When you have learned how to look for symmetry, and how to describe the symmetry which you find, you will constantly be discovering it in the most unexpected places, and observing more and more the role it plays throughout nature and art. (Holden and Singer in Young 1965:232)
When people first think of symmetry, they usually think of perfect mirror imagery or what is sometimes called bilateral symmetry. There is also up and down symmetry based on a horizontal axis. When vertical and horizontal symmetry occur in the same composition, the result is what we call quadrilateral symmetry. The important point with regard to quadrilateral symmetry is that it has two axes, which together provide the composition with a center. Arnheim (1982) has written eloquently and extensively on the power of the center:
Cosmically we find that matter organizes around centers, which are often marked by a dominant mass. Such systems come about wherever their neighbors allow them sufficient freedom. In the vastness of astronomical space the rotating galaxies and the smaller solar or planetary systems are free to create such concentric patterns, and in the microscopic realm so are the atoms with their electrons circling around a nucleus. Even in the crowded world of our direct experience, inorganic and organic matter occasionally has enough freedom to follow its inclination to form symmetrical structures -- flowers, snowflakes, floating and flying creatures, mammalian bodies -- shaped around a central point, a central axis, or at least a central plane. The human mind also invents centric shapes, and our bodies perform centric dances unless this basic tendency is modified by particular impulses and attractions. The earth with what it carries is such a concentric spatial system. (1982: vii).
Arnheim could have added the nucleus of the living cell as a concentric system. The symmetrical pattern of the double helix of DNA is another such concentric system. We would also contend that the human body, the central nervous system, and the psyche all have their centers. As Arnheim shows, the center is certainly a fundamental aspect of aesthetic form, in both that which naturally occurs in the universe and in that which is created by the imaginative powers of human artists. The power of the center, which is found in atoms, DNA molecules, and cells, is also found in the gravitational system of the earth, in the solar system, in the Milky Way galaxy, and in various other portions of the entire universe.
Modern physicists have found that symmetry is a fundamental dimension of the universe. This was emphasized in a presentation by Dennis W. Sciama titled “The Universe as a Whole” at the symposium on the development of “The Physicist's Conception of Nature” held in Italy in 1972 (Mehra 1973). In that presentation, which is chapter two in a book by the same title, Sciama lists four primary properties of the universe: (1) its existence; (2) its evolution; (3) its symmetry; and (4) its singularities.
At the macroscopic level, matter appears in three states: solid, liquid, and gas. Bipolar symmetry plays a key role in this triad. The solid manifestation of matter is primarily crystalline, and all crystals are symmetrical. The stability of the solid is due to its symmetrical structure. In crystalline states, atoms oscillate about positions of equilibrium, and these positions are symmetrical in their spatial relations.
The orderliness of solids is a rather astonishing fact of nature. Physicists have become used to the fact, and they often forget that they do not really know why atoms adopt orderly arrangements. Nevertheless, more than any other property, this orderliness distinguishes solids from liquids: atoms are packed closely together in both, but they have a constantly shifting, disorderly arrangement in a liquid and an orderly arrangement about which they vibrate in a solid. Orderliness of this regularly repeated sort is called crystallinity; anything having crystallinity is a crystal or a collection of crystals. That includes almost all solids. (Holden and Singer in Young 1965:222).
There is some fluctuating, local, temporary order in the liquids -- groups of a hundred atoms become ordered for a short time perhaps -- but order is to be found only here at one instant, there at the next. And when you vaporize the liquid, even those relics of orderliness disappear. (Holden and Singer in Young 1965:224).
The transformation of colored gases or clouds into living forms that occurr in Navajo creation stories exmplifies the primordial transformation of disorder into order. This is also reflective of what Roy Rappoport calls the informing of substance and the substantiation of form (1974: 42-46). In many creation stories, the earth is in a muddy condition because it is lacks form. The imposition of form onto apparent formlessness is the basic feature of creative transformation, and creative transformation is the essence of art and culture. Symmetry is the ultimate form of orderliness and the perfect representation of balance and harmony. However, bipolar symmetry creates dynamic flow (liquid) from order (solid) to disorder (gas) and back again.
The type of order found in the atoms of solids is both symmetrical and repetitive. Atoms in crystals do not organize themselves in just any possible order; their order must be an arrangement that can be repeated infinitely. Figure 22 comes from an excellent book on crystals by Alan Holden and Phylis Singer, Crystals and Crystal Growing. Pattern (B) of this figure clearly illustrates the symmetrical, repetitive order of atoms in a crystal. Pattern (A) is also orderly but not a possible atomic order for atoms in a crystal. The similiarity of the atomic pattern in (B) with Navajo the emblems of Navajo dieties, weaving motifs, and the hooghan design is striking and provocative.
Another important and fascinating aspect of crystals is how they grow and expand:
In the growing of a crystal of alum in solution, aluminum sulfate and potassium sulfate diffuse through the water; and when they reach the surface of the crystal, they join with each other and with some of the water. They adopt positions on the surface that are forced on them by the kind of orderliness confronting them. Settling into those positions, they extend the orderliness outward, and thus the crystal grows. (Holden and Singer in Young 1965: 229).
When different types of crystals conjoin, they form polycrystalline masses. Most of our metals today, forged from the firing process first used by Cro-Magnon artists, are polycrystalline masses.
Probably no more striking and no more beautiful examples of infinite patterns of natural symmetry exist in the universe than those found in snow crystals (Figure 23). Notice the hexagonal and triangular format of these snow crystals. The power of the center is also fundamental to these symmetrical formats.
In Quantum Mechanics by Ashok Das and Adrian C. Melissinos, an entire chapter of 44 pages is devoted solely to the symmetries found in quantum mechanics. This chapter is introduced in the following manner:
The question of symmetry has intrigued man from the earliest times. On the one hand, symmetry is inherently manifest in every facet of nature from crystalline rocks to plant life and living organisms, even if not always exact. On the other hand, man has exploited and introduced symmetry in many of his activities: it is apparent in art, in music, in buildings and machines - to mention but a few examples. One might expect as we examine simpler structures the symmetry properties would be more prevalent, and indeed this is the case. At the other end of the scale the evolution of complex systems with large numbers of constituents, while still obeying the basic principles of symmetry, is primarily governed by statistical laws. The use of symmetries in quantum mechanics is more pronounced than in classical physics where most simple systems can be solved exactly. In quantum mechanics systems are described by state vectors belonging to Hilbert space. In that space the symmetry operations are represented by operators with well-defined properties, and it can be easily shown that for every symmetry operation there exists a corresponding physical observable that is a constant of the motion. Therefore, identifying the symmetries of a system provides helpful information about the state vectors and hence about the system. In the past few decades experimental and theoretical advances have led to the identification of many (unsuspected) new symmetries in nature. (Melissinos 1986: 260).
The concept of global supersymmetry, which implies a relation between particles of different spins, is a necessary ingredient of unification theory, bringing theories of gravitation and particle physics closer together. Supersymmetry was first developed in quantum field theory by Golfand and Likhtman, extended by Volkov and Akulov, and then re-discovered and enlivened by Wess and Zumino.
Bipolar symmetry, sometimes referred to as aysmmetry, has played an important role in physics and in other sciences. The existence of one thing has given rise to the hypothesis that something complementary, juxtaposed, or counterbalanced also exists:
Physicists have rarely gone wrong in banking on the underlying symmetry of nature. For example, when you learn that a changing magnetic field produces an electric field it is a good bet to guess -- and it turns out to be true -- that a changing electric field will produce a magnetic field. For another example: The electron has a so-called antiparticle, that is, a particle of the same mass but of opposite charge. Is it not reasonable to expect that the proton should also have an antiparticle? A 5-GeV proton accelerator was built at the University of California at Berkeley to search for the antiproton. It was found. (Holliday and Resnick 1986: 1132).
The French physicist, Louis de Broglie, discovered the wave nature of matter by postulating an underlying bipolar symmetry of matter with radiation. Radiation exhibited particle-like behavior even though it was thought originally to be solely a wave. If radiation had a bipolar nature, De Broglie posited that matter might also have a bipolar nature. This led him to the discovery that indeed matter can also be wave or particle. Today, the bipolar symmetry of wave and particle is a basic aspect of physics.
Pasteur's first significant scientific acheivement was his discovery of the asymmetry (bipolar symmetry) of tartrate and paratartrate acid. The paratartrate acid was found to be a mixture of two kinds of crystals, some oriented to the right and some oriented to the left in a bipolar symmetrical fashion (Dubos 1960: 240-245). Pasteur felt he had discovered a basic principle of life and the structure of the universe:
Life, as manifested to us, is a function of the asymmetry of the universe and of the consequences of this fact. . . Terrestrial magnetism, the opposition which exists between the north and south poles in a magnet and between positive and negative electricity, are but reultants of asymmetrical actions and movements. . . Life is dominated by asymmetrical actions. I can even imagine that all living species are primordially, in their structure, in their external forms, functions of cosmic asymmetry. (Pastuer quoted in Dubos 1960 reprinted in Young 1965: 246).
Although Pasteur's enthusiastic extension of his discovery of the bipolar symmetry of paratartrate acid to the whole universe may have at the time seemed unwarranted, further research and findings have again and again shown that juxtaposed, bipolar symmetry is ubiquitous. However, the apprehension of bipolar symmetry as a basic part of the structure of the universe did not begin with Pasteur; it existed among the Navajo long before 1850, and it goes back several thousand years in Chinese philosophy, where the dynamics of the universe are understood in terms of the bipolar symmetry of yin and yang. These notions are found in many other ancient and modern cultures throughout the world.
Despite his discovery not being really new, it led Pasteur into many fruitful areas of research:
It was from his conviction that asymmetric molecules are always the product of life that he was led to the study of fermentation, to the recognition that microorganisms play an essential role in the economy of nature, and eventually to his epoch-making discoveries in the field of infectious diseases. (Dubos in Young 1965: 248).
The discovery of the non-conservatioon of parity in certain particle interactions by Tsung Dao Lee and Chen Ning Yang has led to the general acceptence of the notion that bipolar symmetry is inherent in the structure of the universe. Not only have the anti-electron and the anti-proton been discovered, anti-neutrons have also been discovered. The existence of anti-particles was found to be a function of the reversed direction of their spin. The existence of these anti-particles -- a kind of anti-matter -- has provoked much thought and inquiry:
The discovery of the anti-particles did not disturb physicists; on the contrary, it was a pleasing confirmation of the symmetry of the universe. What did disturb them was a quick succession of discoveries showing that the proton, the electron, and the neutron were not the only elementary particles they had to worry about. (Asimov in Young 1965: 254).
Most of the newly discovered particles are unstable and exist for much less than a millionth of a second. The stable and relatively long-lived particles can be divided into three types. One is the photon, which is a unit of electromagnetic radiation; the other two types are the leptons and the hadrons. The leptons are only involved in very weak interactions and cannot hold anything together. These weak interactions are sometimes referred to as weak nuclear force and become manifest only in certain kinds of particle collisions and particle decays. By far the most significant and the most powerful of all interactions occur among the hadrons, and the structure of the hadrons and their interactions are important for us here.
The hadrons subdivide into two types, mesons and baryons, each with their distinct but similar structures. Protons and neutrons are baryon type hadrons. The hadrons consist of particles and antiparticles of opposite charge. The structure of the hadrons reflects a definite symmetry, but this symmetry can only be understood in terms of interactions, not in terms of part to whole subdivision. The symmetry of the hadrons reflects both the power of their structure and the conservation of their energy. Physicists discuss the interactions of the hadrons both in terms of their symmetry and their corresponding patterns of conservation:
Hadrons, for example, carry definite values of isospin and hypercharge, two quantum numbers which are conserved in all strong interactions. If the eight mesons . . . are arranged according to the values of these two quantum numbers, they are seen to fall into a neat hexagonal pattern known as the meson octet. This arrangement exhibits a great deal of symmetry; for example, particles and antiparticles occupy opposite places in the hexagon, the two particles in the centre being their own antiparticles (see Figure 24). The eight lightest baryons form exactly the same pattern which is called the baryon octet (see Figure 25). This time, however, the antiparticles are not contained in the octet, but form an identical anti-octet. The remaining baryon in our particle table, the omega, belongs to a different pattern, called the baryon decuplet, together with nine resonances (see Figure 26). All the particles in a given symmetry pattern have identical quantum numbers, except for isospin and hypercharge which give them their places in the pattern.
The quantum numbers, then, are used to arrange particles into families forming neat symmetric patterns, to specify the places of the individual particles within each pattern, and at the same time to classify the various particle interactions according to the conservation laws they exhibit. The two related concepts of symmetry and conservation are thus seen to be extremely useful for expressing the regularities in the particle world. (Capra 1975:252-254)
The structure of hadrons, both mesons and baryons, is similar to many of the fundamental patterns found in Navajo semiotical geometry, which is reflected in the emblems of the Holy People, in the motifs of Navajo weaving, in the structure of Navajo hooghans, and in the holistic symmetry of Navajo art. All these structures are organized in terms of symmetrical triangles and the power of the center. The bipolarity of positive and negative charge of particles and antiparticles is also reflected in the bipolar symmetry of Navajo art in the frequent alternations of positive and negative space.
Concepts of symmetry and holism pervade modern physics. The new conception of the universe does not have the universe constructed out of dissectable, independent particles. Instead the universe is conceived as a vast interwoven system of interrelated and interacting phenomena. Nothing is independent or fully autonomous; everything is interconnected and thus part of a holistic essence:
We have reversed the usual classical notion that the independent ‘elementary parts’ of the world are the fundamental reality, and that the various systems are merely particular contingent forms and arrangements of these parts. Rather, we say that inseparable quantum interconnectedness of the whole universe is the fundamental reality, and that relatively independently behaving parts are merely particular and contingent forms within this whole. (Bohm and Hiley 1975: 102).
The new science is struggling with unification theory and grand cosmic schemes. The great scientific leaps of this century were integrative and holistic. Einstein discovered the interrelationships of time/space and mass/energy, and he also struggled in his later years to discover the interrelationship among gravitational, electromagnetic, and nuclear forces.
A recent theory called “grand unification” has been able to integrate weak and strong nuclear forces with electromagnetic force. Pati notes that “Within the premises of these ideas there is no intrinsic asymmetry between quarks and leptons. They are regarded as members of one family (multiplet)” (in Ne'eman 1981: 221). Grand unification theory is still lacking adequate experimental verification, but it illustrates the intuition of many physicists, taking a lead from Einstein, that unity and symmetry underlie apparent discontinuity and diversity.
Hermann Weyl developed a theory to unite electromagnetic fields with gravitational fields. P. A. M. Dirac called Weyl's theory “a beautiful synthesis" but it was not acceptable to most physicists because it was contradicted by quantum phenomena. Dirac proposed a different way to view Weyl's theory, using two metrics ds and ds, and thereby eliminating many of the objections to Weyl's theory. Dirac concluded, therefore, that "there is then no objection to Weyl's theory. We can accept it, and get in that way a very beautiful synthesis of the electromagnetic field and the gravitational field” (in Mehra 1973: 53).
Einstein made gigantic leaps when he showed the link between space/time and mass/energy. The third pair in this set was discovered in the quantum, and in the underlying bipolarity of wave and particle:
Quantum theory has shown that particles are not isolated grains of matter, but are probability patterns, interconnections in an inseparable cosmic web. Relativity theory, so to speak, has made these patterns come alive by revealing their intrinsically dynamic character. It has shown that the activity of matter is the very essence of its being. The particles of the subatomic world are not only active in the sense of moving around very fast; they themselves are processes! The existence of matter and its activity cannot be separated. They are but different aspects of the same space-time reality. (Capra 1975: 203).
These formulations put an end to the Great Machine metaphor of classical science as a formulation of the structure and operation of the universe and have given birth to a new kind of science. The new science is built on models of interwoven systems and holistic fields. An emphasis on holism and interwoven essence is, as we discovered in chapter four, also found in some of the major trends in modern American art. This is particularly true of abstract expressionism, chromatic abstraction, color field, and one-image art. The rise of abstract art has an important connection with and similarity to quantum field theory and the development of systems theory in the social and natural sciences.
From physics we learn “that in any given ‘field’ the forces of which it consists will distribute themselves in such a way that the simplest, most regular, most symmetrical organization results” (Arnheim 1968:48). “Field forces” are the underlying dynamics that serve as the vital component of pictorial composition. The effect that these forces have on our visual apprehension is determined by the placement of visual elements (color, value, shape, mass, texture, and line) within a field.
In comparison to all the other visual elements of pictorial composition, color is the most contextual. Our perception of color is influenced significantly by its interaction with neighboring colors and with background elements. Shape and value contrasts are more insistent in our visual perception than is color, but perceptions of value contrasts and shape tend to be individual rather than contextual. This is primarily because of the physiology of perception and to the composition of the retina.
In the retina are two types of photoreceptors of visual impulses: rods and cones. These photoreceptors chemically convert the visual impulses into electrical impulses that are transported to the brain via the optic nerve. These electrical impulses are synthesized in the brain to form a mental conception of that which was perceived. There are three types of cones through which we perceive color, although recent studies indicate the possibility of four types of cones (Fischler and Firschein 1987: 236). One type of cone is most responsive to blue (short wavelengths), a second type to green (middle range wavelengths), and a third type to red (long wavelengths). A cone for the reception of red is considered to be the first color receptor developed in the hominid line. Our blue and green receptors came later.
Value (gradations of black/grey/white) is registered through the rods in the retina. There are 120 million rods per eye and only 6.7 million cones per eye, amounting to 18 times as many rods in the retina as there are cones. This is why the perception of value contrasts tends to dominate color interaction in a pictorial composition (Wertenbaker 1981: 35-36). The significant difference in the number of rods and cones is probably because of cultural and evolutionary factors. Rods that receive value contrasts are what we primarily use in ascertaining shapes. We use the rods to see at night when there is not enough light to perceive color. Chickens, for example, have no rods and, therefore, cannot see at night.
From the rods and cones, visual signals are sent along the optic nerve to the brain. These visual signals first split into two groups, one containing color contrast information and one containing value contrast information. Next, these signals separate into three processing systems, each with a distinct function. “One system appears to process information about shape perception; a second, information about color; a third, information about movement, location and spatial organization” (Livingstone 1988: 78). The shape system produces high-resolution and static perception of form or image, resulting in sharp definitions of individual shapes and objects. The color system has three to four times lower acuity than the shape system. The movement system operates entirely without the influence of color, but all three systems combine in the brain to create an integrated visual perception.
Visual perception results from an integrated and interacting triad, built first on the bipolar distinction between color and value. Spatial organization then mediates and modifies this bipolar system, creating a triad of hue, shape, and movement. Value adds depth to flatness, while color adds planarity to form. By interacting with both flatness and form, movement makes visual perception dynamic. The perception of color is relational, systemic, and contextual, whereas the perception of image (shape or object) is particular, specific, and individual. The perception of movement is directional, variable, and dynamic
Probably the move from the forest to the savannah and the resulting tendency toward diurnalism may have led to the development of cones for the perception of color. Generally, human cultural and social experience has emphasized the recognition of object, shape, and entity over the apprehension of system, interrelatedness, and context. It is easier for us to deal with particular entities (shapes) than it is for us to visualize and conceptualize abstract essences and interrelationships found in holistic and interacting contexts. This tendency, based in our visual apprehension of our environment, has had its impact on the formulation of scientific theories and findings. Our visual and conceptual tendencies have been to see objects, to recognize part-to-whole relationships, and to think in terms of linear cause and effect models. This is why theories based on relativity and quantum phenomena have been so difficult for us to understand and for us to assimilate into our worldview. Nevertheless, we are overcoming these tendencies and are rapidly going in the direction of systemic, cybernetic, symbiotic, ecological, and contextual models.
The rise of abstract art and the development of various systems theories have an important parallel. Abstract art goes beyond the recognition of entity or image and focuses attention on interaction and interrelatedness, striving to uncover and reveal holistic essences. Systems theories have tried to get us beyond linear cause and effect notions and tried to get us to think of phenomena in terms of interaction and interrelationship. It is interesting and significant that abstract art and quantum theory arose at about the same time. Both of these developments preceded the application of various forms of systems theory to the social and biological sciences. Scientific insight and conceptualization have paralleled artistic imagination and aesthetic formulation.
Abstract art, like quantum theory and relativity, has been difficult for those trained in Western intellectual and artistic traditions to understand. The field painters of the last four decades have emphasized abstract and holistic relationships and essences. A special group of these, the color field painters of the 1960s, emphasized the use of color because of its contextual nature and its expressive potential to convey relationships and holistic ideas. They explored color as abstact subject matter for their art, rather than focusing on entities and images. Color lends itself well to interaction and to contextual, holistic compositions.
Color, more than any other visual element, had the potential for extending the general premise of systemic art -- the singularity of the field and the object through holistic configuration. Color field painting most likely took its inspiration from the fundamental nature of the field as discovered in physics in the early part of this century:
In these “quantum field theories,” the classical contrast between the solid particles and the space surrounding them is completely overcome. The quantum field is seen as the fundamental physical entity; a continuous medium which is present everywhere in space. Particles are merely local condensations of the field; concentrations of energy which come and go, thereby losing their individual character and dissolving into the underlying field. (Capra 1975: 210).
From Einstein we get this conclusion: “There is no place in this new kind of physics both for the field and matter, for the field is the only reality” (in Capek 1961: 319).