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.
Sometimes, when using Stella 4d (available here) to make various polyhedra, I lose track of how I got from wherever I started to the final step. That happened with this fractured version of an octahedron.
The mandalas on the eight hexagonal faces of this truncated octahedron were first seen in the blog-post immediately before this one. The six square faces have been hidden, creating holes in their places.
I used Geometer’s Sketchpad and MS-Paint to make this pattern, based on a ring of twenty-two hendecagons. The work of projecting this image onto a polyhedron, as well as making this rotating .gif image, was performed using another program, Stella 4d: Polyhedron Navigator. If you’d like to try Stella for yourself, there’s a free trial download available at http://www.software3d.com/Stella.php.
The blue polygons are regular pentadecagons, and the yellow polygons are irregular dodecagons. There are also equilateral hexagons (orange), red squares, and black concave pentagons.