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 = mc^{2},
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.