I made this solid from the truncated octahedron using Stella 4d‘s “morph duals by truncation” function. It has, as faces, six squares, eight regular hexagons, and 24 isosceles triangles. If you’d like to give Stella a try yourself, for free, the site to visit is http://www.software3d.com/Stella.php.

To make the polyhedron shown above, I started with the great icosahedron, then applied the “morph duals by truncation” function in Stella 4d (a computer program available, with a free trial download, here). The faces of this polyhedron are twenty equilateral triangles, in red, and twelve regular star pentagons, colored yellow. [Later edit: the solid above is called the great icosidodecahedron, and is one of the uniform solids, although I did not realize this immediately.]

Next, I started with the polyhedron above, and applied the same dual-morph operation to it. The resulting polyhedron (shown below) has twenty equilateral triangles and twelve star pentagons (like its “parent”), plus thirty rectangles, which are shown in blue. Here’s the result, with 62 faces, total.

This was made using the “morph duals by expansion” function on a small stellated dodecahedron. I did this using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

I used Stella 4d to make this polyhedron. You can try this program for yourself, for free, at http://www.software3d.com/Stella.php.

This polyhedron’s faces include twelve regular pentagons, twenty regular hexagons, and sixty elongated convex hexagons. I made it using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.