Tetrahedral Cluster of Dodecahedra

Four regular dodecahedra don’t really fit around a point like this, except in four dimensions.

This is an orthogonal projection from 4D to 3D of four dodecahedral cells from a 120-cell. In 4D mathematical space, each of the 600 vertices of the 120-cell are surrounded by four regular dodecahedra. Each of the four lies in a slightly different 3D hyperplane.

When we project to 3D, those dodecahedral cells are necessarily foreshortened. This projection is “vertex-first”, meaning that the central vertex is the center of the projection, so that the cluster of foreshortened cells is perfectly symmetric around the center, with the symmetry of a tetrahedron.