To make a faceted polyhedron, vertices of the original polyhedron are connected in new ways, to create a different set of faces and edges. To make this particular faceting of the truncated icosahedron, I first connected all vertices in the configuration shown below, to make irregular decagonal faces in the interior of the solid.

In this next pic, the decagonal faces formed above are shown in red, and a new set of pentagonal faces is being created.

For a polyhedron to be considered mathematically valid, faces must meet in pairs at each edge. To accomplish that, I had to create another set of pentagonal faces, this one smaller than the last, as shown below.

Here’s the completed polyhedron, with each face-type having its own color.

This next image is of the same polyhedron, but with a different coloring-scheme. In this second version, each face has its own color — except for faces which are parallel, with those faces given the fame color.

I created these images using *Stella 4d: Polyhedron Navigator*, a program which is available here.

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