An Artist Considers Levels in Matter

by Gertrude Myrrh Reagan

Published in Leonardo, July 1990, Vol. 23 No.1

ABSTRACT The idea that particles make atoms, atoms make molecules,and molecules make visible matter—matter that lives and thinks-is basic to a scientific understanding of the universe. While working with hexagons and pondering this hierarchy, the author discovered two surprising circular arrangements of these levels that may shed light on how we think when using these concepts.

THE USE OF A HEXAGONAL MOTIF

The hexagon is one of the basic patterns in nature on flat surfaces. Put three hexagons together, and their 120˚ angles add up to 360˚: they tile a plane. That hexagons are used so little in art and architecture is not surprising, for joining three 120˚ angles requires greater accuracy in drawing and carpentering than joining four 90˚ angles [1]. This underused motif had great appeal to me because I enjoy diagonal lines [2].

I had the desire to work on a large scale, but I had little storage space. Using hexagons, I could work in modules. The size of the hexagon I chose was 18-in across in the longest direction. If hexagons were rendered on cloth, I reasoned, they would be as trouble-free as quilts to store or transport. Batik was the medium I used in the first two works, Potpourri (Fig. 1) and Animal, Vegetable, Mineral (Fig. 2). I devised several ways to control the hot wax resist to do fine, controlled work. Fiber-reactive dyes on cotton were my materials, delicate Indonesian wax pens were my tools. On the third work, Conjecture (Fig. 3), where the hexagons were slightly larger, I devised a way to do the drawing in oil-base printer's ink. Just as paper can be laid on a uniformly inked surface and the drawing can be done with a stylus on the top (back) side of the paper, I discovered that stretched cloth can be similarly used. Next, I batiked the background.

Fig. 1. Potpourri, quilted batik, 72-in triangle, 1973.

Potpourri was quilted in the traditional way. In the other two hangings, each hexagon had a backing and was stuffed with two layers of polyester felt to give it the substance needed for a wall hanging.

My specialty as an artist is representational drawing, but photography has taken over many of the functions of this kind of art. The strategy of many artists has been to invent abstractions. Mine has been to find abstractions in nature to represent, and to place them in new contexts. In so doing, I have learned to observe nature and human artifacts in a fundamental way, looking for patterns and pondering the reasons for them.

Sensitized, I began encountering six-sidedness everywhere. Many shapes were perfectly regular hexagons, but some of the interesting ones were not. I decided to include these in my collection of patterns. As the collection increased in number, my task became that of selecting from a wealth of motifs. Whereas in the first hanging, Potpourri, I used everything from a tortoise shell to a manhole cover design, the second large batik, Animal, Vegetable, Mineral, contained only natural objects.

Fig. 2A. Animal, Vegetable, Mineral, batik, 108-in. triangle, 1977.

Another work, Benzene, (Fig. 2B) rendered in embroidery and patchwork, contains seven conceptualizations of the benzene ring, including the 1865 original. When a phenomenon has many properties, it is difficult to find one diagram or formulation to cover them all. The subject of this hanging was the use of multiple working hypotheses to aid understanding, a subject that has become a major theme in my subsequent work.

Fig. 2B. Benzene, embroidered quilt, 40” diameter, 1978.

The subject of Conjecture is the diagrams that scientists have made to explain their theories about the physical world. This work continues to evolve; the present version is shown in Fig. 3.

For Conjecture I decided that it would be especially interesting to find designs that represent different levels, or scale lengths, of the structure of matter. I freely admit that there is no necessary relation between these levels and hexagons; but, once having decided on a theme for my new hanging, I stumbled onto designs that more or less fit both of these criteria. The method was not rigorous but rather suggestive.

Fig. 3. Conjecture, scratchboard version, 18-in diameter, 1984; revised 1989.

(a) Hexagon I:Space becomes particles.

(b) Hexagon II:Particles become atoms.

(d) Hexagon IV:Molecules be-come matter.

(e) Hexagon V:Matter becomes life.

(f) Hexagon VI:Life makes thought possible.

built of hexagonal groupings at the atomic level.

Starting with the module that I labeled 'space becomes particles' (Fig. 3a), I progressed conceptually through small units of matter to that large and exquisitely organized clump of matter, the brain. Thus, at the sixth hexagon, I arrived at ' life makes thought possible' (Fig. 3f). The six hexagons could have been displayed in a hierarchical column. However, I chose to arrange them in a circle, initially for aesthetic reasons. Then I noticed that the ending hexagon related in an odd way to the beginning one.

Hexagon I: Space Becomes Particles

As I was composing Conjecture, I happened to see a Moebius strip flattened into a hexagonal shape. I chose it to represent the concept of space, in which all things have their existence. The distinction of the Moebius strip is that it has but one surface. To highlight the fact that its 'outside' and 'inside' are identical and continuous, the design on the outside had to fade out to that on the inside. I accomplished this with an arabesque that progresses from a flat pattern to a shaded and seemingly three-dimensional one (Fig. 3a). (The illusion of three-dimensionality is an additional spatial idea depicted in this unit.) I learned that space has surprising qualities. In his essay "Geometrodynamics", John A. Wheeler explained Einstein's idea of space in the following way:

There is nothing in the world except curved empty space. Geometry bent one way here describes gravitation. Rippled another way somewhere else it manifests all the qualities of an electromagnetic wave. Excited at still another place, the magic material that is space shows itself as a particle. There is nothing that is foreign and 'physical' immersed in space [3].

Somewhat easier to contemplate is Jacob Bronowski's assertion that space" is just as crucial a part of nature as matter is, even if (like the air) it is invisible; that is what the science of geometry is about. Symmetry is not merely a descriptive nicety; like other thoughts in Pythagoras, it penetrates to the harmony in nature" [4].

That space and matter are in an odd sense one leads right to the next module in Conjecture.

Hexagon II: Particles Become Atoms

The familiar subatomic particles are electrons, protons and neutrons. Physicists have discovered hundreds of other subatomic particles by accelerating known particles with electromagnetic energy. Many appear to be created out of pure energy. To help physicists work with many different types of particles, Murray Gell-Mann and others formulated the quark theory.

It is believed that quarks, very small theoretical particles having diverse characteristics, combine to make the larger subatomic particles [5]. This theory, and the many that followed, not only grouped the unwieldy population of discovered particles into combinations of a few quarks but also successfully predicted other, undiscovered particles.

There are several diagrams relating to aspects of quark theory. Figure 3b recreates one from the earliest set of diagrams by the inventors of the theory, Murray Gell-Mann and Yuval Ne'eman[6]. (Few quark diagrams are hexagonal. This is but one way these relationships can be shown.) No one has isolated a single quark, but the conjecture has such excellent predictive power that many believe that these particles indeed exist.

Spatial arrays of known entities make the relationships between the entities easy to study. Just as chemists display the elements in the Periodic Table and artists learn color theory from a color wheel, so physicists use this kind of diagram and others to show the attributes of each quark and its relationship to other quarks [7].

Hexagon III: Atoms Become Molecules

In my search for designs, I did not uncover anything within the atom that fit my hexagonal theme: hence my use of quarks. Atoms can, however, bond together into molecules under favorable conditions. Examples of hexagonal groupings abound. Water, mica and benzene are common ones. Figure 3c presents here one version of the benzene ring [8]. An artist cannot possibly convey the spatial proportions in the subatomic world. If the atom were as large in diameter as the dome of St. Peter's Basilica in the Vatican, the nucleus in the center would be the size of a grain of salt. The electrons would be a few particles of dust swirling around it [9]. Nevertheless, the atom behaves much like a solid lump of matter because of powerful attractive forces binding all parts together. On the diagram shown in Fig. 3c, any dot indicating the nucleus would have to be vanishingly small to represent it correctly [10]. Much smaller yet would be the quark.

Hexagon IV: Molecules Become Matter

The hexagon in Fig. 3d shows sulfur molecules having six atoms, each accumulating into lumps large enough to touch—'matter' in the everyday use of the word. Because the arrangement is regular, it forms a crystal [11].

The shape and stacking arrangements of atoms can do much to explain crystal shapes and properties. The differences between, say, carbon as graphite and carbon as diamond no longer mystify us. We now know that carbon atoms are linked together to form planar sheets in graphite. Stacked up, these sheets can slide easily, like stacked papers. In diamonds, the linkages are in three dimensions with rigid angles, like trusses. This is why graphite makes black marks while diamonds make scratches, why 'diamonds are forever' and pencils are not. Two generations ago the seemingly arbitrary properties of minerals could be learned only by rote memorization [12]. As I create ice crystals on cold, wet glass, I reflect in amazement that I am seeing a pattern built of hexagonal groupings at the atomic level.

Hexagon V: Matter Becomes Life

A diagram of how a virus could assemble itself was chosen to represent life in Fig. 3e [13]. Viruses are on the border between the living and the nonliving. They contain a means for penetrating other cells, their own DNA and very little else. The DNA takes over the apparatus of the host cell and uses it as a factory to replicate itself [14].

One of the successes of twentieth-century science has been to discover molecular structures and reactions or mechanisms that make clear how certain biological processes work physically. For example, when the structure of DNA was discovered, the unforeseen benefit was that the way it is put together suggested how it could split apart and replicate to make offspring. "It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material", the discoverers, James Watson and Francis Crick, wrote [15].

That the arrangement of mere molecules explains so much is compelling. It tends to validate the reductionist point of view that all things can be deduced from nothing more than basic building blocks.

Watson, while full of faith that molecular biology "will soon enable us to understand all the basic features of the living state" [16], nevertheless admitted that the molecules in question were so large and complex, and so numerous, "that the structure of a cell will never be understood in the same way as that of water or glucose molecules"[17]. Will it be the enormity of the task, ignorance or some other factor that will limit our understanding?

Hexagon VI: Life Makes Thought Possible

I believed that conjecture about the activity of the brain was an important area of investigation, one warranting inclusion in Conjecture. In my original version of the work (a wall hanging; Fig.4), a geometric puzzle based on 30˚_600˚ angles was used to represent mental activity.

Fig. 4. Conjecture, monoprint and batik, 45-in diameter, 1977

This did not satisfy me because it had nothing to do with levels in matter. When doing the scratch board version (Fig. 3), I was pleased to find a repeating pattern for this keystone position (Fig. 3f) in a design recorded by Roger Shepard [18]. Shepard is a psychologist interested in spontaneous images produced by the brain, especially geometric ones, during dreams, twilight states, and so on. In this case, he drew a design that he 'saw' while putting pressure on his closed eyes. This design, while not especially handsome, excited me, for it seemed to emanate from the organization of neurons, chemicals and electrical activity itself.

In 1867 Sir John Herschel asked,

What are these Geometrical Specters?and how, and in what department of the bodily or mental economy do they originate? . . . If it be true that the conception of a regular geometrical pattern implies the exercise of thought and intelligence, it would almost seem that in such cases as those above adduced we have evidence of a thought, an intelligence, working within our own organization distinct from that of our own personality [19].

Roger Shepard, after quoting these remarks, goes on to say,

Perhaps, then, it is no accident that a number of creative insights in science have taken the form of a regular, repeating, or symmetrical pattern in space. . . . Suggestive examples include. . . Maxwell's space-filling cylinder-and-ball model of the electromagnetic field. This particular model has a direct correspondence to the regular hexagonal tessellation of the plane that has emerged in spontaneous images and may even be related to under-lying neuroanatomical structures [20].

The idea that the physical structure of the brain, the world it describes, the mathematics it invents, and the aesthetic pleasure it feels are all related was put forward by Kant in the eighteenth century [21]. While the mind must include not only the cells and chemicals but also the processes that are carried on by them [22], the physical apparatus, itself constrained by the rules of how things go together, enables and limits what processes can take place.

I find this idea of a larger unity with nature enchanting. I would like to believe that we only appear to be separate from that which we observe, that in a fundamental and demonstrable way everything is connected [23J. As new instrumentation allows us to study neural structures and activities, it will be interesting to see whether this conjecture is vindicated. It might suggest why advances in mathematics, a mental activity based on structured relationships, open doors to discoveries of relationships in the physical world. In the case of the brain, factors such as 'information', the patterns of impulses coursing around over the dendrites and axons of the nerve cells, are essential to explaining mental processes. The true mystery is consciousness. E. Roy John suggests that it results from mental activity at another level of complexity: the "emergent property of consciousness. . . . [This] subjective experience may actually be a property of a certain level of organization of matter" [24]. Is this the mental equipment needed to think about and even imagine space?

Fig. 5.Ouroboros, scratchboard, 8 x 10-in, 1989 (based on a paper construction , 8 x 10-in 1977)

A STRANGE LOOP

We now have come full circle to 'thought imagines space', as represented by the Moebius strip. The linkage reminded me of the disciplines fit closely resembles Douglas Hofstadter's strange loop: "The 'Strange Loop' phenomenon occurs whenever, by moving upward (or downward) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started" [26]. Until I read Hofstadter's description of this, about two years after finishing Conjecture, I had no words to describe the structure I had created! This unexpected bonus made the circle of hexagons more than an art object. I had started the work with a liking for the hexagon and with criteria from which to choose. I had ended with a new appreciation of how our concepts of matter studied in various disciplines fit together. That it looped around into paradox itself became a tool for thought, a conjecture to be manipulated (Fig. 5).

One day I heard a statement to the effect that events are nothing until they are observed. What could this possibly mean? Subatomic particles' behavior, I learned, is inherently random. Randomness and probability in the description of the behavior of matter were two controversial twentieth-century additions to classical physics [27]. They were controversial because they undermined the idea of determinism. Usually unimportant in the normal world, the indeterminacy of quantum physics can occasionally be noticed in the everyday world. For instance, we speak of the statistical half-life of radioactive decay of certain elements. Thus, at any given moment the path of an electron enters a range of possibilities, and the exact outcome will not be known until after it has been observed. John A. Wheeler declared, "No phenomenon is a phenomenon until it is an observed phenomenon" [28]. Some physicists believe "that the idea of material reality without consciousness is unthinkable,” according to Heinz Pagels, who added, however, that most find this solipsism as disagreeable as the puzzle of whether the tree's fall in the forest makes a sound when no one is present [29].

Fig. 6. Reductionism ad Absurdum, scratchboard, 8 x 10-in, 1983

However, the notion that "events are nothing until they are observed" struck me as being similar to reductionist statements I had heard all my life. I laughed. Just as one might think about the expanding universe backward to arrive at the Big Bang, so I ran my strange loop structure in reverse with respect to this observation to see what would result. The structure that I devised (Fig. 6) is similar to Fig. 5, but it illustrates a paradox I had noticed; consequently, the categories are a little different.

This loop assigns an order to several assumptions that have been fruitful for scientific inquiry. Embodying the reductionist prejudice that physical processes, when thoroughly understood, provide an adequate explanation for most phenomena, these assumptions have helped to disprove the reality of the supposedly spontaneous generation of life and elan vital. Now, the idea of a mind residing in the brain appears naive: a confusion of categories similar to confusing the Capitol building with the government [30]. The successes of reductionism have been many. One mentioned earlier was that the very structure of DNA suggests how replication of genes occurs.

The reductionist assertions fit together in my loop as follows: Man is nothing but an animal; an animal is nothing but matter; matter is nothing but atoms; atoms are nothing but particles; particles are nothing but events. But wait—events are nothing until known by man! What is this creature,on which the existence of the matter in the universe apparently depends? Thus, to construct a world view entirely on these statements, no matter how useful they may be individually, leads the inquirer into paradox. Paradoxes are helpful when they serve to dramatize that a formulation is incomplete or inconsistent. In this case, the inconsistency may offer comfort to the reader tired of hearing that man is nothing but an animal (or worse, "a dollar's worth of chemicals").

THE HIERARCHY OF LEVELS

Conjecture is based on the hierarchy of the structure of matter. Scientists describe this in a factual way, but I cannot help but behold it in amazement. If particles are nothing but events, why does anything hold together and endure? Jacob Bronowski declared that units at one level form stable structures that then become the building blocks for the next higher level. The chemist need only deal with atoms and need not use or even understand quark theory to solve problems [31].

Perhaps logical structures that are similar to each other exist at all levels. How else can one explain the usefulness of mathematics in physics, chemistry, biology, geology and astronomy? Just as visual patterns at many levels of matter are based on familiar motifs such as the hexagon, the spiral, the helix and the sine wave, so it looks as if logical structures underlie what happens and what we can know.

Mathematicians, who get aesthetic thrills from elegant simplicity in their work, find that even in cases where mathematics is used for description, as in physics, the choice of the more elegant of two formulations is often more accurate. Instead of the word description I should more properly use the phrase"the mathematics is isomorphic to, or fits, the physical phenomenon" [32]. The mental structure has a correspondence to the physical. Finding what corresponds is a more mysterious process than translating a face into a painted portrait when the correspondence has predictive power. Dirac predicted the existence of the positron before it was observed, because the equation he solved had two solutions, one for electrons and another for an unknown particle bearing the opposite charge [33]. There are many examples of seemingly useless mathematical contrivances that found applications in science later on[34].

Most unexpected was stumbling upon the strange loop. This tool for paradox raised peculiar questions about what can be known. How concrete is matter, if it all comes down to 'events' and consciousness of them? How can consciousness take place in such an ephemeral 'material'? During the thinking, what exists in the banter between neurons to provide the imagination, engrossing the thinker, or you, the reader in conjecture?

References and Notes

1. "The Hanna House", House Beautiful (January1963) pp. 54-59. An account of the ordeal of finding carpenters competent to build a Frank Lloyd Wright house based on 60˚ angles.

2. The tessellating property of the hexagon is evident in the structure of mica and of the honeycomb. The honeycomb lattice is what the bubble expert calls a minimal surface configuration—that is, bees can make the honey cells with a minimum of wax. See Stephan Hildebrandt and Anthony Tromba, Mathematics and Optimal Form (New York: Scientific American Books. 1985) pp. 155-156.

3. Quoted in Peter Stevens, Patterns in Nature (Boston: Little, Brown, 1974) p. 6.

4. Jacob Bronowski, The Ascent of Man (Boston: Little, Brown, 1973) p. 161.

5. Heinz R. Pagels, The Cosmic Code (New York: Bantam, 1983) p. 203.

6. Drawing inspired by Murray Gell-Mann and Yuval Ne'eman, The Eightfold Way (New York: W. A. Benjamin, 1964) p. 308; see also Pagels [5] p. 202.

7. Conversation with Daryl Reagan, Stanford Linear Accelerator Center, Stanford, CA, 1988.

8. Drawing based on illustration, p. 175, in John E.Hearst and James B. Ifft, Contemporary Chemistry (San Francisco: W. H. Freeman, 1976); by permission. In the wall hanging version of Conjecture (Fig.4), an 'electron density map' was chosen as the model for this particular module. I found the map to be particularly beautiful, but it proved to be a group of molecules strung together instead of atoms creating molecules.

9. Gary Zukav, The Dancing Wu Li Masters (New York: Morrow, 1979) p. 57.

10. Zukav [9] p. 57: 'The nucleus of an atom as high as a fourteen-story building would be about the size of a grain of salt. ... The electrons revolving around this nucleus would be about the size of dust particles! The dome of Saint Peter's Basilica in the Vatican has a diameter of about fourteen stories. Imagine a grain of salt in the middle of the Dome of Saint Peter's with a few dust particles revolving around it."

12. Conversation with Philip B. King, U.S. Geological Survey, Menlo Park, CA, 1956.

13. Drawing based on illustration, p. 221, in Philip Hanawalt and Robert Haynes, The Chemical Basis of Life: An Introduction to Molecular Biology (San Francisco: W. H. Freeman, 1953); by permission.

14. Burton S. Gutman and Johns W. Hopkins, Understanding Biology (New York: Harcourt Brace Jovanovich, 1983) p. 69.

15. J. D. Watson and F. H. Crick, "Molecular Structure of Nucleic Acids", Nature 171 (April 1953) p.738.

16. J. D. Watson, Molecular Biology of the Gene, 3rd Ed. (Menlo Park, CA: W. A. Benjamin, 1976) p. 56.17. Watson [16] p. 69.

18. Roger N. Shepard, "Externalization of Mental Images and the Act of Creation", in Visual Learning, Thinking, and Communication (New York: Academic Press, 1978) p. 177.

19. Quoted in Shepard [18] p. 181.

20. Shepard [18] p. 182.

21. Pagels [5] pp.301-302.

22. Gregory Bateson, Mind and Nature (New York:Dutton, 1979) p. 103.

23. Worth mentioning is Gregory Bateson's "pattern which connects", found throughout nature—not a fixed affair, but a dance of interacting parts.'The pattern which connects is a metapattern. It is a pattern of patterns. It is that metapattern which defines the vast generalization that, indeed, it is patterns which connect" (Bateson [22] pp.12-13).

24. Quoted in Richard Restak, The Brain: The Last Frontier (Garden City, NY: Doubleday, 1979) p. 350.

25. Erich Neumann, The Origins of Consciousness (New York: Random House, 1954) pp. 5-15; Plates 2-9. There are many independent occurrences of ouroboros figures, ancient and modern.

26. Douglas Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (New York: Basic Books, 1979) p.10.

27. Pagels [5] pp. 90-91.

28. Quoted in Pagels [5] p. 76.

29. Pagels [5] p. 76.

30. Restak [24] pp. 9-10.

31. Bronowski [4] pp. 348-349: 'The stable units that compose one level or stratum are the raw material for random encounters which produce higher configurations, some of which will chance to be stable. So long as there remains a potential of stability which has not become actual, there is no other way for chance to go. Evolution is the climbing of a ladder from simple to complex by steps, each of which is stable in itself." This he calls 'stratified stability'.

32. Michael Heller, Questions to the Universe: Ten Lectures on the Foundations of Physics and Cosmology (Tucson, AZ: Pachart, 1987) 'p. 3. But Paul Davies,"What are the Laws of Nature?", in John Brockman, ed., The Reality Club #2 (New York: Lynx Communications, 1989), explores this idea and many others, including one that the mathematical relations that mathematicians discover must have existed before the Big Bang.

33. Pagels [5] p. 214.

34. Pagels [5] p. 270.

Glossary

conjecture (personal definition)—an explanation that exists first as a thought, or guess, that may later find a basis in fact.

fiber-reactive—dyes that bind with the molecules in natural fibers and become part of them. Most fiber-reactive colors work in cold water, preserving waxed batik designs.

ouroboros—a snake growing by eating its tail; a symbol of infinite regress.

solipsism—the notion that an unobserved phenomenon does not exist.

strange loop—used here in the limited sense cited in Hofstadter [26].

scratchboard—a board coated with fine plaster, in which the artist may either draw black areas or carve white ones.

An Artist Considers Levels in Matter

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