What is the smallest number of dimensions in which we can embed a pentagon with integer vertices? More generally, for an n-gon, the argument below suggests (I think) that we need at least p-1 dimensions where p is the largest prime factor of n. We can do a hexagon in three dimensions, but not, I think, in two. Cris On May 25, 2013, at 11:09 PM, Bill Gosper wrote:
Let's see. If a Platonic solid K can be embedded in R^3 with integer vertices, then the center will be a rational point, so by an integer expansion it, too, will be integer. So by an integer translation we can assume K has integer vertices and center.
Then the isometry group Isom(K) will be a subgroup of GL(3,Z), so in particular GL(3,Z) must have an element g of order 5. Then a primitive 5th root of unity must be an eigenvalue of g's matrix. But such roots of unity have a minimal polynomial equal to (x^5-1)/(x-1) = an integer polynomial of 4th degree, so can't be an eigenvalue of an integer 3x3 matrix.