Random exploration of the three regular polytopes

There are just three regular polytopes (the hypercube, cross polytope and the simplex) in Euclidean (n>4) space. We calculate their dimensions, including the distance from the centroid to the periphery in a random direction-that of a white Gaussian-noise vector. As n/spl rarr//spl infin/, this distance becomes very predictable. It differs from the distance near which almost all of the volume and surface of the polytope lie.