I’m going to start this experiment with a single octahedron, with faces in two colors, placed so that two faces which share an edge are always of different colors.
Next, I will augment the red faces — and only the red faces — with identical octahedra.
The regions with four blue, adjacent faces look as though they might hold icosahedra — but I checked, and they don’t quite fit. I will therefore continue the same process — augmenting only the red faces with more octahedra of the original type.
I’ve now decided that I definitely like this game, so I’ll keep playing it.
Immediately above, at the fourth of these images, some of the octahedra have started to overlap slightly, but I’m choosing to not be bothered by that — I’m continuing the now-established pattern, just in order to see where it takes me.
The regions of overlap are now far more obvious, but I’m continuing, anyway. Why? Because this is fun, that’s why! Right now, Stella 4d, the program I use to do these polyhedral manipulations, is chugging away on the next one. (This program is avilable at http://www.software3.com/Stella.php.) Ah, it’s ready — here it is!
Rather than repeat this process again, I now have another question: what would the convex hull of this figure look like? (A convex hull of a non-convex polyhedron is the smallest convex polyhedron which can contain a given non-convex polyhedron.) With Stella 4d, that’s easily answered.
I must admit this: that was nothing like what I expected — but such unexpected discoveries are a large part of what makes these polyhedral investigations with Stella 4d so much fun. And now, to close this particular polyhedral journey, I will have Stella 4d produce, for me, the dual of the convex hull shown above. (In case you aren’t familiar with duality regarding polyhedra, it describes the relationship between the octahedron, with which this post began, and the familar cube. Basically, with duals, faces and verticies are “flipped” over edges, although that is an extremely informal and imprecise way to describe the at the process.)
And with that, my friends, I bid you good night!