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Re: tessalation of a sphere

• To: mathgroup at smc.vnet.net
• Subject: [mg9283] Re: tessalation of a sphere
• From: weber at math.uni-bonn.de (Matthias Weber)
• Date: Mon, 27 Oct 1997 02:46:54 -0500
• Organization: RHRZ - University of Bonn (Germany)
• Sender: owner-wri-mathgroup at wolfram.com

In article <62pedv$acs at smc.vnet.net>, "I. Inanc Tarhan"
<tarhan at piranha.cis.upenn.edu> wrote:

> Hi all,
> 
> Would anybody have information on, or pointers to the subject of
> representing a sphere as a number of equidistantly spaced points on its
> surface (i.e., if you can fold open the sphere without any distortions,
> the points will be hexagonally "close" packed)?  Specifically, I am
> trying to represent the surface of a sphere as a collection of
> identical equilateral triangles. 
> 
> Any good ways of doing this with Mathematice?
> 

There is a much more general theorem of Aleksandoff stating that this is
possible if there are at most 6 triangles around each vertex as the
boundary of a convex body in R^3. Hence if you actually want to do it
with a given combinatorial triangulation, this problem reduces to
solving a bunch of linear equations, which Mathematica should be able
to do. I don´t know about ready code doing this. A general reference
for your problem are Voronoi diagrams. There is a vast literature about
them, just search the net.

Matthias Weber