In three-dimensional space, there are five Platonic and thirteen Archimedean polyhedra, plus numerous other shapes, in several categories. The whole collection can appear to be quite a confusing jumble — until, and unless, you start surveying four-dimensional polytopes, known as polychora.

There are six regular polychora, and they are analogous to the five Platonic solids. Each three-dimensional cell is regular, and all are identical, within a single one of these six. When peering beyond these six, however, things can get very confusing, very quickly.

The software I used to generate this image, Stella 4d, has a built-in library of polyhedra and polychora. You can examine it as a free trial download at http://www.software3d.com/stella.php. Today, motivated by curiosity, I went surveying, using this program, into the more complex polychora — beyond the six regular ones — which have different polyhedra as cells, looking for one I could (try to) understand, and which appealed to me aesthetically.

The one I settled on for this post is known as 165-Srix, as well as the small rhombated 600-cell, a/k/a the cantellated 600-cell. It has 600 cells which are cuboctahedra, shown here in yellow, 120 more which are icosidodecahedra, shown here in blue, and 720 cells which are regular pentagonal prisms.

I must admit this: I’m more than a little jealous of those who seem to be able to easily understand these four-dimensional shapes. I am definitely not one of them.

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