On Tue, May 18, 2010 at 6:00 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Suppose we define a slightly generalized type of regular "spherical" polyhedron (RSP) as follows.
(on say the unit sphere), equipped with a mapping
F: X -> S^2
that is an isometry when restricted to any one P_k. Note that the P_k are not necessarily either disjoint or distinct, and they may have interior angles > pi. Note that the polygons P_k are subsets of X but not of S^2.
X must satisfy:
* For any edge e of any one polygon, there are exactly two polygons that have e as a full edge. Two such polygons sharing exactly one full edge are called "adjacent". No two adjacent polygons have the same image under F.
If I was making the definition, I would add the condition here that the image under X of the two polygons adjacent to e includes an open neighborhood of every point in the image of e except possibly the vertices e connects. Or roughly speaking, the two faces adjacent to e are on "opposite sides" of the edge. Did you intend to include this condition? Are there polygons that satisfy your definition that do not satisfy this condition? Andy andy.latto@pobox.com