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Symmetry

PRINCETON UNIVERSITY PRESS

BILATERAL SYMMETRY

If I am not mistaken the word symmetry is used in our everyday language in two meanings. In the one sense symmetric means something like well-proportioned, well-balanced, and symmetry denotes that sort of concordance of several parts by which they integrate into a whole. Beauty is bound up with symmetry. Thus Polykleitos, who wrote a book on proportion and whom the ancients praised for the harmonious perfection of his sculptures, uses the word, and Durer follows him in setting down a canon of proportions for the human figure. In this sense the idea is by no means restricted to spatial objects; the synonym "harmony" points more toward its acoustical and musical than its geometric applications. Ebenmass is a good German equivalent for the Greek symmetry; for like this it carries also the connotation of "middle measure," the mean toward which the virtuous should strive in their actions according to Aristotle's Nicomachean Ethics, and which Galen in De temperamentis describes as that state of mind which is equally removed from both extremes: [TEXT NOT REPRODUCIBLE IN ASCII]

The image of the balance provides a natural link to the second sense in which the word symmetry is used in modern times: bilateral symmetry, the symmetry of left and right, which is so conspicuous in the structure of the higher animals, especially the human body. Now this bilateral symmetry is a strictly geometric and, in contrast to the vague notion of symmetry discussed before, an absolutely precise concept. A body, a spatial configuration, is symmetric with respect to a given plane E if it is carried into itself by reflection in E. Take any line l perpendicular to E and any point p on l: there exists one and only one point p' on l which has the same distance from E but lies on the other side. The point p' coincides with p only if p is on E. Reflection in E is that mapping of space upon itself, S: p [right arrow] p', that carries the arbitrary point p into this its mirror image p' with respect to E. A mapping is defined whenever a rule is established by which every point p is associated with an image p'. Another example: a rotation around a perpendicular axis, say by 30°, carries each point p of space into a point p' and thus defines a mapping. A figure has rotational symmetry around an axis l if it is carried into itself by all rotations around l. Bilateral symmetry appears thus as the first case of a geometric concept of symmetry that refers to such operations as reflections or rotations. Because of their complete rotational symmetry, the circle in the plane, the sphere in space were considered by the Pythagoreans the most perfect geometric figures, and Aristotle ascribed spherical shape to the celestial bodies because any other would detract from their heavenly perfection. It is in this tradition that a modern poet addresses the Divine Being as "Thou great symmetry":

    God, Thou great symmetry,
    Who put a biting lust in me
    From whence my sorrows spring,
    For all the frittered days
    That I have spent in shapeless ways
    Give me one perfect thing.

Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection.

The course these lectures will take is as follows. First I will discuss bilateral symmetry in some detail and its role in art as well as organic and inorganic nature. Then we shall generalize this concept gradually, in the direction indicated by our example of rotational symmetry, first staying within the confines of geometry, but then going beyond these limits through the process of mathematical abstraction along a road that will finally lead us to a mathematical idea of great generality, the Platonic idea as it were behind all the special appearances and applications of symmetry. To a certain degree this scheme is typical for all theoretic knowledge: We begin with some general but vague principle (symmetry in the first sense), then find an important case where we can give that notion a concrete precise meaning (bilateral symmetry), and from that case we gradually rise again to generality, guided more by mathematical construction and abstraction than by the mirages of philosophy; and if v.-c arc lucky we end up with an idea no less universal than the one from which we started. Gone may be much of its emotional appeal, but it has the same or even greater unifying power in the realm of thought and is exact instead of vague.

I open the discussion on bilateral symmetry by using this noble Greek sculpture from the fourth century B.C., the statue of a praying boy (Fig. 2), to let you feel as in a symbol the great significance of this type of symmetry both for life and art. One may ask whether the aesthetic value of symmetry depends on its vital value: Did the artist discover the symmetry with which nature according to some inherent law has endowed its creatures, and then copied and perfected what nature presented but in imperfect realizations; or has the aesthetic value of symmetry an independent source? I am in clined to think with Plato that the mathematical idea is the common origin of both: the mathematical laws governing nature are the origin of symmetry in nature, the intuitive realization of the idea in the creative artist 's mind its origin in art; although I am ready to admit that in the arts the fact of the bilateral symmetry of the human body in its outward appearance has acted as an additional stimulus.

Of all ancient peoples the Sumerians seem to have been particularly fond of strict bilateral or heraldic symmetry. A typical design on the famous silver vase of King Entemena, who ruled in the city of Lagash around 2700 B.C., shows a lion-headed eagle with spread wings en face, each of whose claws grips a stag in side view, which in its turn is frontally attacked by a lion (the stags in the upper design are replaced by goats in the lower) (Fig. 3) . Extension of the exact symmetry of the eagle to the other beasts obviously enforces their duplication. Not much later the eagle is given two heads facing in either direction, the formal principle of symmetry thus completely overwhelming the imitative principle of truth to nature. This heraldic design can then be followed to Persia, Syria, later to Byzantium, and anyone who lived before the First World War will remember the double-headed eagle in the coats-of-arms of Czarist Russia and the Austro-Hungarian monarchy.

Look now at this Sumerian picture (Fig. 4). The two eagle-headed men are nearly but not quite symmetric; why not? In plane geometry reflection in a vertical line l can also be brought about by rotating the plane in space around the axis l by 180°. If you look at their arms you would say these two monsters arise from each other by such rotation; the overlappings depicting their position in space prevent the plane picture from having bilateral symmetry. Yet the artist aimed at that symmetry by giving both figures a half turn toward the observer and also by the arrangement of feet and wings: the drooping wing is the right one in the left figure, the left one in the right figure.

The designs on the cylindrical Babylonian seal stones are frequently ruled by heraldic symmetry. I remember seeing in the collection of my former colleague, the late Ernst Herzfeld, samples where for symmetry's sake not the head, but the lower bull-shaped part of a god's body, rendered in profile, was doubled and given four instead of two hind legs. In Christian times one may see an analogy in certain representations of the Eucharist as on this Byzantine paten (Fig. 5), where two symmetric Christs are facing the disciples. But here symmetry is not complete and has clearly more than formal significance, for Christ on one side breaks the bread, on the other pours the wine.

Between Sumeria and Byzantium let me insert Persia: These enameled sphinxes (Fig. 6) are from Darius' palace in Susa built in the days of Marathon. Crossing the Aegean we find these floor patterns (Fig. 7) at the Megaron in Tiryns, late helladic about 1200 B.C. Who believes strongly in historic continuity and dependence will trace the graceful designs of marine life, dolphin and octopus, back to the Minoan culture of Crete, the heraldic symmetry to oriental, in the last instance Sumerian, influence. Skipping thousands of years we still see the same influences at work in this plaque (Fig. 8) from the altar enclosure in the dome of Torcello, Italy, eleventh century A.D. The peacocks drinking from a pine well among vine leaves are an ancient Christian symbol of immortality, the structural heraldic symmetry is oriental.

For in contrast to the orient, occidental art, like life itself, is inclined to mitigate, to loosen, to modify, even to break strict symmetry. But seldom is asymmetry merely the absence of symmetry. Even in asymmetric designs one feels symmetry as the norm from which one deviates under the influence of forces of non-formal character. I think the riders from the famous Etruscan Tomb of the Triclinium at Corneto (Fig. 9) provide a good example. I have already mentioned representations of the Eucharist with Christ duplicated handing out bread and wine. The central group, Mary flanked by two angels, in this mosaic of the Lord's Ascension (Fig. 10) in the cathedral at Monreale, Sicily (twelfth century), has almost perfect symmetry. [The band ornament's above and below the mosaic will demand our attention in the second lecture.] The principle of symmetry is somewhat less strictly observed in a n earlier mosaic from San Apollinare in Ravenna (Fig. 11), showing Christ surrounded by an angelic guard of honor. For instance Mary in the Monreale mosaic raises both hands symmetrically, in the orans gesture; here only the right hands are raised. Asymmetry has made further inroads in the next picture (Fig. 12), a Byzantine relief ikon from San Marco, Venice. It is a Deësis, and, of course, the two figures praying for mercy as the Lord is about to pronounce the last judgment cannot be mirror images of each other; for to the right stands his Virgin Mother, to the left John the Baptist. You may also think of Mary and John the Evangelist on both sides of the cross in crucifixions as examples of broken symmetry.

Clearly we touch ground here where the precise geometric notion of bilateral symmetry begins to dissolve into the vague notion of Ausgewogenheít, balanced design with which we started. "Symmetry," says Dagobert Frey in an article On the Problem of Symmetry in Art, "signifies rest and binding, asymmetry motion and loosening, the one order and law, the other arbitrariness and accident, the one formal rigidity and constraint, the other life, play and freedom." Wherever God or Christ are represented as symbols for everlasting truth or justice they are given in the symmetric frontal view, not in profile. Probably for similar reasons public buildings and houses of worship, whether they are Greek temples or Christian basilicas and cathedrals, are bilaterally symmetric. It is, however, true that not infrequently the two towers of Gothic cathedrals are different, as for instance in Chartres. But in practically every case this seems to be due to the history of the cathedral, namely to the fact that the towers were built in different periods. It is understandable that a later time was no longer satisfied with the design of an earlier period; hence one may speak here of historic asymmetry. Mirror images occur where there is a mirror, be it a lake reflecting a landscape or a glass mirror into which a woman looks. Nature as well as painters make use of this motif. I trust, examples will easily come to your mind. The one most familiar to me, because I look at it in my study every day, is Hodler's Lake of Silvaplana.

While we are about to turn from art to nature, let us tarry a few minutes and first consider what one may call the mathematical philosophy of left and right. To the scientific mind there is no inner difference, no polarity between left and right, as there is for instance in the contrast of male and female, or of the anterior and posterior ends of an animal. It requires an arbitrary act of choice to determine what is left and what is right. But after it is made for one body it is determined for every body. I must try to make this a little clearer. In space the distinction of left and right concerns the orientation of a screw. If you speak of turning left you mean that the sense in which you turn combined with the upward direction from foot to head of your body forms a left screw. The daily rotation of the earth together with the direction of its axis from South to North Pole is a left screw, it is a right screw if you give the axis the opposite direction. There are certain crystalline substances called optically active which betray the inner asymmetry of their constitution by turning the polarization plane of polarized light sent through them either to the left or to the right; by this, of course, we mean that the sense in which the plane rotates while the light travels in a definite direction, combined with that direction, forms a left screw (or a right one, as the case may be). Hence when we said above and now repeat in a terminology due to Leibniz, that left and right are indiscernible, we want to express that the inner structure of space does not permit us, except by arbitrary choice, to distinguish a left from a right screw.

I wish to make this fundamental notion still more precise, for on it depends the entire theory of relativity, which is but another aspect of symmetry. According to Euclid one can describe the structure of space by a number of basic relations between points, such as ABC lie on a straight line, ABCD lie in a plane, AB is congruent CD. Perhaps the best way of describing the structure of space is the one Helmholtz adopted: by the single notion of congruence of figures. A mapping S of space associates with every point p a point p': p [right arrow] p'. A pair of mappings S, S': p [right arrow] p', p' [right arrow] p, of which the one is the inverse of the other, so that if S carries p into p' then S' carries p' back into p and vice versa, is spoken of as a pair of one-to-one mappings or transformations. A transformation which preserves the structure of space — and if we define this structure in the Helmholtz way, that would mean that it carries any two congruent figures into two congruent ones — is called an automorphism by the mathematicians. Leibniz recognized that this is the idea underlying the geometric concept of similarity. An automorphism carries a figure into one that in Leibniz' words is "indiscernible from it if each of the two figures is considered by itself." What we mean then by stating that left and right are of the same essence is the fact that reflection in a plane is an automorphism.

Space as such is studied by geometry. But space is also the medium of all physical occurrences. The structure of the physical world is revealed by the general laws of nature. They are formulated in terms of certain basic quantities which are functions in space and time. We would conclude that the physical structure of space "contains a screw," to use a suggestive figure of speech, if these laws were not invariant throughout with respect to reflection. Ernst Mach tells of the intellectual shock he received when he learned as a boy that a magnetic needle is deflected in a certain sense, to the left or to the right, if suspended parallel to a wire through which an electric current is sent in a definite direction (Fig. 14). Since the whole geometric and physical configuration, including the electric current and the south and north poles of the magnetic needle, to all appearances, are symmetric with respect to the plane E laid through the wire and the needle, the needle should react like Buridan's ass between equal bundles of hay and refuse to decide between left and right, just as scales of equal arms with equal weights neither go down on their left nor on their right side but stay horizontal. But appearances are sometimes deceptive. Young Mach's dilemma was the result of a too hasty assumption concerning the effect of reflection in E on the electric current and the positive and negative magnetic poles of the needle: while we know a priori how geometric entities fare under reflection, we have to learn from nature how the physical quantities behave. And this is what we find: under reflection in the plane E the electric current preserves its direction, but the magnetic south and north poles are interchanged. Of course this way out, which reestablishes the equivalence of left and right, is possible only because of the essential equality of positive and negative magnetism. All doubts were dispelled when one found that the magnetism of the needle has its origin in molecular electric currents circulating around the needle's direction; it is clear that under reflection in the plane E such currents change the sense in which they flow.

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Excerpted from Symmetry by Hermann Weyl. Copyright © 1980 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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