Created with Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.
I used three programs to make this: Stella 4d, Geometer’s Sketchpad, and MS-Paint. You can try Stella for free at http://www.software3d.com/Stella.php.
The 18th stellation of the rhombicosidodecahedron, shown above, is also an interesting compound. The yellow component of this compound is the rhombic triacontahedron, and the blue-and-red component is a “stretched” form of the truncated icosahedron.
This was made using Stella 4d, which you can try for free right here.
These two polyhedra are the icosidodecahedron (left), and its dual, the rhombic triacontahedron (right).
One nice thing about these two polyhedra is that one of them, the rhombic triacontahedron, can be used repeatedly, as a building-block, to build the other one, the icosidodecahedron. To get this started, I first constructed one edge of the icosidodecahedron, simply by lining up four rhombic triacontahedra.
Three of these lines of rhombic triacontahedra make one of the icosidodecahedron’s triangular faces.
Next, a pentagon is attached to this triangle.
Next, the pentagonal ring is surrounded by triangles.
More triangles and pentagons bring this process to the half-way point. If we were building a pentagonal rotunda (one of the Johnson solids), this would be the finished product.
Adding the other half completes the icosidodecahedron.
All of these images were created using Stella 4d: Polyhedron Navigator. You may try this program yourself, for free, at http://www.software3d.com/Stella.php. The last thing I did with Stella, for this post, was to put the finished model into rainbow color mode.
I made this using Stella 4d: Polyhedron Navigator. You may try this software, for free, at http://www.software3d.com/Stella.php.
The blue figure below is the rhombic triacontahedron. It has thirty identical faces, and is one of the Catalan solids, also known as Archimedean duals. This particular Catalan solid’s dual is the icosidodecahedron.
I use a program called Stella 4d (available here) to transform polyhedra, and the next step here was to augment each face of this polyhedron with a prism, keeping all edge lengths the same.
After that, I created the convex hull of this prism-augmented rhombic triacontahedron, which is the smallest convex figure which can enclose a given polyhedron.
Another ability of Stella is the “try to make faces regular” function. Throwing this function at this four-color polyhedron above produced the altered version below, in which edge lengths are brought as close together as possible. It isn’t possible to do this perfectly, though, and that is most easily seen in the yellow faces. While close to being squares, they are actually trapedoids.
For the next transformation, I looked at the dual of this polyhedron. If I had to name it, I would call it the trikaipentakis icosidodecahedron. It has two face types: sixty of the larger kites, and sixty of the smaller ones, also.
Next, I used prisms, again, to augment each face. The height used for these prisms is the length of the edges where orange kites meet purple kites.
Lastly, I made the convex hull of the polyhedron above. This convex hull appears below.
The simplest way I can explain faceting is that it takes a familiar polyhedron’s vertices, and then connects them in unusual ways, so that you obtain different edges and faces. If you take the convex hull of a faceted polyhedron, it returns you to the original polyhedron.
This was created using Stella 4d, software available (including as a free trial download) right here: http://www.software3d.com/Stella.php.