Thanks to Fred, Bill, and Andy for pointing out my blunder with the allegedly 4-triangle Moebius band. --------------------------------------------------------- A natural extension of the question of what dimension a simplicial complex embeds in, affinely on simplices (AOS), is this: QUESTION: Given a simplicial complex K and its abstract symmetry group, what is the smallest N such that K embeds AOS in R^N, such that each original symmetry of the embedded complex is realized by a rotation of R^N ??? 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? 2. Ditto for the lovely torus someone mentioned -- given by a simplicial complex of 14 triangles (6 per vertex). Its symmetry group has order 84. (This complex is the dual of the amazing 7-hexagon tiling of the torus.) 3. Likewise for the beautiful rendition of the 3-holed torus as 56 triangles (7 per vertex). Its symmetry group has order 336. (This complex is the dual of the gorgeous 24-heptagon tiling of the surface of genus 3.) (Or, for 2. and 3. one could just ask to realize as rotations the *orientation- preserving* symmetry groups -- of orders 42, and 168 (= the second-smallest non-abelian simple group), respectively.) --Dan