Principal Shadows of the 12 Pentagonal Regular 4-Dimensional Objects (Polytopes)

One method of visualizing a 4-dimensional object is to construct its 3-diraensional shadow. This involves imagining a point source of light at infinite distance in the dimensional space in which the object is located, shining on the object and casting a shadow on a 3-dimensional hyperplane behind it. This shadow will have three dimensions. If the object is a polytope, the shadow is a polyhedron.

There are 16 polytopes in four dimensions that are regular, that is, the cells that bound them are all regular polyhedra, all of the same type, with the same configuration at each vertex. Of the 16 regular polytopes, 12 are pentagonal: They all have the same symmetry group, denoted by [3,3,5], with order 14400.

Because they are so complicated, these pentagonal forms are especially difficult to visualize. Of the available techniques of visualization, the one which best displays their basic structure is probably the shadow method.

This paper shows the principal shadows of the 12 pentagonal polytopes, which are the shadows with the highest possible degree of symmetry. Interesting features of these shadows and the ways that they are related to each other are presented.