# A Faceted Truncated Octahedron with Twenty Faces

The twenty faces of this polyhedron are six small blue squares, six interpenetrating, larger red squares, and eight irregular, interpenetrating yellow hexagons. I made it using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php. The squares are easy to find, but it can be a challenge to see the yellow hexagons. In the .gif below, all of the yellow faces but one are hidden, which should make it easier to see where the hexagons are positioned, relative to the squares.

# Space-Filling Truncated Octahedra in a Rhombic Dodecahedral Cluster

The truncated octahedron is well-known as the only Archimedean solid which can fill space, by itself, without leaving any gaps. The cluster below shows this, and has the overall shape of a rhombic dodecahedron.

It’s easier to see the rhombic dodecahedral shape of this cluster when looking at its convex hull:

I stumbled upon this zonohedron by adding zones to a truncated octahedron, based on its faces, edges, and vertices. It was created using Stella 4d, which you may try for free at http://www.software3d.com/Stella.php. To the best of my recollection, this is the only zonohedron I have seen which includes rhombi, hexagons, octagons, and, of course, the red hexadecagons.

# A Twisted Expansion of the Truncated Octahedron

I made this variant of the truncated octahedron using Stella 4d: Polyhedron Navigator. You may try this program for free at http://www.software3d.com/Stella.php.

# A Space-Filling Lattice of Truncated Octahedra

Truncated octahedra are among the special polyhedra which can fill space without leaving any gaps. There are others, as well. This image was created using Stella 4d, software you may try, for yourself, right here. There is a free “try it before you buy it” download available.

# Bowtie Cubes in a Polyhedral Honeycomb

This polyhedron has been described here as a “bowtie cube.” It is possible to augment its six dodecagonal faces with additional bowtie cubes. Also, the bowtie cube’s hexagonal faces may be augmented by truncated octahedra.

These two polyhedra “tessellate” space, together which square pyramidal bifrustrums, meeting in pairs, which fill the blue-and-green “holes” seen above. This last image shows more of the “honeycomb” produced after yet more of these same polyhedra have been added.

This pattern may be expanded into space without limit. I discovered it while playing with Stella 4d, software you may try for free at this website.

# The Compound of the Truncated Octahedron and the Rhombic Dodecahedron

I created this compound using Stella 4d, software you may try at this website.

# Two Compounds with Pyritohedral Symmetry: the Icosidodecahedron / Truncated Octahedron Compound, and the Rhombic Triacontahedron / Tetrakis Cube Compound

Stella 4d, a program you can try here, was used to create these two compounds. Both have pyritohedral symmetry: the symmetry of a standard volleyball. The two compounds are also duals.

# Two Polyhedral Compounds: the Dodecahedron / Truncated Octahedron, and Its Dual, the Icosahedron / Tetrakis Cube

That’s the compound of the dodecahedron and the truncated octahedron above. Shown next is its dual, the compound of the icosahedron and the tetrakis cube. Both compounds were made using Stella 4d: Polyhedron Navigator, which you may try here.