Q u o t e the C z e c h writer Karel (~apek[8] on his visit to the mineral collection o f the British M u s e u m • . .But I must speak again about crystals, shapes, colors. There are crystals as huge as the collonade of a cathedral, soft as mould, prickly as thorns; pure, azure, green, like nothing else in the world, fiery, black; mathematically exact, complete, like constructions by crazy, capricious scientists, or reminiscent of the liver, the h e a r t . . . There are crystal grottos, monstrous bubbles of mineral mass, there is fermentation, fusion, growth of minerals, ar- chitecture and engineering a r t . . . Egypt crystallizes in pyramids and obelisks, Greece in Fig. 4. Illustration for phyllotaxis. (Photograph by the author.) Limits of perfection columns; the middle ages in gilly-flowers; London in grimy c u b e s . . . Like secret mathe- matical flashes of lightning the countless laws of construction penetrate the matter. To equal nature it is necessary to be mathematically and geometrically exact. Number and phantasy, law and abundance--these are the living, creative strengths of nature; not to sit under a green tree but to create crystals and to form ideas, that is what it means to be at one with nature! This is then exactly the point where Bart6k and the crystals meet. While the c o m p o s e r invariably refused to discuss the technicalities o f his work, he liked to state " W e create after N a t u r e " and he meant it literally. As a crystal is being built from molecules, and the energetically most favorable arrangement is being achieved, the molecules c o m e into close, touching range with each other. C o m p a r e d with the free molecules, their interactions m a y have perturbing effects on their structures. One o f the simplest consequences m a y be the lowering o f their original symmetry. Discussing the structural consequences o f densest packing in molecular crystals, the following explanation is attributed to crystallographer Aleksandr Kitaigorodskii, " T h e molecule also has a body. W h e n this b o d y is hit, the molecule feels hurt all o v e r . " This analogy emphasizes the importance o f spatial requirements o f the molecules in building the molecular crystal rather than the peculiarities o f its electronic structure which would be o f greater importance in a more chemical behavior as in a chemical reaction. Personifying the molecule obviously had an appeal to the scientist, and it also has an appeal for children who generally like to do so with various objects. W h e n some years ago the a u t h o r ' s daughter was asked in the kindergarten about her father's occupation, she said he cured sick molecules. The tendency to make metaphors, however, seems to diminish with c o m i n g o f age. S o m e valuable things m a y be lost in the educational process. It is much easier to get through to children the notion that there is much more to s y m m e t r y than what we call bilateral s y m m e t r y . Adults seem to be more narrow-minded and more indoctrinated. Not everything is perfect, however, even with the most symmetrical molecules. Concerning their s y m m e t r y , that is. W h e n the s y m m e t r y o f a molecule is described, it is usually the motionless, frozen molecule that is meant. This structure would correspond to the m i n i m u m energy. The molecules are never motionless, however. Even if they could be cooled to the absolute coldest temperature o f 0 K or - 2 7 3 . 1 6 ° C , they would not c o m e to a standstill. The molecular vibrations often lead to some instantaneous distortion or lowering o f the molecular s y m m e t r y . This is true for the relatively rigid molecules and even more so for the very flexible molecules. Intramolecular motion m a y even lead to a continuous permutation o f the atoms in the molecule. Imagine a ring molecule o f five atoms with four being in one plane and the fifth atom sticking out o f the plane. This arrangement lowers the s y m m e t r y o f the five-member ring to a Fig. 5. Henri Matisse: Dance. (The Hermitage, Leningrad.) Reproduced by kind permission from The Hermitage. 6 IsTVAN HARGrVrAI mere symmetry plane. The point is that this molecule may be performing such intramolecular motion that during every second a million times or more the sticking out position switches from one atom to the next and to the next and so on. Consider now five dancers in a circle instead o f the five atoms (see Fig. 5). Let them make a j u m p one after the other in a quick succession. I f we take photographs with very short exposures, we can catch various configurations, including even the most symmetrical one in which all the five dancers are on the ground. On the other hand, a longer exposure leads to a blurred picture all around the circle. The apparent symmetry of the dancing group obviously depends on the length o f the exposure used, and also, of course on the speed of their movement. The molecular structures and crystal structures represent two well-separated cases from the point o f view o f their symmetry properties. The molecule is characterized by point-group symmetry as it has at least one unique point in the whole structure. Crystal structures are characterized by space-group symmetry or translational symmetry as they have no unique point in their structure. When point-group and space-group symmetries are compared, it is not obvious how to distinguish between higher and lower symmetries. Within each domain, however, and the molecules and crystals are merely examples, there is a hierarchy o f symmetries. Increasing the symmetry beyond some limits may lead, however, to sterility and certainly diminishes the information content. Scientific instruments with ever increasing perfection may filter out important peculiarities which do not conform with the general pattern. Perfect symmetry may be aimless, and it irritates many. Perhaps symmetry considerations could facilitate relating the perception o f structures and the world of emotions? On the level of analogies this seems to be possible as is illustrated by a poem by Ann Wickham[9]: GIFT TO A JADE For love he offered me his perfect world. This world was so constricted and so small It had no loveliness at all, And I flung back the little silly ball. At that cold moralist 1 hotly hurled His perfect, pure, symmetrical, small world. Geometrical symmetry is strict: it allows for no " d e g r e e s " o f symmetry. Something is either symmetrical or not. What may be called material symmetry., on the other hand, implies a continuous spectrum o f the degrees o f symmetry. The term material symmetry here refers generally to non-geometrical symmetry and may be applied to real material objects as much as to any abstraction. The human face is an obvious example o f bilateral symmetry. However, none o f us has a perfectly symmetrical face (Fig. 6). It may be a matter o f flattery on the part of the painter to show more perfect symmetry than there is, or this may even be demanded by the paintee. Old religious paintings or contemporary political portraits may show more facial symmetry than there is. Any personality cult produces very symmetrical face images. However, minor asym- metries may have a strong appeal and the notion about the beauty of a face may also be changing. There was a minor uproar some time ago when a Budapest theater critic praised the acting o f an actress and remarked also on her pretty, modern face. In fact, hers was conspicuously asymmetric, so people were wondering whether their faces were modern or old-fashioned. There is considerable interest currently in the origin and meaning of facial asymmetry. Another fascinating s y m m e t r y / a s y m m e t r y relationship exists between our hands (Fig. 7). Distinguishing between left and right has definite connotations in almost all fields o f human activity. The importance o f chirality is ever growing in the sciences. Even human attitude toward handedness is evolving. Figure 8 shows two pictures from classrooms o f the University o f Connecticut. The older classrooms of the Chemistry Department have homochiral chairs only, designed for right-handed students. The more contemporary classrooms o f the Mathematics Department are furnished with heterochiral chairs to accommodate both the right-handed and the left-handed students. Limits of perfection 7 Fig. 6. Eszter Hargittai in front of a shop-window, 1984. (Photograph by the author.) Fig. 7. Tomb in the Jewish cemetery, Prague. (Photograph by the author.) 8 ISTV,/tN HARGITTAI Fig. 8(a). Classroom in the Chemistry Department, University of Connecticut. 1984. Fig. 8(b). Classroom in the Mathematics Department, University of Connecticut, 1984. (Photographs by the author.) A r t i s t i c c r e a t i o n as r e f l e c t i o n m a y v a r y in a l i m i t l e s s range. The two p h o t o g r a p h s in F i g . 9 illustrate this point. C o n s t r u c t i o n m a c h i n e r y on the b a n k o f the D a n u b e is shown in one o f the p h o t o g r a p h s . Its r e f l e c t i o n in the D a n u b e is c a p t u r e d in the other. I m a g i n e a m i r r o r - p e r f e c t w a t e r surface with much m o r e l i k e n e s s , o r g a l e c o n d i t i o n s d e s t r o y i n g any trace o f r e s e m b l a n c e . The o r i g i n o f r e f l e c t i o n i t s e l f as seen by Jean Effel is d e p i c t e d in Fig. 10. The s c i e n c e s o f the 20th century have o p e n e d up a new w o r l d , one which had p r e v i o u s l y been i n a c c e s s i b l e to m a n ' s instruments, let alone his senses. To p a r a l l e l this d e v e l o p m e n t , artistic e x p r e s s i o n is c o p i n g , o r is at least a t t e m p t i n g to do so, in reflecting o u r w o r l d on an Limits of perfection 9 Fig. 9(a). Building construction machinery on the bank of the Danube. Fig. 9(b). Reflection of the building construction machinery in the Danube, (Photographs by the author.) entirely new level. It is not that the artist is expected to bend over the screen o f an electron microscope and paint " a f t e r n a t u r e " , but the newly discovered domains and p h e n o m e n a must and do find their reflections in artistic expression. It may be considered symbolic that X-ray crystallography and the Black Square o f Kazimir Malevich were born about the same time. The picture from 1915 is reproduced in Fig. 11 along with the title page o f a later Malevich work. Current progress mandates the expansion o f the well-established frameworks o f the sym- metry concepts, One o f the cradles o f modern s y m m e t r o l o g y , crystallography is transforming itself to embrace all structural science on the atomic level(10]. Liquids, colloids, amorphous 10 ISTVAN HARGITrAI ( / - Institution du reflet L e s r i v e r a i n s a u r o n t , g r a c i e u s e m e n t , p o r t r a i t ~t l ' a q u a r e l l e . . . l e u r Fig. 10. The creation of reflection, by Jean Effel. [La Creation du Monde]. ("Those who dwell by the river will have their portrait, gracefully, in water c o l o r s . . . " ) . (Reproduced by kind permission from Mme. Jean Effel.) solids c a n n o t be put into the e x i s t i n g " p e r f e c t " systems even t h o u g h t h e y are not without structure. W e are w i t n e s s i n g their e m a n c i p a t i o n . We quote A n n W i c k m a n [ 9 } again: THE WOMAN AND HER INITIATIVE Give me a deed, and l will give a quality. Compel this colloid with your crystalline. Show clear the difference between you and me By some plane symmetry, some clear stated line. These bubblings, these half-actions, my revolt from unity. Give me a deed, and 1 will show my quality. John Bernal was the p i o n e e r o f g e n e r a l i z e d c r y s t a l l o g r a p h y , and B e l o v noted in his o b i t u a r y " . . . h i s last e n t h u s i a s m was for the laws o f l a w l e s s n e s s . " Did A n n W i c k h a m ' s m e t a p h o r s p a r a l l e l B e r n a l ' s d i s c o v e r i e s ? A l t h o u g h several d i s c i p l i n e s apart, g e o g r a p h i c a l l y and c h r o n o - l o g i c a l l y t h e y o p e r a t e d in c l o s e range ( L o n d o n ) . In a n y c a s e the n o n - c l a s s i c a l , i r r e g u l a r , u n s t a b l e , u n u s u a l , u n e x p e c t e d are g a i n i n g i m p o r - Limits of perfection , i 11 Fig. 1 l(a). Kazimir Malevich: Black square (1915?). crn . m Blfl'm~K Ig20 Fig. 1 l(b). Kazimir Malevich: Suprematism, 34 drawings, UNOV1S, Vitebsk, 1920, Title page. 12 ISTVAN HARGrVrAl tance in the sciences. The " m o r p h o l o g y o f the amorphous" is being investigated and " m a p p e d " [ l 1]. The 1978 Nobel laureate physicist Philip Anderson stated that ~'the next decade is very likely to be the most 'disordered' decade in theoretical physics. "112] The increasingly recognized importance o f non-periodic structures makes even more con- spicuous the absence or near-absence o f the three-dimensional space groups outside the world o f crystals. The h o n e y c o m b built by the bees is a notable exception. Its regular hexagonal structure is then partially copied by the construction o f the concrete base o f off-shore oil platforms. There is unmistakable similarity to the h o n e y c o m b in the hexagonal joints o f basaltic sheets resulting from contraction during cooling. These are examples that meet the eye and there m a y be others hidden. H o w much visible and how much less visible s y m m e t r y is there in the arts? There is a lot o f visible s y m m e t r y in B a c h ' s music and there is also a lot in B a r t 6 k ' s - - n o t so visible but well- established by research. There is a wealth o f s y m m e t r y p h e n o m e n a in other music as well as in other arts, not only in paintings and sculptures whose s y m m e t r y properties are most c o m m o n l y considered. One o f the less frequently perceived symmetries is inversion, a combination o f applying a two-fold rotation axis and a s y m m e t r y plane. Here is an example from Hungarian author Frigyes K a r i n t h y ' s short story entitled " T h e same in m a n " [ 1 3 ] represented by some edited fragments. There are three characters: Bella the beloved lady, Fox the e m p l o y e e and Bella's suitor S~indor w h o is also F o x ' s boss. The editing means to present the two meeting in a parallel w a y rather than consecutively. BELLA FOX Sometimes 1 just gaze before me without Sometimes I just gaze before me without thinking of a n y t h i n g . . , thinking of a n y t h i n g . . . S,~NDOR/BOSS Bella! If only you knew how beautifully you expressed y o u r s e l f . . . BELLA Sometimes I have the feeling that I'd like to be somewhere else than I am. 1 can't say where, somewhere I haven't been be- fore. On my money? Then you'd better go to a lunatic asylum, that's where cases like you are t r e a t e d . . . FOX 1 often have the feeling that I'd like to be somewhere else than I am. I don't know where, anywhere, somewhere, 1 haven't been before. S,~NDOR/BOSS Bella, how true, how w o n d e r f u l . . . How The nuthouse, man, the nuthouse. That's did you put it? Let me engrave it in the where you belong. records of my m i n d . . . There is inversion in James R e s t o n ' s description o f New Zealand in his Letter from Wellington, Search for End o f the Rainbow[14]: Nothing is quite the same here. Summer is from December to March. It is warmer in the North Island and colder in the South Island. The people drive on the left rather than on the right. Even the sky is different--dark blue velvet with stars of the Southern Cross--and the fish love the hooks. Wellington and Madrid are approximately connected by a straight line going through the center o f the Earth which is then the inversion center for them. It is too bad that the journalist did not date his letter from Madrid. The black-and-white ,variation is the simplest case o f color s y m m e t r y and it is also the simplest example for antisymmetry (see Fig. 12). The relationship between matter and antimatter is another example. " O p e r a t i o n s o f antisymmetry transform objects possessing two possible values o f a given property from one value to the o t h e r " [ 15]. According to this general definition antisymmetry can be given broad interpretation and application. Geometrically less strict but in their atmosphere truly black and white antisymmetries are presented in Fig. 13. Another literary example is taken from K a r i n t h y ' s writing, this time to illustrate antisym- metry. It is edited from a short story entitled " T w o D i a g n o s e s " [ 1 6 ] . The same person Dr. Same goes to see a physician at two different places. At the recruiting station he would obviously Q ] C | -,, s 14 ISTVAN HARGITTA~ • ii Fig. 13(a). Victor Vasarely: "P62-Basilan", 1951. (Reproduced by kind permission from the artist.) Fig. |3(b). Neizvestnii: N. S. Khrushchev's tomb, Novodevichi cemetery, Moscow. Limits of perfection 15 Fig. 14. Two restaurants in downtown Washington, D. C.: the Sans Souci and a McDonald's. (Photograph by the author.) Fig. 15. Logo of sporting goods store in downtown Boston, MA. (Photograph by the author.) 16 ISTVAN HARGITTAI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ !~i 1 Fig. 16. A breed apart. Poster. (Reproduced by kind permission from Merrill Lynch Company.) Fig. 17. Military jets and a sea gull, off Bod0, Norway. (Photograph by the author.) Limits of perfection 17 like to a v o i d b e i n g d r a f t e d , w h i l e at the i n s u r a n c e s o c i e t y he w o u l d like to a c q u i r e the b e s t p o s s i b l e t e r m s f o r his p o l i c y . H i s a n s w e r s to the i d e n t i c a l q u e s t i o n s o f the t w o p h y s i c i a n s are r e l a t e d b y a n t i s y m m e t r y . DR. SAME PHYSICIAN DR. SAME At the recruiting station At the insurance society Broken-looking, sad, ruined human Young athlete with straightened back, wreckage, feeble masculinity, haggard flashing eyes. eyes, wavering movement. How old are you? O l d . . . very old, indeed. Coyly, On, my gosh, I ' m almost ashamed of i t . . . I ' m so s i l l y . . . Your I.D. says you're thirty two. With pain. To be old is not to be far from To be young is not to be near the cradle, the c r a d l e - - b u t near the coffin, but far from the coffin. Are you ever dizzy? D o n ' t mention dizziness, please, Doctor, Quite often, sorry to say. Every time I ' m or else i'11 collapse at once. I always aboard an airplane and it's up-side-down, have to walk in the middle of the street, and breaking to pieces. Otherwise, because if 1 look down from the curb, I n o t . . . become dizzy at once. T w o r e s t a u r a n t s s t a n d side b y side in d o w n t o w n W a s h i n g t o n , D. C. O n e is t h e o n e - o f - a - k i n d e x c l u s i v e S a n S o u c i , the o t h e r is a M c D o n a l d ' s o f the f a m o u s f a s t - f o o d c h a i n . T h e a n t i s y m m e t r y p l a n e a p p e a r s p h y s i c a l l y as a v e r t i c a l wall b e t w e e n t h e t w o r e s t a u r a n t s ( F i g . 14). A n t i s y m m e t r y m a y b e p o w e r f u l in f o c u s i n g a t t e n t i o n , s h o w i n g c o n t r a s t , e m p h a s i z i n g a p o i n t . F i g u r e 15 s h o w s t h e l o g o o f a s p o r t i n g g o o d s store in d o w n t o w n B o s t o n . T h e a n t i s y m m e t r y p l a n e e m p h a s i z e s that b o t h w i n t e r a n d s u m m e r sport f a n s are w e l c o m e . A p o s t e r o f t h e M e r r i l l L y n c h i n v e s t m e n t c o m p a n y is r e p r o d u c e d in F i g . 16. T h e r e is a h o r i z o n t a l a n t i s y m m e t r y p l a n e in " a b r e e d a p a r t " . M i l i t a r y j e t s a n d a sea g u l l f l y o n the t w o sides o f a n i m a g i n a r y a n t i s y m m e t r y p l a n e in o u r u l t i m a t e e x a m p l e (Fig. 17). REFERENCES 1. A. Koestler, The Sleepwalkers, The University Library. Grosset and Dunlap, New York (1963). 2. J. Kepler, Strena, seu De Nive Sexangula, t61 I. English translation, The Six-cornered Snowflake. Clarendon Press, Oxford (1966). 3. R. E. Benfield and B. E G. Johnson, The structures and fluxional behaviour of the binary carbonyls; A new approach. Part 2. Cluster carbonyls Mm(COL (n = 12, 13, 14, 15, or 16). J. Chem. Soc. Dalton Trans., 1743- 1767 (1980): 4. R. J. Gillespie, Molecular Geometry. Van Nostrand Reinhold Co., London (1972). 5. H. R. Jones and R. B. Bentley, Electron-pair repulsions, a mechanical analogy. Proc. Chem. Soc., 438-440 (1961). 6. G. Niac and C. Florea, Walnut models of simple molecules. J. Chem. Educ. $7, 429-429 (1980). 7. N. V. Belov, Ocherki po strukturnoi mineralogii. Nedra, Moskva (1976). 8. K. (~apek, Anglick~ Listy. (~eskoslovensk# Spisovatel, Praha (1970). The English version cited here was kindly prepared by Dr. Alan L. Mackay, Birkbeck College (University of London), 1982. 9. A. Wickham, Selected Poems. Chatto and Windus, London (1971). 10. A. L. Mackay, Crystallography--the continuous re-definition of the subject. Indian J. of Pure and Appl. Phys. 19, 765-768 (1981). 11. B. B. Mandelbrot, The Fractal Geometry of Nature. Freeman, New York (1983). 12. V. F. Weisskopf, Contemporary Frontiers in Physics. Science 203, 240-244 (1979), 13. F. Karinthy, Grave and Gray, Selections from his Work. Corvina Press, Budapest (1973). 14. J. Reston, Letter from Wellington. Search for end of the rainbow. International Herald Tribune, Thursday, May 7, 4-4 (1981). 15. A. L. Mackay, Expansion of Space-Group Theory. Acta Cryst. 10, 543-548 (1957). 16. E Karinthy, Selected Works (in Hungarian). Sz6pirodalmi, Budapest (1962). The English translation was kindly checked by Dr. R. B. Wilkenfeld, Professor of English, University of Connecticut, 1984. 17. Gy. Lengyel, Handiwork. New Techniques--New Solutions (in Hungarian). Kossuth, Budapest (1975). CAH~I~12~I/2(B)-C