thanks for raising questions about the humbubbahedron. B gave me some info about recent work in filling space with tetrahedrons. R. Buckminster Fuller did some original work in space-filling solids, and space-enclosing solids. The humbubbahedron is closely related to the latter, not so much the former. I have no idea how humbubbahedrons pack, or how tetrahedrons fit into them. It's more about enclosing a space with a bunch of sticks all of the same length, or with identical triangles.

Bucky is perhaps most famous for the geodesic, which is the shape of the classic soccer ball with 12 black pentagonal panels and 20 white hexagonal panels. The humbubbahedron is perhaps the anti-geodesic. A geodesic is derived from an icosahedron with 20 regular triangle faces, by clipping the points. Five triangles meet at each point of an icosahedron, so when you truncate it you create pentagons at the points, and the triangles become hexagons.

The humbubbahedron is produced by the opposite process. You generate pyramids on each face of a dodecahedron. I think if you do the same thing to the pentagons of a geodesic you get your icosahedron back. By using regular triangles to "cumulate" a dodecahedron, you get a figure with 60 triangle faces all the same size and shape. Cumulating a polyhedron can be in or out, i.e. concave or convex. The humbubbahedron is therefor an externally (convexly) regularly cumulated dodecahedron. As with the geodesic, I think it's interesting enough for a name of its own, hence "humbubbahedron".

A humbubbahedron wouldn't make a good soccer ball. It's pointy. All the points of a geodesic are in the same sphere. A humbubbahedron has joints in two concentric spheres. Both figures only have a single edge length. The humbubbahedron however, is all equilateral triangles. At 60 faces, it's a lot bigger than an icosahedron, with 20. I suspect the humbubbahedron is extremely strong, being all triangles, and being somewhat "corrugated", so to speak.

The model I gave Rep. C. is the second one I've made, and the first one precisely made enough to show the subtle symmetry of the hills and valleys of the humbubbahedron. Making it was an extension of playing around with a drafting compass, as most of us have often done, making hexagonal clusters of circles. One pyramid of a humbubbahedron is a hexagon with one triangular segment missing. 12 of those can be arranged in a single figure to fold up into a humbubbahedron. The one you've seen spanned about 2/3 of a standard CVS posterboard.

DC has a famous pentagon, an oval, a well known ellipse. I think we're ready for a humbubbahedron.

Of course, the shape has been discussed before. There's a picture of a humbubbahedron, called a 60 face star deltahedron, here. That's what it SHOULD look like.