Monday, March 26, 2007

Klein’s quartic and me

I’ve been chattering happily to lots of people over the last few weeks about Klein’s quartic curve and how it might (and probably will) provide some very interesting and fruitful directions for my research. So I figured it would be worthwhile to post a little about all that here. There is no shortage of websites about this surface (it’s called a curve because it only takes one complex parameter to describe it near a given point, but one complex parameter looks like two real parameters, so it looks like a surface to us); two places in particular to go for more information are John Baez’s page, which he based on several entries of his column This Week’s Finds in Mathematical Physics, and MSRI’s online version of The Eightfold Way (you can either order a paperback, or download most of the essays in the book), which was commissioned in honor of the unveiling of Helaman Ferguson’s sculpture “The Eightfold Way” (a picture of the whole sculpture is available in Mathworld’s entry on Klein’s quartic). I’ve been inspired a great deal by the essays in this book. I’m going to take a slightly different approach to introducing the surface, which may help to illuminate some of what my area of mathematics deals with.

Start with a regular heptagon (7-sided polygon). Already this is interesting, because this is the first regular polygon that can’t be constructed by the Greek straightedge-and-compass method. Gauss and Wantzel proved exactly which polygons could be constructed by this method; Gauss showed that an n-gon could be constructed if n has certain kinds of factors, and Wantzel showed that the only constructible polygons were those satisfying Gauss’s condition. Now consider all the triangles you can make whose vertices are vertices of the heptagon. Of these, exactly one has three different side lengths, if you consider congruent triangles to be the same. The heptagon is (almost) the unique polygon with this property. (The “almost” is because the hexagon also has this property, but the triangle you get from it is just the familiar 30–60–90 triangle.) The triangle you get has angles of π/7, 2π/7, and 4π/7, and acquired the illustrious name of the heptagonal triangle in a 1973 paper by Bankoff and Garfunkel. I’ll call it H.

This is where my people come in. Because the angles of H are rational multiples of π, we can assemble a bunch of copies to make an interesting surface. Do this by first reflecting H across its longest side, then rotating both H and the reflected triangle by multiples of 2π/7 around the vertex of their smallest angle, to get the picture below.

Already pretty, isn’t it? Now notice that as I’ve numbered the sides of this nonconvex 14-gon (tetradecagon, if you want to get fancy), sides with the same number are parallel. This parallelism is important to people like me. Now you build the surface in a way that’s difficult to actually manage if you’re doing this with, say, a piece of cloth. “Glue” each pair of parallel sides; if we were making this from cloth, this would mean simply sewing those edges together. All of the “innie” vertices end up coming together at one point, as do all of the “outie” vertices. This surface is flat almost everywhere; the two points obtained from the innie and the outie vertices are the only points that can’t be flattened out. These are called cone points; however, unlike the cones we learned about in high school geometry, the angle at each of these points is more than 2π. In fact, we can just count up what the angle is for each. Each outie vertex has the 2π/7 angle of two copies of H, which makes 4π/7, and there are seven such vertices, so this point has an angle of 4π. Likewise, the point where all the innie vertices come together has an angle of 8π. Notice that the π/7 angles of the copies of H all ended up in the middle of the picture, and there are fourteen of them, so they just make up a regular point with an angle of 2π around it. We say this surface is obtained by unfolding H.

This process of “unfolding” a polygon with angles that are rational multiples of π is important in the study of billiards in such polygons. Billiards here just means an ideal abstraction of the game we all know; instead of a ball, you have a point, which moves in a straight Euclidean line inside the polygon (which is the “table”) until it hits a side, at which point it follows the law often stated “the angle of incidence equals the angle of reflection.” This is a very nice situation to study, but it would be even nicer if we could get rid of those kinks in the point’s path where it hits a side. We do this by reflecting the polygon instead of the path. If the angles of the polygon are rational multiples of π, then any such sequence of polygons, obtained by reflecting as we follow a billiard path, will eventually come back to a polygon that’s a translation of the original. Since from this perspective the point has just been moving in a straight line in the plane, it will enter this translated copy going exactly the same direction as it would be going if it had just been bouncing around inside the original table. So, instead of making that last reflection, we glue the side the point crosses over along this path to the corresponding side of the original triangle. Do this starting in enough different directions, and you get a surface like we did above. A theorem proved by William Veech in the 1980s says that by studying certain features of the geometry of the surface we can conclude a great deal about the behavior of billiard paths in the original polygon (such as which ones are periodic, i.e., eventually come back to the same point with the same direction, and which ones eventually “fill up” the table, i.e., come arbitrarily close to any point of the table).

Now maybe it doesn’t seem like we’ve gotten to anything deserving to be called “Klein’s quartic” yet. But it turns out that the flat geometry above comes very naturally from the geometry of the surface we’re looking for. In the paper by Karcher and Weber in The Eightfold Way, they show that this geometry can be used to derive the equation x3y + y3z + z3x = 0 , where x, y, and z are complex variables, and that any of the coordinates x, y, or z can be used to construct this flat geometry. This equation, of degree four, is where Klein’s quartic gets its name. (There’s some projective geometry at play here, too; if (x,y,z) provides a solution to the equation, then so does (ax,ay,az), because the polynomial on the left of the equation is homogeneous, so we consider any solutions related by a multiple in this way to be the same point. This is going a bit farther than I’d really like to explain here, so I’ll hope you’ll either know exactly what I’m talking about and can make the necessary adjustments, or you’ll take the fourth degree equation and just trust that it gives the name “quartic” to a certain surface representing its solutions.)

A surface described by an equation like the one we’ve got can be given many different “flat structures” resembling the one we started with on Klein’s quartic. All of these different ways of making the surface flat form an interesting collection of objects to study. I’ve chosen to study Klein’s quartic in particular because it has many beautiful internal symmetries, and these are reflected (no pun intended) in some of the flat structures on it. For example, the flat structure coming from H has a clear 7-fold symmetry. But another, composed of one large cylinder and three equal small cylinders, shows a 3-fold symmetry, visible in the equation by permuting x, y, and z.

I’ll leave the description at that for now and get back to studying the surface.