compound of five cubes | RobertLovesPi.net

The most common depiction of the compound of five cubes uses solid cubes, each of a different color:

This isn’t the only way to display this compound, though. If the faces of the cubes are hidden, then the interior structure of the compound can be seen. An edges-only depiction, still keeping a separate color for each cube, looks like this:

If these thin edges are then thickened into cylinders, that makes a third way to depict this polyhedral compound. It creates a minor problem, though: edges-as-cylinders looks awful without vertices shown as well, and the best way I have found to depict vertices, in this situation, is with spheres. With vertices shown as spheres, however, a sixth color, only for the vertex-spheres, is needed. Why? Because each vertex is shared by six edges: three from a cube of one color, and three from a second cube, of a different color.

Finally, here are all three versions, side-by-side for comparison, and with the motion stopped.

All images in this post were created using Stella 4d: Polyhedron Navigator, software you may try for free at this website.

An Examination of the Compound of Five Cubes

To make the compound of five cubes, begin with a dodecahedron, as seen above. Next, add segments as new edges, and let them be all of the diagonals of all the dodecahedron’s faces. Then, remove the pentagonal faces, as well as the original edges. What’s left is five cubes, in this arrangement.

Using polyhedral manipulation software called Stella 4d (available at www.software3d.com/Stella.php), these five cubes can be removed one at a time. The first removal has this result:

That left four cubes, so the next removal leaves three:

And then only two:

And, finally, only one remains:

Because their edges were pentagon-diagonals for the original dodecahedron, each of these cubes has an edge length equal to the Golden Ratio, (1 + √5)/2, times the edge length of that dodecahedron.