[A post I intended to but, I think, didn't send to s.m.r. File is dated
1/18/95 -- djr]
Newsgroups: sci.math.research
Subject: Re: Equal subdivision of spherical surface
In article <3fhj5m$s7i@mordred.gatech.edu>,
Paul Goldsman wrote:
>I believe that there is no known way to subdivide the surface of a
>sphere into greater than 120 congruent (ie, equal area, equal shape)
>divisions. The 120 is accomplished by bisecting each side of each face
>of a spherical icosihedron, 6*20=120.
You're forgetting about slicing the sphere like an orange with many
segments.
Perhaps this is what you have in mind: you want to describe the surface
of the sphere as a finite union of pieces overlapping only on the boundary,
with the property that given any two of them there is a rotation of the
sphere which carries each patch to another and in the process carries
your first to your second. If so, look at the collection of rotations
which interchange the pieces. It will be a finite subgroup of the group
SO(3) of rotations. In that case it is known that the group must be
either the group of the orange sections or the group of one of the Platonic
solids, and so yes, the number of sections is limited to 120 if you
disallow having them all reach an axis.
dave