I came across this little beauty while exploring stellations of the triakis octahedron, which is the dual of the truncated cube. Its three components are each eight-faced trapezohedra, and it showed up as the sixth in that stellation-series.
Stella 4d: Polyhedron Navigator was used to make this rotating image. You may try it for free right here.
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:
Both images here were made using Stella 4d, which you can try for free right here.
I made these using Stella 4d, which you can try for free here.
In the first version of this compound shown here, the great stellated dodecahedron is shown in yellow, while the small stellated dodecahedron is shown in red.
In the next version, each face has its own color, except for those in parallel planes, which have the same color.
Finally, the third version is shown in “rainbow color mode.”
All three of these images were created using Stella 4d: Polyhedron Navigator, software you can try for free right here.
This is the truncated cube, which is one of the Archimedean solids.
To make a faceted version of this solid, one must connect at least some of the vertices in different ways. Doing that creates new faces.
This faceted version of the truncated cube includes eight blue equilateral triangles, eight larger, yellow equilateral triangles, and eight irregular, red hexagons. It’s easy to spot the yellow and blue triangles, but seeing the red hexagons is harder. In the final picture here, I have hidden all faces except for three of the hexagons, so that their positions can be more easily seen.
I made all three of these images using Robert Webb’s program called Stella 4d: Polyhedron Navigator. It is available for purchase, or as a free trial download, at http://www.software3d.com/Stella.php.
This is an expansion of the last post here. It may be possible to continue this tiling outward indefnitely, forming an aperioidic tiling — or it may not. I am simply uncertain about this