This essay was originally printed in the catalogue titled FINITY/INFINITY published by the New York Academy of Science, New York City, and accompanied the Ronald Davis show held there in 1986.

---

Theoretical physics is a highly abstract discipline. Mathematics is its language, logic its method. But theoretical physicists are human, too, and our brains tire from the effort of thinking about things that hover on the edge of inconceivability. And so, to relieve the mental strain, we sometimes attach concrete images to the technical terms and mathematical symbols of our craft. Thus I think of electrons as fuzzy yellow tennis balls, of trajectories of photons as undulating blue lines, and of quarks as colored glass marbles. Our images, like those of painters, are derived from the simple things we see in the world around us. That isn't really surprising, for physics, like all creative endeavors, engages the imagination, and the word "imagination" comes, not by coincidence, from "image."

Physics conjures up images and, conversely, images stimulate thoughts about physics. The paintings of Ronald Davis are especially inviting to this kind of meditation because they display just the right blend of realism and abstraction. In particular, it seems to me that Davis's work can be seen as a metaphor for the way theoretical physicists think about the world. To learn about the workings of a physicist's mind, without having to delve into the actual theories, we can do no better than to turn to Albert Einstein, who recorded some of his profound insights into his own mental processes. Davis's work, through its imagery, helps us to get a glimpse of the great physicist's thinking. At the same time, Einstein's way of seeing the world illuminates Davis's art.

Einstein was primarily a visual thinker. He rarely thought in words at all, and mathematics did not come naturally to him-he used it only to the extent that he had to. The special theory of relativity, for example, is couched in terms of high school algebra, while the later, much more sophisticated theory of gravity requires a formalism that took Einstein ten arduous years to learn. The basic objects of his thinking were visual images. Gerald Holton, in an essay entitled "On Trying to Understand Scientific Genius," quotes Einstein's own description of his mental activity in words that apply equally well to Davis's work:

What, precisely, is "thinking"? When, upon reception of sense-impressions, memory pictures emerge, that is not yet "thinking." And when such pictures form a series, each member of which calls forth another, this too is not yet "thinking." But when a certain image turns up in many such series, then-precisely by its return–it becomes an ordering element–a concept.... It is by no means necessary that a concept must be connected with a recognizable sign or word.... All our thinking is of this nature of a free play with concepts.

And elsewhere Einstein elaborates:

This combinatory play seems to be the essential feature of productive thought before there is any connection with logical construction in words or other kinds of signs (such as mathematical symbols) which can be communicated to others. The elements mentioned above are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the associative play is sufficiently established.

The term "play" occurs often in Einstein's writings about the creative process. He played with concepts the way a dog worries a bone and the way Davis plays with the visual possibilities inherent in an arch or a box. But for both Einstein and Davis play is serious. The intensity with which Einstein juggled the same sparse set of concepts–elativity, symmetry, continuity, atomicity–for an entire lifetime is echoed in the singleminded concentration with which Davis explores his own set of themes. This may be play, but the approach is not playful. Davis and Einstein look at the world with childlike, eyes, but they are the eyes of grave and deeply thoughtful children.

This play with concepts, for both men, is guided by a simple purpose: to get it right. Einstein, when he formulated special relativity, did not set out to revolutionize physics. All he had wanted to do was reconcile a seemingly trivial inconsistency in classical physics. As a teenager, he tried to imagine what he would see if he rode along a beam of light at its own speed, and later he found out that mechanics and optics gave different answers to the question. It wasn't a very pressing problem but, as in everything else he did, Einstein was deter mined to get it right. Davis, too, is concerned with simple objects, and the seemingly inconsequential problems they pose. Where do the lines meet? Is the shadow here a little darker, or should it be lighter? How do the planes overlap? Are they parallel, or not? These are questions that face all painters, but because he concentrates on them more sharply, Davis has to answer them more precisely And invariably he gets them right. Right, not in the simplistic sense of verisimilitude, but by the more exacting standards of artistic integrity. The cogency of Davis's work reminds us of the ease with which Einstein's theory of 1905, with the famous E = mc², in spite of its strangeness, convinced the majority of his colleagues that it must be right.

Davis' painted objects may be seen as metaphors for Einstein's images or concepts. For the fiberglass pieces this relationship is straightforward. As a child learning geometry, Einstein felt that "the objects with which geometry deals seemed to be of no different type than the objects of sensory perception, which can be seen and touched." Euclid's constructions were, for Einstein, tangible objects. Yet they are not simple objects that can be easily described in words. Like Davis's objects, they are fine and subtle things in which solidity and palpability compete with an essential ineffability. One can imagine Einstein, as a boy, seeing the entire proof of the Pythagorean theorem appear before his inward eye in the form of Davis's Sawtooth (1970). It is the proof of the theorem, not merely the statement, that so appears. The statement is a simple fact, easily verbalized and memorized. The proof, on the other hand, is an intricate process that requires active thinking: "First you draw this auxiliary triangle, and then that one, and then you notice their connection In looking at Sawtooth, your eye and brain are similarly compelled into action, comparing, measuring, visualizing hidden spatial relationships, jumping from three dimensions to two and back again and, finally, with a sigh of satisfaction, concluding that it is right. Just like the Pythagorean theorem.

The objects of Davis's later paintings and lithographs, as well as the concepts of Einstein's mature mind, are more complex, and their relationship to one another more tenuous. The objects are elusive. Their size, for example, is ambiguous, owing to the absence of scale. The planes can be seen either as paper-thin three-dimensional walls, or as true two-dimensional surfaces. The shadows have, to some extent, become detached and have acquired an independent existence. Thus the objects hover enigmatically between concreteness and abstraction. They share this quality with the mathematical models physicists use to imitate the world. Consider, for example, the earth's gravitational field, a concept that can be defined mathematically and used to make predictions, but that can't be understood intuitively the way a rock can, or a volume of air, or even space can. We imagine gravity all around us, affecting our every move, but we don't really know what it is. The gravitational field, and the other devices that physicists construct to model the world, are located, like Davis' objects, halfway between reality and imagination.

The objects of Davis's later paintings and lithographs, as well as the concepts of Einstein's mature mind, are more complex, and their relationship to one another more tenuous. The objects are elusive. Their size, for example, is ambiguous, owing to the absence of scale. The planes can be seen either as paper-thin three-dimensional walls, or as true two-dimensional surfaces. The shadows have, to some extent, become detached and have acquired an independent existence. Thus the objects hover enigmatically between concreteness and abstraction. They share this quality with the mathematical models physicists use to imitate the world. Consider, for example, the earth's gravitational field, a concept that can be defined mathematically and used to make predictions, but that can't be understood intuitively the way a rock can, or a volume of air, or even space can. We imagine gravity all around us, affecting our every move, but we don't really know what it is. The gravitational field, and the other devices that physicists construct to model the world, are located, like Davis' objects, halfway between reality and imagination.

A conspicuous element common to Einstein's thinking and Davis's painting is the frame of reference. Einstein drew an astonishing wealth of far reaching conclusions from careful attention to the simple fact, obvious to painters, that the description of a physical phenomenon depends on the observer's point of view. Without exception Einstein's first question about a new problem was: What is the frame of reference? Where is the observer? So it is with Davis. The device he uses to define the frame of reference in Frame and Beam (1975), as well as in most other works of this period, is the snapline. As a guide for construction, builders stretch a string drenched in chalk tightly against a flat surface. When they pull the string up, and let it snap back against the surface, it leaves a clear, straight chalk line, or snapline. With dry pigment substituting for chalk, Davis constructs multi-colored grids from snaplines, according to the rules, selectively interpreted, of perspective. To clarify spatial relationships even further, Davis always paints from a fixed point of view, above the object. The consistency of this perspective draws attention to the position of the painter and reminds us that in art, as well as in physics, the observer cannot be entirely detached from the observed. Ron Davis is always there, an unseen cicerone behind our shoulder, pointing out the subtleties of his vision.

In Brick (1983) a change has occurred that parallels Einstein's progress from the special theory of relativity in 1905 to the general theory in 1916: The global frame of reference has become local. The single rigid framework that spans all space has given way to a portable frame carried by each object. In the general theory of relativity, every massive object in the universe determines how space and time are configured in its own immediate vicinity. When all these different private frames of reference, which point in different directions, are connected together smoothly, the result is a web called curved space-time. Brick shows the gridlines around one object and invites speculation about how they would continue to the cosmos in the background, and then on to infinity.

The frame of reference, besides anchoring objects in place, plays an active role as part of the fabric of mathematics. The snaplines belong to the apparatus of projective geometry, and thus to the whole world of mathematics. They carry us off to realms of pure reason where human senses are irrelevant and infinity acquires meaning. Davis' objects help us visualize mathematical relationships. But which is fundamental, the abstract formalism, or its material representations? The rationalist position holds that mathematical truth exists independently of real examples. The empiricist counters that abstractions, like space and shape, must be derived from real phenomena. In terms of Davis' paintings, we ask: Which is primary, the objects or the lines? Are the objects merely flimsy bits of plywood or cloth stretched between gridlines like warning flags on guy wires or, on the contrary, are the lines actually defined by the edges of the objects? Which holds up which? We can assume either position, and even switch purposely from one to the other, thereby changing our reaction to a painting. Davis encourages this mental exercise by the balance he maintains between object and frame of reference.

And again, there is a parallel to Einstein's way of thinking about the world. In response to the charge, "Einstein's position .... contains features of rationalism and extreme empiricism...," Einstein replied, "This remark is entirely correct .... A wavering between these extremes appears to me unavoidable."

The difference between theoretical physics and mathematics lies in their tests for validity. Even as the physicist constructs his most elegant mathematical edifice, he keeps in mind the real world in all its messiness, confusion, elusiveness, and stubbornness. Into its tumultuous welter of phenomena he must eventually plunge his carefully wrought model to check whether it has any predictive value. If it doesn't, he must discard it. The mathematician is spared that ordeal. His criterion is spelled out by G. H. Hardy in A Mathematician's Apology: "The mathematician's pattern, like the painter's or the poet's, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way; Beauty is the first test." On this authority, mathematicians might claim Ron Davis as a kindred spirit. But he paints like Albert Einstein thought, and Einstein, although his theories rank with the great monuments of mathematics, was a physicist. The wildness of nature was always in his mind, and it is always present in Davis's paintings, as in Frame Float (1975), in the form of the background behind the geometrical forms.

Objects, snaplines, and background constitute the three major elements of Davis' paintings, and we may see them as metaphors for the theoretical models, the mathematical apparatus, and the uncontrollable phenomena of the natural world that together comprise physics. Of the three, mathematics is the most artificial and controlled. Snaplines can be placed at will, and even the rules of perspective are largely arbitrary. Nature, on the other hand, goes her own way: we do not control her laws. The most natural element in Davis' paintings is the splashing of paint in the background and near the snaplines. The colors are selected with exacting care, but the droplets fall where they must. Their patterns are not like the patterns of mathematicians, but like those of the world of physics. Mathematical models, finally, mediate between the realms of mathematics and nature. The gravitational field, for example, is a mathematical construct just as surely as it is an observed fact. Much of the power of Davis' paintings derives from his ability to make this connection. His objects are rigid geometrical constructs, but they are also inextricable parts of the relaxed play of color around them.

Relative strengths of the three elements do vary from example to example, as they do in physics, but you can't look at a Davis painting without being aware of all three. By consciously varying the importance we attach to each of the elements, we can mentally manipulate the painting in an almost uncanny way. This game is reminiscent of Bernard Berenson's insistence on learning to associate tactile values with retinal impressions of paintings in order to gain "the illusion of being able to touch the figures" in Renaissance art, and thus to appreciate them. Only, in the case of Davis, the effort of the game is more cerebral than muscular.

The harmony between the objects and the background reflects the relationship between a mathematical model and the real phenomena. In Frame and Beam, the link between the objects and the great green splotch, which could be an exploding galaxy or a bursting amoeba, is provided by the gridlines. The lines themselves can be thought of as functions in an equation or, more empirically, as light rays. They are first established, defined, and manipulated in, structures we can control-the frame and beam themselves, regarded as mathematical equations or optical instruments. And then the lines are thrust forth and extrapolated to the almost inaccessible region where they impose order on a random natural event. In another example, the same hues that are separated on the model in Invert Span (1979) blend into each other in the surrounding background. And further, the surface of the object in Brick is treated in a manner that mimics the cosmic background, but doesn't copy it.

There is no fixed prescription for the relationship between object and surroundings, the way there are prescriptions for the construction of gridlines that date back to Renaissance perspective. The physicist recognizes this variability as an echo of the multitude of ways in which he tries to model nature. Some models are approximate, but universal; others precise, but of limited applicability. Some are mathematically rigorous, but unrealistic; others just the opposite. In theoretical physics, no less than in painting, there are many ways to come to terms with nature.

What remains constant, however, is the style. The great theoretical physicists have styles that are as personal and unique as those of the great painters. When Johann Bernoulli, a Swiss physicist of the eighteenth century, saw an anonymous solution to a difficult problem, he exclaimed: "Tanquam ex ungue leonem" (The lion is known by his clawprint! ). He had spotted the unmistakably imperial manner of Isaac Newton. Both Einstein's style, and Ron Davis', are marked by a peculiar blend of concreteness and abstraction, of empiricism and rationalism. Their affinity is rooted in the power of their visual imagination and the unfathomable common origins of artistic and scientific creativity. Both men are equipped with a kind of x-ray vision that allows them to see through the material objects before them to the underlying mathematical structure. And both are adept at expressing their deeply felt sense of awe at the beauty of the hidden order they discover there.

In Brick (1983) a change has occurred that parallels Einstein's progress from the special theory of relativity in 1905 to the general theory in 1916: The global frame of reference has become local. The single rigid framework that spans all space has given way to a portable frame carried by each object. In the general theory of relativity, every massive object in the universe determines how space and time are configured in its own immediate vicinity. When all these different private frames of reference, which point in different directions, are connected together smoothly, the result is a web called curved space-time. Brick shows the gridlines around one object and invites speculation about how they would continue to the cosmos in the background, and then on to infinity.

The frame of reference, besides anchoring objects in place, plays an active role as part of the fabric of mathematics. The snaplines belong to the apparatus of projective geometry, and thus to the whole world of mathematics. They carry us off to realms of pure reason where human senses are irrelevant and infinity acquires meaning. Davis' objects help us visualize mathematical relationships. But which is fundamental, the abstract formalism, or its material representations? The rationalist position holds that mathematical truth exists independently of real examples. The empiricist counters that abstractions, like space and shape, must be derived from real phenomena. In terms of Davis' paintings, we ask: Which is primary, the objects or the lines? Are the objects merely flimsy bits of plywood or cloth stretched between gridlines like warning flags on guy wires or, on the contrary, are the lines actually defined by the edges of the objects? Which holds up which? We can assume either position, and even switch purposely from one to the other, thereby changing our reaction to a painting. Davis encourages this mental exercise by the balance he maintains between object and frame of reference.

And again, there is a parallel to Einstein's way of thinking about the world. In response to the charge, "Einstein's position ... contains features of rationalism and extreme empiricism...," Einstein replied, "This remark is entirely correct ... A wavering between these extremes appears to me unavoidable."

The difference between theoretical physics and mathematics lies in their tests for validity. Even as the physicist constructs his most elegant mathematical edifice, he keeps in mind the real world in all its messiness, confusion, elusiveness, and stubbornness. Into its tumultuous welter of phenomena he must eventually plunge his carefully wrought model to check whether it has any predictive value. If it doesn't, he must discard it. The mathematician is spared that ordeal. His criterion is spelled out by G. H. Hardy in A Mathematician's Apology: "The mathematician's pattern, like the painter's or the poet's, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way; Beauty is the first test." On this authority, mathematicians might claim Ron Davis as a kindred spirit. But he paints like Albert Einstein thought, and Einstein, although his theories rank with the great monuments of mathematics, was a physicist. The wildness of nature was always in his mind, and it is always present in Davis's paintings, as in Frame Float (1975), in the form of the background behind the geometrical forms.

Objects, snaplines, and background constitute the three major elements of Davis' paintings, and we may see them as metaphors for the theoretical models, the mathematical apparatus, and the uncontrollable phenomena of the natural world that together comprise physics. Of the three, mathematics is the most artificial and controlled. Snaplines can be placed at will, and even the rules of perspective are largely arbitrary. Nature, on the other hand, goes her own way: we do not control her laws. The most natural element in Davis' paintings is the splashing of paint in the background and near the snaplines. The colors are selected with exacting care, but the droplets fall where they must. Their patterns are not like the patterns of mathematicians, but like those of the world of physics. Mathematical models, finally, mediate between the realms of mathematics and nature. The gravitational field, for example, is a mathematical construct just as surely as it is an observed fact. Much of the power of Davis' paintings derives from his ability to make this connection. His objects are rigid geometrical constructs, but they are also inextricable parts of the relaxed play of color around them.

Relative strengths of the three elements do vary from example to example, as they do in physics, but you can't look at a Davis painting without being aware of all three. By consciously varying the importance we attach to each of the elements, we can mentally manipulate the painting in an almost uncanny way. This game is reminiscent of Bernard Berenson's insistence on learning to associate tactile values with retinal impressions of paintings in order to gain "the illusion of being able to touch the figures" in Renaissance art, and thus to appreciate them. Only, in the case of Davis, the effort of the game is more cerebral than muscular.

The harmony between the objects and the background reflects the relationship between a mathematical model and the real phenomena. In Frame and Beam, the link between the objects and the great green splotch, which could be an exploding galaxy or a bursting amoeba, is provided by the gridlines. The lines themselves can be thought of as functions in an equation or, more empirically, as light rays. They are first established, defined, and manipulated in, structures we can control-the frame and beam themselves, regarded as mathematical equations or optical instruments. And then the lines are thrust forth and extrapolated to the almost inaccessible region where they impose order on a random natural event. In another example, the same hues that are separated on the model in Invert Span (1979) blend into each other in the surrounding background. And further, the surface of the object in Brick is treated in a manner that mimics the cosmic background, but doesn't copy it.

There is no fixed prescription for the relationship between object and surroundings, the way there are prescriptions for the construction of gridlines that date back to Renaissance perspective. The physicist recognizes this variability as an echo of the multitude of ways in which he tries to model nature. Some models are approximate, but universal; others precise, but of limited applicability. Some are mathematically rigorous, but unrealistic; others just the opposite. In theoretical physics, no less than in painting, there are many ways to come to terms with nature.

What remains constant, however, is the style. The great theoretical physicists have styles that are as personal and unique as those of the great painters. When Johann Bernoulli, a Swiss physicist of the eighteenth century, saw an anonymous solution to a difficult problem, he exclaimed: "Tanquam ex ungue leonem" (The lion is known by his clawprint! ). He had spotted the unmistakably imperial manner of Isaac Newton. Both Einstein's style, and Ron Davis', are marked by a peculiar blend of concreteness and abstraction, of empiricism and rationalism. Their affinity is rooted in the power of their visual imagination and the unfathomable common origins of artistic and scientific creativity. Both men are equipped with a kind of x-ray vision that allows them to see through the material objects before them to the underlying mathematical structure. And both are adept at expressing their deeply felt sense of awe at the beauty of the hidden order they discover there.

— HANS CHRISTIAN VON BAEYER, 1986

Hans Christian von Baeyer is Professor of Physics at The College of William and Mary in Williamsburg, Virginia. He is the author of the award winning book Rainbows, Snowflakes and Quarks and is Contributing Editor of the Academy's magazine, The Sciences.

SELECTED READINGS

Bernstein, Jeremy. Einstein. New York: The Viking Press, 1973.

Bronowski, Jacob. The Origins of Knowledge and Imagination. New Haven, Conn.: Yale University Press, 1978.

Calder, Nigel. Einstein's Universe. New York: The Viking Press, 1979.

Elderfield, John. "New Paintings by Ron Davis," Artforum, March 1971, pp. 32-34.

Fine, Ruth E. Gemini G.E.L.: Art and Collaboration. Exhibition catalogue, National Gallery of Art, Washington, D.C. New York: Abbeville Press, 1984.

Fried, Michael. "Ronald Davis: Surface and Illusion," Artforum, April 1967, pp. 37-41.

Hardy, G. H. A Mathematician's Apology. Cambridge, England: Cambridge University Press, 1940.

Holton, Gerald J. "On Trying to Understand Scientific Genius," in Thematic Origins of Scientific Thought: Kepler to Einstein. Cambridge, Mass.: Harvard University Press, 1973.

Kessler, Charles. Ronald Davis Paintings 1962-76. Exhibition catalogue, The Oakland Museum, Oakland, California, 1976.

Marmer, Nancy. "Ron Davis: Beyond Flatness," Artforum, November 1976, pp. 34-37.

von Baeyer, Hans Christian. Rainbows, Snowflakes and Quarks. New York: McGrawHill Book Company, 1984. The The New York Academy of Sciences, 1986. All rights reserved.