It’s related to the icosahedron obviously and it’s symmetrical, but it’s not a semi-regular polyhedron since neither the triangles nor the hexagons are regular. I think it’s what you get when you take a truncated icosahedron (soccerball pattern) and then truncate it again. It may not be obvious from the picture, but it’s edges form ten great circles each dividing a sphere in half, and so it could be considered a family with the octahedron, cuboctahedron, and icosidodecahedron, which do the same thing with three, four, and six great circles. In fact they could be considered corresponding to the four Platonic solids that have three, four, six and ten opposing vertices and faces.
Anyway, as this long-winded explanation shows, it would really help if I could knew this beast’s name.
It’s a tricky one; I agree that it might be a twice-truncated icosahedron, but I’m having a little trouble visualising it from the picture provided. Afraid I’ve got to run off so can’t chase this down more, but there’s a rather magnificent Encyclopedia Polyhedra that might be of use to you; it has a whole bunch of VRML 3D models to view if you’ve got the right browser plugin. Hope this helps…
Searching that site found the image used near the bottom of this page. The caption there is “Sphere intersecting the infinite cells of the stellated icosahedron”.
Wonderful site DB! The closest I found was quasi-regular polyhedra, which my figure doesn’t quite meet the technical defintion of, but shares the property of the edges forming great circles. It’s similar in appearance to the dodecadodecahedron, except it’s convex.
According to the Balcones Forge newsletter from April 2008 (warning: PDF), it’s an Icosidodecahedron. We made one out of 5 strips of steel. There are some pictures if you care to download it.
But seriously, looking at the OP’s example, it looks like a spherical dodecadodecahedron, which legend holds can only be destroyed in the very fires that forged it.
