6. SYMMETRY–ASYMMETRY, TRANSFORMATION AND COMPLEMENTARITY IN PICTORIAL SYMBOLS AND FOOTPRINTS

One of the most basic attributes of symbols is the actual duality of the symbol and the symbolized. In principle, the symbol differs from the symbolized and is not dependent upon the latter for its existence, since symbols can denote entities that are not perceptual–fictitious or hypothetical entities. In most symbol systems the connection between the symbol and the symbolized is fundamentally a convention which becomes entrenched through the use of the symbols, as with all linguistic and formal symbol systems. In figurative art too, there is in principle a duality of the symbol and the symbolized, only that here the connection between the two is not the outcome of convention and habit, but is inherent. This internal linkage between the figurative symbol and the object symbolized by it, springs from the fact that the symbol in this case is built on a certain symmetry between the two. It is easy to ascertain this fact by looking at some of the very earliest pictures: those of handprints. Since these were created by blowing a spray of paint over the hand, it cannot be denied that there is symmetry between the shape of the hand and the shape of the negative image produced by the spraying. The principle of symmetry between symbol and symbolized is also preserved in the more complex pictures produced not by spraying but by drawing the contour of the depicted animal. Besides figurative painting, the only area in which an inherent connection of this kind is to be found between the sign and the signified is that of footprints. Since all footprints are produced by the imprint of the foot on the ground, it cannot be denied that there is symmetry between the form of the foot and the shape of the footprint it created. It is an instructive fact that in footprints and in handprints the configuration is of the same type: symmetry that is a mirror image or reflective transformation of the shape of the object that produced the footprint or the picture. It is this transformation that produces the duality between the symbolized and the picture, or between the sign and the signified in the case of footprints. However, it also exemplifies a much more general principle: that there is no symbolization or signification without transformation of one sort or another, and this principle is already evidenced for the first time in a quite advanced manner in footprints literacy. The fact that figurative art and footprints are based on a certain symmetry between the object and the symbol or the sign that represents it, is of great importance on at least two levels: firstly, it is this fact that makes figurative art readable at all periods and in all places, even tens of thousands of years after it was produced, as with the very earliest prehistoric art. We can evaluate better the universal readability of figurative art, if we bear in mind that contemporary 'abstract art', founded precisely on the breaking and negation of this symmetry, cannot be read by anyone, including the artists who created it. This symmetry is also the factor that makes footprints readable not only to the skilled tracker who read them soon after they were made, but to us as well. This is actually so even when footprints were made millions of years ago and have been preserved only because of special conditions and circumstances, as in the case of the footprints left by hominids and animals at Laetoli in northern Tanzania (Leakey, 1987). These tracks were produced some 3.5 million years ago at least, and one of these tracks is especially relevant to the topic of footprints literacy, because it seems to have been produced as the result of a certain hominid treading exactly in the footprints of another hominid with larger feet who walked before him. The result is a double track such that there is superimposition of the smaller footprints of the second subject upon the larger footprints of the first subject. Here it should be mentioned that indeed most researchers believe that there is in fact superimposition of the small footprint upon the larger one, but there are others who do not accept this interpretation and maintain that it is only a single footprint. In my analysis I shall take as my point of departure the assumption accepted by most, that there is superimposition of two footprints here. According to research by L.M. Robbins it is impossible to walk in someone else's footprints by chance, and her conclusion is therefore that the second subject deliberately walked in the footprints of the first (Robbins, 1987). Nevertheless, the fact that the second hominid trod exactly in the footprints of the man in front of him, does not mean that he read those signs as footprints and therefore, even if that is very likely, it is impossible to maintain with certainty that hominids could already read footprints 3.5 million years ago. What supports the hypothesis that the second hominid could apparently read footprints is the fact that both for the purpose of identification of the marks or prints in front of him as the same type of visual patterns, and for him to be able to step exactly and consistently in those prints, he must have had the capacity of pattern recognition at a complex level that actually includes almost all of the cognitive attributes required for the reading of footprints: connectivity, discrimination, selection, abstraction, classification and generalization: this hominid must have had some latent understanding of symmetry–asymmetry, inclusion–exclusion, transformation–invariance, complementarity of ground and figure, hypothetical thinking, comparison, recursiveness, causality, and others. But we cannot infer from his treading in the man's footprints that he had any understanding of the reference relation between a mark and the thing it indicates, which is necessary for the reading of footprints, but unnecessary for treading in the same kind of pattern. However, if the supposition is correct, it has of course weighty implications regarding the cognitive structures of these hominids, and one of them is the startling resemblance between their mode of thinking and our mode of thinking today, despite the fact that their brain was only about half the size of ours. That is, although their thinking was mostly visual and ours is mostly verbal, we share the same mindprints. If this is true, then we should begin to think of our remote ancestors as visual sages rather than as non-verbal savages.

This matter obviously raises the question whether the primates too are capable of reading footprints. Indeed, this question has bothered me for many years, whereas I have found in the literature no direct reference to the matter at all. Mary Leakey mentions the possibility that the footprints at Laetoli were created by a process similar to that by which chimpanzees and gorillas customarily play the game called 'Follow the leader' (Leakey, 1981, 1987). On this subject she relied upon Schaller (1963), but he does not say anywhere in his book that the gorillas step each within the footprints of the one in front. Of course the fact that in this game they walk "in the steps of the leader's" does not mean that they pay any attention to the footprints, especially since they usually live in places where there is abundant and tall grass, so that there is little exposed soil upon which footprints can be seen. Furthermore, in this game they are so close to one another that they have no need to look for the one in front of them, and this game therefore provides no proof for footprints literacy among apes. Nevertheless, even if someone were to prove that they customarily trod in exactly the same place as the subject in front of them, this would still not constitute evidence that they read the graphical pattern in the soil as footprints of the subject walking in front. As we have already seen, in order to read the signs on the ground as footprints, many and highly complex cognitive mechanisms are required. So far as I know, there is still no proof that primates are capable, or incapable, of reading footprints. My feeling is that chimpanzees almost certainly can read footprints, for there are experiments carried out in completely different contexts, from which it can be deduced that they indeed have at least a considerable part of the cognitive qualifications required for reading footprints. Nevertheless, anyone who attempts to examine this matter empirically will have to expect very difficult methodological problems. As a matter of fact, a few years ago I canceled at the last moment an experiment I was to have carried out at the Ramat Gan Safari Park in Israel, using two young chimpanzees, with the aim of putting to the test their capacity to read footprints. The chief reason for this cancellation was that I found no way of proving that they were reading the marks as footprints. That is to say, I might have succeeded in teaching them to distinguish between their keeper's footprints, which it was intended should lead to boxes containing bananas, and my own, which were larger and would have led to empty boxes; but I could find no way of proving that they saw the prints of their keeper's feet or my own as transformations of the shape of a foot, and not as visual marks that could as well have been triangles and circles or any other two objects. Another, more amusing but diverting, problem was that the two chimpanzees were very young and most of the time preferred to hug and kiss their motherly young keeper than to cooperate in any game.

Cognitive evolution is not the central issue of this essay, but it is important at this stage only to indicate the possibility that footprints literacy is a much earlier human achievement than we tend to think. Secondly, the symmetry between the symbol and the symbolized in figurative art, like that between the footprint and the foot, is precisely the means of connection unique to art and to footprints. Indeed, art and footprints differ in this from all other kinds of symbols. That is to say, symbolization produces grouping and classification of things, and this requires abstraction, analysis and synthesis of the aspects and attributes common to the objects grouped under the same symbol. The symbols make the connectivity between the members of the class in different ways: a verbal symbol such as 'bull' tells us nothing about the common attributes of the entities that belong to the class ‘bulls’; and if we wish to know what the attributes of a bull are, we have to refer to our own knowledge and memory. By contrast, the pictorial symbol representing a bull represents explicitly the most basic visual attributes of all bulls. That is to say, what is special in the mode of connectivity of the pictorial symbol is that this connectivity is achieved by displaying a particular figurative common denominator of the class of bulls it describes. This visual common denominator is a particular symmetry or isomorphism present among all bulls connected by the symbol, but the same symmetry is also preserved concisely in the symbol itself! We find the same characteristic in footprints as well: there is symmetry between the shape of the feet of all bulls of a certain kind, and the same symmetry is also preserved, albeit reversed, in every footprint that one of these bulls leaves on the ground. Each footprint is therefore a means both of connection and of classification for all living creatures that possess a foot that fits or is symmetrical with a given footprint. We have seen that in footprints and in figurative art, symmetry is the connecting principle (Avital, 1996). Since this principle of connection by symmetry must have appeared in footprints literacy long before it appeared in art, it is highly probable that in art this principle is a generalization and elaboration of the same principle found in footprints literacy. The strength of the connection by symmetry, can be seen from the fact that in early hunting civilizations, and even those of them that still exist today, the connection between the animal and its footprints, or between the handprint or footprint and the person they belong to, is an iconic connection between the footprint and the whole animal or person, and not only his foot or hand. This connection is in fact so strong that members of these cultures tend to confuse the two: such a footprint or picture might represent the creature it belongs to, but is more often a substitute for the creature. Thus, for instance, the Mehinaku tribe that lives on the banks of the Xingu River, a tributary of the Amazon, has a tragi-comic story about a man whose sweetheart ran away from him, and who looked for her in vain. "All he could find was her footprint, and he had sex with that." (Gregor, 1985).

It has already been noted that connectivity–disconnectivity or codis, is one of the most basic mindprints in the hierarchy of mindprints, and it is thus already clear that symmetry–asymmetry is a certain variant of codis. It is therefore not surprising that connectivity in general, and connection by symmetry in particular, are among the cornerstones of human intelligence and culture. These characteristics appear not only in the reading of footprints and art, but in different forms in all symbol systems. So for example, pictures and footprints connect bulls in a similar way to that in which the equations of Newton connect the stars: in both cases the connection is made by symmetries, except that in the first case the symmetry is figurative, and in the second case it is formal and relational. In a paper the starting point of which is symmetry in biology, Gregory Bateson maintains that symmetry is the connecting pattern, and he explains: "The pattern which connects is a meta-pattern. It is a pattern of patterns. It is that meta-pattern which defines the vast generalization that indeed it is pattern which connects." (Bateson, 1978. His emphases.) This fine insight of Bateson's helps us to a more profound understanding of art and footprints: figurative painting is a system of pictorial or visual universals. However, since every such symbol and each of its sub-symbols is built upon a particular symmetry in relation to the part of the body it depicts, painting is thus a hierarchy of symmetries. A painting is therefore a system of connecting patterns, and the whole picture is a meta-pattern connecting all of these patterns and all of the objects it describes. Because of this attribute, footprints and pictures do not belong to the same level of being as all objects, but they are a priori at least one level higher than the world of objects. Footprints and pictures, like words, concepts, numbers, equations, models and theories, belong to the world of ideas or patterns and not to the world of objects; they are of the kind of mindstuff that makes possible the complementarity of mind and reality. Complementarity is another fundamental attribute of footprints and figurative pictures.

Footprints are marked by the contours of the lack of soil, following the inclusion relationships between foot and Mother Earth. Like every inclusion relationship, the relationship between the foot and the footprint is one of yin and yang; it is a relationship of complementarity. This relationship is also present in the graphic structure of the footprint as mark. Thus it is impossible to read a footprint without the background upon which it exists, for the footprint is a gestalt or complementary unity of figure-ground. However, this gestalt is constructed from a complementarity of symmetry and asymmetry together. For on the one hand, the pattern of the footprint is symmetrical to the foot; but on the other hand, the ground which surrounds the footprint and which is an inseparable part of it, is asymmetrical to the foot. The fact that we are much more aware of the pattern of the footprint from its outer boundaries inward, does not mean that the space surrounding the pattern of the footprint from its outer boundaries and outward is irrelevant or less important to the reading of the footprint. In a way, footprints literacy requires that the tracker-hunter should be able to think in paradoxical terms: he must be capable of identifying the unity of a thing and its opposite; or the complementarity of symmetry–asymmetry, and not only symmetry. In a sense the symmetry between the pattern of the foot and the reversed pattern of the footprint is derived from the complementarity of the two patterns. That is to say, the pattern that connects is a symmetry that includes both the symmetry and its complementarity, asymmetry. We observed this paradoxical attribute in codis, which is built upon the complementarity of connectivity–disconnectivity. We also find the same complementarity of figure-ground explicitly, in prehistoric paintings of handprints and footprints in which the contours of those parts were generated by spraying. In a less explicit form we can observe this complementarity in the generalization of this principle to pictures based upon contour, whether emphasized as in early paintings, or deliberately blurred as in later paintings. That is to say, no figurative picture is possible without the integration of the principle of complementarity into its structure. It is possible that the complementarity–mutual exclusiveness that is revealed in both the reading of footprints and in art is indeed one of the cornerstones of human intelligence and perhaps even of Being at all levels. Tens of thousands of years after the emergence of painting and in a completely different context, Lao Tzu in China, and Heraclitus in Greece assumed complementarity as a cornerstone of their philosophies, and in our century Niels Bohr did the same in physics. When he argued that light is a wave phenomenon and a particle phenomenon at one and the same time, he was thinking in the terms of a real hunter; except that he was a hunter-tracker of 'little thoughts' rather than of little animals. In a sense, footprints literacy and general systems theory are two ends of the same issue; the first deals with the symmetry between graphic patterns and various animals, and the second, which is a meta-theory, deals with the symmetry between the various symmetries upon which are based the main areas of knowledge created by man. Totemism, art, mythology, philosophy and science are only intermediate links between these two poles, and all of them assume symmetries of one sort or another as the foundation for the connections and distinctions that they create.