# 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.

# Augmenting the Dodecahedron with Great Dodecahedra

These two polyhedra are the dodecahedron (left), and the great dodecahedron (right).

Since the faces of both of these polyhedra are regular pentagons, it is possible to augment each of the dodecahedron’s twelve faces with a great dodecahedron. Here is the result. I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

# Augmenting the Icosahedron with Great Icosahedra

These two polyhedra are the icosahedron (left), and the great icosahedron (right).

Since the faces of both of these polyhedra are equilateral triangles, it is possible to augment each of the icosahedron’s twenty faces with a great icosahedron. Here is the result. I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

# Augmenting the Octahedron with Octahedra, Repeatedly.

This is an octahedron. If you augment each face of an octahedron with more octahedra, you end up with this. One can then augment each triangular face of this with yet more octahedra. Here’s the next iteration: This could, of course, go on forever, but one more step in the series is all you will see here. I don’t want to get caught in an infinite loop. Performing various manipulations of polyhedra is easy with Stella 4d: Polyhedron Navigator, which I used to make all five of these rotating images. If you’d like to try this program for yourself, just check out http://www.software3d.com/Stella.php.

# The Compound of Five Cubes, Augmented with Thirty Snub Cubes: Three Versions This cluster-polyhedron was made with Stella 4d, software you can try at this website. Above, it is colored by face-type, referring to each face’s position within the overall cluster. In the image below, the original compound of five cubes contained one cube each, of five colors, and then each snub cube “inherited” its color from the cube to which it was attached. In the next version, the colors are chosen by the number of sides of each face. # A Great Icosahedron, Augmented with Twenty Icosahedra The polyhedral clusters above and below use different coloring-schemes, but are otherwise identical. Invisible, in the center, is a great icosahedron. Each of its faces has been augmented by a (Platonic) icosahedron. Both images were created using Stella 4d, software you can try here.

# The Greatly Augmented Rhombicosidodecahedron I call this variant on the rhombicosidocahedron “greatly augmented” because it was formed by augmenting each pentagonal face of a central rhombicosidodecahedron with great dodecahedra, while each triangular face is augmented with great icosahedra. It was made using Stella 4d, which may be found here.

# Icosidodecahedra, Icosahedra, and Dodecahedra

If one starts with a single icosidodecahedron, and then augments its pentagonal faces with dodecahedra, and its trianguar faces with icosahedra, this is the result. This figure has gaps in it where two pentagons and two triangles meet around a vertex. If one puts icosidodecahedra in those gaps, this is the resulting figure. Next, once again, the pentagonal faces are augmented with dodecahedra, and the triangular faces with icosahedra. These virtual polyhedral models were all built using Stella 4d, available at http://www.software3d.com/Stella.php.

# An Icosahedron, Augmented with Twenty Triangular Cupolae, Together with Its Dual After making the above polyhedron using Stella 4d (a program you can try for free at www.software3d.com/Stella.php), I checked its dual, which is shown below. I was surprised at its appearance, for it resembles a stellated polyhedron, even though it was created by a completely different process. 