On Tue, Jun 16, 2015 at 9:40 AM, James Propp <jamespropp@gmail.com> wrote:
For purposes of my analogy, one should delete the cone point itself, to avoid the infinite curvature there. The resulting manifold then has zero curvature everywhere, but it has fewer symmetries than the plane that gave rise to it by cutting and gluing. (How should one wallpaper the inside of a teepee for maximal mathematical elegance? Unclear.)
Cut out either 90 degrees for a square tiling (or 60 degrees for an equilateral triangular tiling). Then at the one defect, you'll have three squares (or five triangles) meeting at a vertex, but everywhere else you'll have four (or six). Something similar should work on the hyperbolic plane. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com