Symmetric four-dimensional polytope and visualization method in four, eight, and sixteen dimensions using Hopf maps

Inspired by, and using methods of optimization derived from classical three-dimensional electrostatics we have found a four-dimensional polytope, new to our knowledge, with a high degree of symmetry in terms of the lengths of sides--64 of the 80 vertices have twelve nearest neighbors with the same four nearest neighbor distances, and the other 16 vertices have ten nearest neighbors with distances that are two of the four nearest neighbor distances for the set of 64 vertices. We give and illustrate a simple geometric method to visualize this configuration and other configurations in four, eight, and sixteen dimensions.