I should have written << As an abstract regular polytope, the 57-cell has as its group of symmetries L(2,19), aka PSL(2,19). The lowest dimension* of a permutation representation of this group is 19. So to embed the 2-skeleton of the 57-cell (171 dodecahedra suitably identified) in R^n so that its symmetries are all elements of O(n), we'd need n >= 19.

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The main difference is that the original 20's should be 19's. Sorry about that. --Dan ___________________ * I.e., on 20 points. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele