Saturday, June 5, 2010

Klein's Quartic Curve - Greg Egan

"Klein's quartic curve is a surface of genus 3, which is to say that it is like a 3-holed torus. As well as having that topology, the surface has a metric (a definition of distances and angles) of constant negative curvature, which means it has the local geometry of the hyperbolic plane. By drawing a 14-sided polygon in the hyperbolic plane and indicating that certain edges need to be connected to each other, you can specify the geometry of Klein's quartic curve completely. There are a variety of other ways to specify the curve, most of them involving algebra and/or complex analysis. I've written a simple account of the construction of the curve via the wonderful quartic equation for it, u3v+v3w+w3u=0, in this page on Klein's quartic equation."

5 out of 5

No comments:

Post a Comment