A Parallelogram-Expanded Snub Dodecahedron

The snub dodecahedron may be thought of as a dozen regular pentagons, surrounded and separated by a sea of triangles. In this expansion of that Archimedean solid, thirty parallelograms are added to the mix, also surrounded by triangles. In the image above, coloring is by face type — for example, the yellow triangles are those triangles which share an edge with a pentagon. Other triangles have other colors.

The image shown below is of the same polyhedron, but with a different coloring-scheme. In it, all triangles are given the same color, even when their shapes are slightly different.

This polyhedron has an interesting all-pentagon dual, which is shown below. This dual has sixty each of both the small and large pentagons, for a total of 120 faces.

I used Stella 4d: Polyhedron Navigator to create these polyhedra, and to make these rotating images. This program may be tried for free at http://www.software3d.com/Stella.php.

Modifying a Snub Dodecahedron

The polyhedron shown above is the snub dodecahedron, which is one of the Archimedean solids. The one shown below is its first stellation.

Next, each of the yellow faces was augmented with a tall prism.

The next step was to form the convex hull of the polyhedron above.

Finally, I used Stella 4d‘s “try to make faces regular” function on this convex hull. The result is below, and has 242 faces.

If you would like to try Stella 4d, the software I used to do all of this, simply visit http://www.software3d.com/Stella.php, and look for the free trial download.

An Expanded Snub Dodecahedron

To make this polyhedron, I started with a snub dodecahedron. Next, I augmented all triangular faces of it with prisms, then took the convex hull of the result. Finally, I used Stella 4d‘s “try to make faces regular” function on the convex hull.

An expanded snub dodecahedron

If you’d like to try the trial version of Stella for yourself, the website to visit is http://www.software3d.com/Stella.php.