I made this using Stella 4d, which you can try for free here.
Here’s a truncated icosahedron, one of the thirteen Archimedean solids.
The next image shows this solid with its hexagonal faces augmented by prisms.
This augmented polyhedron has an interesting dual:
Finally, here’s this dual shown in “rainbow color mode.”
These images were created with Stella 4d, a program you can try for free right here.
I made this using Stella 4d: Polyhedron Navigator, a program you can try for free at this website. It shows Hexagon the Cat riding in circles on the twenty hexagonal faces of a rotating truncated icosahedron. We don’t know of a cat named Pentagon, so I hid the twelve pentagonal faces.
This particular truncated icosahedron has an edge length of one. I may build one with a longer edge length at some point; this would have the effect of shrinking the white edges, and magnifying the orange and blue faces, as fractions of the overall model. The individual Lux square pieces are identical, except for their color.
If you’d like to try Lux Blox for yourself, the site to visit is http://www.luxblox.com.
Here’s my starting point: the truncated icosahedron, one of the thirteen Archimedean solids.
Next, each face is augmented by a prism, with squares used for the prisms’ lateral faces.
The convex hull of the polyhedron above yields what can be called an expanded truncated icosahedron, as shown below:
Could these faces be made regular, and the polyhedron still hold together? I checked, using Stella 4d‘s “try to make faces regular” function. Here’s the result:
As you can see, the faces of this polyhedron can be made to be regular, but this forces the model to become non-convex.
To try Stella for yourself, for free, just pay a visit to http://www.software3d.com/Stella.php. The trial version is a free download.
The components of this toroid are sixty rhombic triacontahedra, as well as ninety rhombic prisms with lateral edges three times as long as their base edges. I made this using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.
I made this using Stella 4d: Polyhedron Navigator, a program you can try for free at http://www.software3d.com/Stella.php.
The yellow-and-red polyhedron in the compound below is the truncated icosahedron, one of the Archimedean solids. The blue figure is its dual, the pentakis dodecahedron, which is one of the Catalan solids.
The next image shows the convex hull of this base/dual compound. Its faces are kites and rhombi.
Shown next is the dual of this convex hull, which features regular hexagons, regular pentagons, and isosceles triangles.
Next, here is the compound of the last two polyhedra shown.
Continuing this process, here is the convex hull of the compound shown immediately above.
This latest convex hull has an interesting dual, which is shown below. It blends characteristics of several Archimedean solids, including the rhombicosidodecahedron, the truncated icosahedron, and the great rhombicosidodecahedron.
This process could be continued indefinitely — making a compound of the last two polyhedra shown, then forming its convex hull, then creating that convex hull’s dual, and so on.
All these polyhedra were made using Stella 4d: Polyhedron Navigator, which you can purchase (or try for free) at http://www.software3d.com/Stella.php.
The polyhedron above is the truncated icosahedron, widely known as the pattern for most soccer balls. In the image below, the faces and edges have been hidden, leaving only the vertices.
To make a faceted version of this polyhedron, these vertices must be connected in novel ways, creating new edges and faces. There are many faceted versions of this polyhedron, of which seven are shown below.
I used Stella 4d to make these polyhedral images, and you’re invited to try the program for yourself at http://www.software3d.com/Stella.php.
This is one of many possible facetings of the truncated icosahedron. I made it using Stella 4d, which you can try for yourself at this website: http://www.software3d.com/Stella.php.