[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