Here’s an engaging moving picture from RobertLovesPi. The Platonic solids — cubes, pyramids, octahedrons, icosahedrons, and dodecahedrons — are five solid shapes each with the same regular convex polygon as their face. This is a nice two-dimensional rendering of a three-dimensional projection of a “hyperdodecahedron”. It’s made of 120 dodecahedrons, in a four-dimensional space. And it’s got the same kind of structure that Platonic solids have, being made of the same regular convex polyhedron for each face.
Remarkably, I learn from Mathworld, the shape is three-colorable. That is, suppose you wanted to assign colors to each of the corners in this four-dimensional shape. They’re all green circles here, but they don’t have to be. There are a lot of these corners, and they’re connected in complicated ways to one another. But you could color in every one of them, so that none if them is connected directly to another of the same color, using only three different colors.
This is the hyperdodecahedron, or 120-cell, one of the six four-dimensional analogs of the Platonic solids. It’s been shown on this blog before, but this image has one major change: a much slower rotational speed. It is my hope that this will help people, including myself, with the difficult task of understanding four-dimensional objects.
This image was created using Stella 4d, a program you can try, as a free trial download, at this website.)