On 11/25/06, Daniel Asimov <dasimov@earthlink.net> wrote:
... 1. E.g., the projective plane can be constructed by identifying the faces of the icosahedron by the antipodal map, giving a simplicial complex K of 10 triangles, having a symmetry group of order 60 (= the smallest non-abelian simple group?).
So . . . what is the lowest-dimensional Euclidean space into which this embeds AOS, such that all 60 symmetries are realized by rotations?
This is presumably the triangulation referred to by Cervone in the quote posted earlier about Brehm's theorem: I cannot at the moment visualise how this construction fails to immerse in 3-space (for which apparently at least 9 vertices are required)! If the embedding in 4-space can be equilateral --- Brehm's paper must discuss this --- then the symmetry would follow. But it somehow doesn't look entirely plausible --- for example, the (eutactic star driven) projection of a hypercubic lattice into a Penrose tiling is down from 5 dimensions, rather than 4. So for what it's worth, I'd put my money on 5-space here. WFL