The cluster above was formed by starting with one rhombicosidodecahedron, and then augmenting each of its thirty square faces with another rhombicosidodecahedron. If you examine the single rhombicosidodecahedron below, though, you’ll see that this can be done in two different ways, each of which produces a cluster with the same degree of symmetry as the original. (If less symmetry is acceptable, there are far more than two different ways to obtain such clusters, but that does not interest me.)

When new rhombicosidodecahedra are attached to the square faces of this central rhombicosidodecahedron, they new ones can be oriented such that pentagons are placed above pentagons, and triangles above triangles — or the new ones can be oriented the other way, so that pentagons are placed above triangles, and triangles above pentagons. One of these possibilities produces the cluster at the top of this post. The other possibility produces the cluster shown below.

All of these polyhedral manipulations were performed using a program called *Stella 4d: Polyhedron Navigator*, which may be purchased (or tried for free, as a trial download) at http://www.software3d.com/Stella.php.

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