In addition to the sixty-two regular faces, this polyhedron also has two sets of sixty isosceles trapezoids each, shown in different shades of blue. That’s 182 faces in all. I made it using Stella 4d, which you can try for free at www.software3d.com/Stella.php.
I previously blogged a different version of this symmetrohedron, one which sacrifices octagon regularity for regularity of all the other faces. Both polyhedra were created using Stella 4d: Polyhedron Navigator, which you can try for free right here. In this version, the twenty red hexagons are equiangular, and the sixty green faces are isosceles trapezoids.
In this symmetrohedron, all faces are regular, except for the green isosceles triangles. If these triangles were a little closer to being regular, this would be a near-miss to the Johnson solids, but that is not the case. I made this (starting with the last polyhedron in the post right before this one) using Stella 4d: Polyhedron Navigator, a program you can try for free at http://www.software3d.com/Stella.php.
Symmetrohedra are polyhedra which have some form of polyhedral symmetry, as well as having regular polygons for many (not necessarily all) of their faces. I found these two symmetrohedra while playing around with Stella 4d: Polyhedron Navigator — a program you can try for free at this website. Each of these symmetrohedra have 74 faces, with twelve of them being regular decagons.
The only irregular faces in this polyhedron are the quadrilaterals (kites and rectangles). I made it using Stella 4d, which you can try for yourself — for free — at http://www.software3d.com/Stella.php.
This zonohedron is based on the icosidodecahedron / rhombic triacontahedron compound — more specifically, on its edges. Twelve faces are regular decagons, twenty are regular hexagons, sixty are squares, and the only irregular faces are the thirty equilateral octagons. That’s 122 faces in all.
I made this using Stella 4d: Polyhedron Navigator. You may try this program for yourself, for free, at http://www.software3d.com/Stella.php.