The Compound of the Truncated Isocahedron and the Pentakis Dodecahedron, with Related Polyhedra

The yellow-and-red polyhedron in the compound below is the truncated icosahedron, one of the Archimedean solids. The blue figure is its dual, the pentakis dodecahedron, which is one of the Catalan solids.

Pentakis dodecahedron and truncated icosahedron

The next image shows the convex hull of this base/dual compound. Its faces are kites and rhombi.

Convex hull of trunctaed icosahedron slash pentakis dodecahedron compound

Shown next is the dual of this convex hull, which features regular hexagons, regular pentagons, and isosceles triangles.

dual of Convex hull of trunctaed icosahedron slash pentakis dodecahedron compound

Next, here is the compound of the last two polyhedra shown.

dual and base compound of Convex hull of trunctaed icosahedron slash pentakis dodecahedron compound

Continuing this process, here is the convex hull of the compound shown immediately above.

Convex hull

This latest convex hull has an interesting dual, which is shown below. It blends characteristics of several Archimedean solids, including the rhombicosidodecahedron, the truncated icosahedron, and the great rhombicosidodecahedron.

Dual of Convex hull

This process could be continued indefinitely — making a compound of the last two polyhedra shown, then forming its convex hull, then creating that convex hull’s dual, and so on.

All these polyhedra were made using Stella 4d: Polyhedron Navigator, which you can purchase (or try for free) at

Seven Different Facetings of the Truncated Icosahedron

Trunc Icosa.gif

The polyhedron above is the truncated icosahedron, widely known as the pattern for most soccer balls. In the image below, the faces and edges have been hidden, leaving only the vertices.

Trunc Icosa vertices only

To make a faceted version of this polyhedron, these vertices must be connected in novel ways, creating new edges and faces. There are many faceted versions of this polyhedron, of which seven are shown below.

Faceted Trunc Icosa

Faceted Trunc Icosa 8

Faceted Trunc Icosa 7

Faceted Trunc Icosa 5.gif

Faceted Trunc Icosa 4.gif

Faceted Trunc Icosa 3

Faceted Trunc Icosa 2.gif

I used Stella 4d to make these polyhedral images, and you’re invited to try the program for yourself at

A Faceted Truncated Icosahedron

Faceted truncated icosahedron

This is one of many possible facetings of the truncated icosahedron. I made it using Stella 4d, which you can try for yourself at this website:

A Truncated Icosahedron, Formed By Silver Pipes, and Gold Fastenings

Trunc Icosa gold and silver

I made this precious-metal version of the truncated icosahedron using Stella 4d, a program which is available here:

A Cluster of 33 Truncated Icosahedra

Augmented Trunc Icosa

There is one truncated icosahedron at the center of this cluster, and each of its 32 faces is augmented with another truncated icosahedron, for a total of 33. I built this cluster using Stella 4d, software available here.

A Pyritohedral Coloring-Scheme for the Truncated Icosahedron

pyritohedral coloring of the truncated icosahedron

While the polyhedron above, informally known as the “soccer ball,” has icosidodecahedral symmetry, its coloring-scheme does not. Instead, I colored the faces in such a way that the coloring-scheme has pyritohedral symmetry — the symmetry of a standard volleyball. This rotating image was made with Stella 4d, a program you can buy, or try for free, right here: