On Tuesday, March 2, 2004, at 11:12 AM, Michael Kleber wrote:
Does anyone know the pedigree of the following line of reasoning? I haven't defined "by continuity," but I think the intent will be clear from the proof. [...] There you have it: the polyhedron so constructed clearly has twenty equilateral triangular faces meeting five at each vertex.
Another proof, I believe, can be cobbled from the idea of a cube, each of whose faces has been bisected into two rectangles so that none of 6 new lines meet each other. Now each of these 6 new lines can be symmetrically shrunk toward its midpoint, while allowing the cube to hinge identically along all 6 shrinking lines. By continuity there will be a moment that each of the 12 resulting pentagons is regular, and it can be shown that at this point all dihedral angles are equal. Of course details need to be filled in, but I like this approach because of its symmetry. --Dan Asimov