So I wrote a final paper last night that was supposed to be a response to an essay in art theory. It was about the use of grids in modern art. I waited until the last minute, obviously, to write the paper and decided to go to a three hour calc-III, RIT-sponsored, free-pizza study session before writing the paper. Well it was originally a procrastination-induced decision but it turned out to be a fantastic idea. I haven’t gotten this excited about a paper in a while. I was asked to share it with sunday and wednesday, but it was originally written to impress a really intense RIT professor/art intellect with a Ph. D in philosophy and a generally overwhelming understanding of almost everything, not a conversation starter to a casual audience. I realized today that this is the kind of stuff I try to bring to conversation with people because it’s what I’m all about, but I get so excited and flustered that it’s generally impossible. And I come off as pretty insane. So I took the essay and made a book report of it; made it more casual. You know, less of the ultimately/consequently/therefore/henceforth essay noise and more conversation. More real. Enjoy.
Rosalind Krauss analyzed the contemporary use of the grid in modern art in her 1979 essay Grids. Among other statements, she refers to the grid as, “a ubiquitous presence within modernism,” “a barrier to language,” and a structure that, “declares the space of art at once autonomous.” The grid is without a doubt an indispensable factor that modernism depends on, both visually and philosophically. Modernist ideas lend themselves to the claim that there is order in everything; that all mediums-—whether be it painting, sculpture, architecture, photography, etc.—-are distinct and verifiably separate entities that should never merge.
It’s interesting to point out that these modern, orderly, structured ideas accordingly resemble the grid itself and the properties of a Cartesian (x/y) system. It’s like how each x integer’s infinitely-extending vertical line on a traditional Cartesian grid, say x=2 and x=3, will forever be intersected, at a right angle, by each y integer’s infinitely-extending horizontal line within the space, and therefore are considered parallel. x=2 and x=3 will eternally be separated by one whole integer worth of space and will likewise never touch in either direction, out to infinity. The Greek mathematician Euclid justified this in his fifth postulate.
Krauss explains that,
“Logically speaking, the grid extends, in all directions, to infinity. Any boundaries imposed upon it by a given painting or sculpture can only be seen—according to this logic—as arbitrary. By virtue of the grid, the given work of art is presented as a mere fragment, a tiny piece arbitrarily cropped from an infinitely larger fabric. Thus the grid operates from the work of art outward, compelling our acknowledgement of a world beyond the frame.”
It’s crucial to realize that during the same general time as what has–at least in the textbooks–become the most important and influential upon any art to follow, the Renaissance was also at it’s climax during the same time as when Galileo theorized that the world wasn’t flat and that we weren’t the center of the universe. He was jailed, ridiculed and his ideas generally weren’t accepted until years and years after his death. But it was this huge misconception of a flat planet, a notion that’s totally exploited in Renaissance art (and okay, mainly due to the camera obscura as perspective tool) that radically became the principal motif in modern art more than five centuries later. We all know art inherently mimics the societal principles of its time, and as modernism almost immediately followed the Industrial Revolution–a time of shifting from the organic to a staggeringly impure and synthetic atmosphere for human existence–the ideals of the art world also radically began to demand this reciprocated shift back to the pure and orderly ideals, the ideals within Renaissance art.
Here’s another chief element to consider: the concepts of zero and infinity are both human inventions. They both ultimately stem from the scientific enlightenment of a spherical world, and are still (relatively) in their infancy: they’re only about six-hundred year-old ideas. And logically, neither zero nor infinity can be proven to exist in the native physical world, only the theoretical. So from Galileo’s first claim of a spherical, orbiting world, ultimately birthed a new world of physics and mathematics. We thought we had everything figured out when in fact we had maybe one-tenth of it. That’s even pushing it. So these new ideas were appropriately labeled as Non-Euclidean Geometry. As Non-Euclidean Geometry developed and expanded its knowledge, we began to realize that not only is the world round, but most previous physical phenomena could be accurately explained and understood with these Non-Euclidean principles, furthering the proof that Cartesian laws were only useful in a small-scale setting, and that parallel lines don’t exist. But now that we had invented infinity (and a small-scale setting had now become, say, Earth), Cartesian ideals didn’t do much good. So an infinite number of infinitely-parallel and perpendicular lines as such that Euclid had postulated in about 300 B.C. had proven to be relatively (but by no means entirely) useless, in the big picture.
So rather, the laws of the circular and spherical dominated. The two branching subdivisions of Non-Euclidean Geometry, hyperbolic and elliptical, respectfully state that two lines will either be infinitely growing apart from each other or will intersect infinitely many times, but regardless, both will be in an infinite state of change. It sounds more realistic even in as vague of terms as those, doesn’t it? The world exists not within the horizontal and vertical ideals of the traditional Cartesian grid, but rather the radial and angular, as obeyed in the Polar grid (that which is the ultimate basis of any Non-Euclidean model). These ideas were generally dismissed, as the modernists (perhaps unintentionally, perhaps not) called for a relapse back to the pure. It is of no surprise then, that the grid that begins to reappear in work during modernism—-that which Krauss refers to—-is strictly the horizontal and vertical grid that so well defined the new, false, city-environment that pursued after the Industrial Revolution. It’s also of no surprise that Krauss claims that,
“the grid’s mythic power is that it makes us able to think we are dealing with materialism (or sometimes science, or logic) while at the same time it provides us with a release into belief (or illusion, or fiction).”
Jackson Pollock, a distinct Abstract Expressionist painter, painted with deliberate intentions of breaking rules of pure, modern ideals. His process was staggeringly un-pure: using industrial materials, laying canvases on the floor, removing the brush (and ultimately the ‘hand of the artist’) almost entirely. It’s also interesting to question what role physics and gravity-—two natural and uncontrollable elements of the real world—-are inherently given in this process.
Studies conducted by notable M.I.T. authors have shed new light on Pollock’s paintings, illustrating how within them exist properties of mathematical fractals. Fractals are geometric shapes that repeat their entire shape, but in smaller copies within the whole; they are organic, naturally occurring entities and can be found in snowflakes, mountains, ferns, among others. They’re directly linked to and support much of what developed into Chaos Theory, a theory that aims to understand and explain much of the natural world (such as weather and clouds, to keep it simple). Fractals are (okay, with the exception of the straight line, which is only formally accepted to be fractal-esque) entirely non-Euclidean. The identifying property of fractals is called self-similarity, and it’s (once more!) interesting to point out how easily that single idea can be paralleled to most identifying characteristics of post-modernity.
It is an important distinction to point out when you’re criticizing the use of the grid, and what role it exactly plays in twentieth century modern art: the Euclidean/modern relationship and the Non-Euclidean/postmodern relationship. Though Krauss does bring the grid up under a positive and rational light (example, “by it’s very abstraction, the grid conveyed one of the basic laws of knowledge—the separation of the perceptual screen from that of the ‘real’ world… science began to yield [art’s] opposite,”). The Cartesian grid in modern art, and what it suggests not only visually but also what deeper-rooted connotations that it carries within its intrinsic mathematical place, definitely reveal a whole new perspective on it’s presence. It’s a stretch, but its presence not only further the connection between science and art, but ultimately—-and perhaps entirely instinctively/subconsciously—-further the commonalities between left and right hemispheres of the human brain, between the logical and the theoretical, and perhaps between you and I. Infinity is hard to wrap your mind around. So is zero, if you think enough about it–as less of a number and more of a concept. Everything and nothing can (and usually will) make someone feel really small and insignificant. Ian Stewart wrote something that has stuck with me and I’ll do my best to paraphrase it. Math is a product of human minds but not bendable to human will. It’s not imaginary though, imagination is a human characteristic. It’s real but in a metaphysical sense. And I think that’s the most important part.
I can write forever about this stuff. And I’m sure this won’t be the last time I shove math on here, but for now I’ll leave you with this quote (again, from Ian Stewart in his Letters to a Young Mathematician) and hopefully turn this rant into conversation.
You asked me whether you would have to give up your sense of beauty to study mathematics, whether everything would become just numbers and equations to you, laws and formulas. Rest assured, Meg, I don’t blame you for asking this, since it’s unfortunately a very common idea, but it couldn’t be more wrong. It’s exactly the opposite of the truth.
What math does for me is this: It makes me aware of the world I inhabit in an entirely new way. It opens my eyes to nature’s laws and patterns. It offers an entirely new experience of beauty.
When I see a rainbow, for instance, I don’t just see a bright, multicolored arc across the sky. I don’t just see the effect of raindrops on sunlight, splitting the white light from the sun into its constituent colors. I still find rainbows beautiful and inspiring, but I appreciate that there’s more to a rainbow that mere refraction of light. The colors are, so to speak, a red (and blue and green) herring. What require explanation are the shape and the brightness. Why is a rainbow a circular arc? Why is the light from the rainbow so bright?
You may not have thought about those questions. You know that a rainbow appears when sunlight is refracted by tiny droplets of water, with each color of light being diverted through a slightly different angle and bouncing back from the raindrops to meet the observing eye. But if that’s all there is to a rainbow, why don’t the billions of differently colored light rays from billions of raindrops just overlap and smear out?
The answer lies in the geometry of the rainbow. When the light bounces around inside a raindrop, the spherical shape of the drop causes the light to emerge with a very strong focus along a particular direction…
The rainbow that you see and the rainbow that I see are created by different raindrops. Our eyes are in different places, so we detect different cones, produced by different drops.
Rainbows are personal.
Some people think that this kind of understanding “spoils” the emotional experience. I think this is rubbish. It demonstrates a depressing sort of aesthetic complacency. People who make such statements often like to pretend they are poetic types, wide open to the world’s wonders, but in fact they suffer from a serious lack of curiosity: they refuse to believe the world is more wonderous than their own limited imaginations. Nature is always deeper, richer, and more interesting than you thought, and mathematics gives you a very powerful way to appreciate this.
Q.E.D.; Goodnight.