Like everyone else I know, when winter comes I think of Plato’s theory of ideal forms. If I say circle or square, you know what I mean, though in some sense you have never seen a circle—the shape in the plane where all the points are exactly equidistant from the center—because in reality everything is always at least a little off. Plato thought truth ought to work this way, that what we understand as truth is always an approximation to an ideal form.
Winter is the Platonist’s season. We search for a Christmas tree with a form in mind and, among the hundreds at the farm stand or parking lot, we seek the one that comes closest to matching the ideal green cone that we carry in our mind’s eye. We bring it home and set it up, taking care to line up its central axis with the vertical, so that its rotational symmetry is apparent. Then we hang glass spheres on it, and drape it evenly with lights to accentuate its shape. Christmas trees may have arrived on the scene a couple thousand years after Euclid, but he certainly would have understood our appreciation for them.
Some of us decorate our houses with strings of lights, outlining the house, drawing in the roofline, the eaves, the doors, and the windows with a pencil of light. At night, the flip of a switch reveals not the house but an abstraction of it—an ideal form of “home” that is as simple as a child’s sketch.
There’s the snowman: the human form given in three spheres. It is a sort of absurdist abstraction: the top sphere makes sense, and we can stretch to consider the middle one to have captured the salient properties of those of us with more orbic midsections. But I don’t know what to make of the bottom sphere. There may be some work to do here, which Euclid or Archimedes might have gotten to if it had snowed more often in Greece. The ratios of the spheres matter. A stack of three equally sized white spheres might read as tennis balls in a sleeve or cocktail onions on a toothpick. After considerable investigation, I have discovered that if the proportions of the diameters are 5:4:2.5 (from bottom to top) then the form unambiguously reads as a snowman, with or without carrot, coal, sticks, scarf, or hat. We have then a stack of three white spheres that signify archetypal “winter” quite clearly. I challenge you to signify any other time of year with such simple geometry.
Then there’s ice. You can walk on it, walk around it, measure it, sculpt it. This is water, the exemplar of the formless, given form. The icicle, yet another ideal form of winter, is the drip frozen—actually, drip upon drip. A single solid is the entire history of a tiny stream. We cannot see but we can certainly imagine what has happened on a molecular level. When the temperature dropped, the water molecules, once free-flowing and independent, arranged themselves in an orderly, symmetrical fashion: a crystal. This crystallization on the grander scale can be seen in the snowflake, perhaps nature’s most inspirational gift to the Platonist. Into your hand fall example after example of intricate, Platonic glory, each more miraculous than the last. How can you look at such symmetry and not suppose, like the ancients, that a theory of ideal forms would allow you describe the basic structure of the universe?
What is snow if not the ideal material? We can shape it, build shelter with it, throw it, slide on it, jump into it, and eat it. It is made of crystals, and those crystals refract and scatter the light upon them, ultimately reflecting it back to us. It does so without prejudice as to wavelength, and so we see white, the color of light. And what a white it is. The white of a fresh snow under a blue sky? It is the white that the paint-maker aspires to.
Most mathematicians spend much of their time in either an explicit or an implicit search for these ideal forms, in geometry, in numbers, in the symbol dance of algebra. Looking for these forms is often just the process of abstraction—trying to see the thing in its most elemental of characteristics, and no more.
Winter seems to do this for us. Consider the deciduous trees in winter. Bereft of leaves, they hardly move in the wind. They stand in their fractal nakedness—each branch is itself a copy of the tree. In fact, their branching patterns can be all that distinguishes two species from one another. Mathematically describe their branching patterns (done in two or three numbers), and you have described the tree.
I have had people ask me what it is like to do research in mathematics, and perhaps the answer is that it is like a snowstorm. As the snow falls, the light dims and the world goes gray. Local distinctions are lost, sharp curves disappear, and the world is made softer, quieter, and simpler. When the sun comes out, the way we see the world has been transformed to a place of startling clarity and simplicity. A ski area in summer is a rugged, foreboding place, full of crags, rocks and brush, rough to look at and to hike. But the snow falls, and fills the holes and softens the points, and the jagged becomes smooth. A snow-covered hill is a mathematician’s dream come to this earth. All detail is gone, and there is nothing but the surface itself. This is the joy captured by the helicopter shot of the lone skier in the untrammelled backcountry bowl. At that moment the skier experiences a pure surface, and there is nothing but the contour of the the slope. The skier might be a drop of milk rolling down the side of a ceramic breakfast bowl. The snow-covered hills resemble nothing so much as the abstract surfaces that mathematicians draw for themselves on blackboards (with white chalk)—the simplest set of curves, which only convey shape. The snow-covered world is an abstraction of the world that lies underneath: the details are smoothed over, the color is removed, all that is left is an essence of shape. These are the forms that one can work with. This is how the mathematician thinks. This is what she does, in her minds eye, to the world around her.
Gregory Buck is a professor at mathematics at Saint Anselm College. He wrote about the mathematics of the beach this summer.
Photograph by Chris Steele-Perkins/Magnum.
