Random exploration of the three regular polytopes

There are just three regular polytopes in Euclidean (n>4)-space. Their dimensions are determined, including the distance from the centroid to the periphery in a random direction-that of a white Gaussian noise vector. As n→∞, this distance becomes very predictable. It differs from the distance near which almost all of the volume and surface of the polytope lie. The discrete set of different signals that might be transmitted during any signaling interval may be represented by a set of points in such a space. To each of these signal points belongs a Voronoi-polytope decision region. White Gaussian noise in the transmission channel will add random contributions to the coordinates of the transmitted-signal point, moving it a somewhat random distance in a uniformly distributed random direction and causing a reception error if it moves the signal point outside its Voronoi polytope