# Repeated Augmentations of a Dodecahedron With More and More Dodecahedra

Here’s a single dodecahedron.

A new “cluster polyhedron” can be made by augmenting each pentagonal face with another dodecahedron.

If you can do it once, you can do it again, augmenting each pentagon with a new dodecahedron.

Once more.

I made these polyhedral clusters using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

# A Rhombic Triacontahedron, Vertices Surrounded By Smaller Rhombic Triacontahedra, and Its Interesting Dual

The first image shows a central yellow rhombic triacontahedron, with smaller, blue rhombic triacontahedra attached to each of its thirty-two vertices. The second polyhedron shown is the dual of the first one, with colors chosen by the number of sides per face in the second image — pentagons red, and triangles yellow. The convex hull of this second polyhedral complex shown would be an icosidodecahedron, itself the dual of the rhombic triacontahedron.

I use software called Stella 4d: Polyhedron Navigator to make the rotating polyhedral images on this blog. You can try Stella for yourself, for free, at http://www.software3d.com/Stella.php.

# Spectral Icosahedra

Stella 4d: Polyhedron Navigator has a “put models on vertices” function which I used to build this cluster of 61 icosahedra. If you’d like to try this software for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.

# Spectral Dodecahedra

Stella 4d: Polyhedron Navigator has a “put models on vertices” function which I used to build this cluster of 101 dodecahedra. If you’d like to try this software for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.

# A Dozen Truncated Icosahedra, Surrounding a Central Dodecahedron

Created using Stella 4d, which you can try for yourself, for free, at http://www.software3d.com/Stella.

# Cluster of 33 Icosidodecahedra

There’s one icosidodecahedron at the center of this cluster, with more icosidodecahedra attached to each of the central figure’s 32 faces. In the first version, the coloring is simply based as the number of sides for each face.

In the next picture, the coloring is by face-type (position in the overall cluster).

The last image shown here has the cluster in “rainbow color mode.”

I used Stella 4d to make these — a program you may try for free right here.

# An Icosahedron Augmented with Twenty Great Icosahedra, together with the Dual of this Cluster-Polyhedron

The cluster-polyhedron above was formed by augmenting a central isocahedron with twenty great icosahedra. The dual of this cluster is shown below.

Both these images were created using Stella 4d, which you may try for free at http://www.software3d.com/Stella.php.

# Augmenting, and Then Reaugmenting, the Icosahedron, with Icosahedra

A reader of this blog, in a comment on the last post here, asked what would happen if each face of an icosahedron were augmented by another icosahedron. I was also asked what the convex hull of such an icosahedron-cluster would be. Here are pictures which answer both questions, in order.

While the icosahedron augmented by twenty icosahedron forms an unusual non-convex shape, its convex hull is simply a slightly “stretched” version of the truncated dodecahedron, one of the Archimedean solids.

The reader who asked these questions did not ask what would happen if the icosahedron-cluster above were to be augmented, on every face, by yet more icosahedra. However, I got curious about this, myself, and created the answer: the following cluster of even-more numerous icosahedra. This could be called, I suppose, the “reaugmented” icosahedron.

Finally, here is the convex hull of this even-larger cluster. No one asked for it; I simply got curious.

To accomplish the polyhedron-manipulation and image-creation for this post, I used a program called Stella 4d: Polyhedron Navigator, which is available at http://www.software3d.com/Stella.php. A free trial download is available there, so you can try the software before deciding whether or not to purchase it.

# Cluster of Six Rhombicosidodecahedra

To make this cluster, start with a cuboctahedron, then augment each of its square faces with rhombicosidodecahedra. Although the cuboctahedron has cuboctahedral symmetry, this cluster does not — rather, it has tetrahedral symmetry. I created this using Stella 4d, which is available here.