This beautiful shape is one of my favorites, and I love its name: the rhombicosidodecahedron. It sounds as fancy as it looks! But I'd left it out of the story until now, since you can't get it by chopping corners off a Platonic solid. Still, it fits very nicely into the story, along with some other beauties. Now it's time to understand them! #4d
Hey, John, have you heard that now they have come up with something more than half again as hard as a diamond? Its called Wurtzite boron nitride!Dec 2, 2012
No, I hadn't heard about that! In addition to the the hexagonal crystals we commonly see, ice has a closely related cubic form whose crystal pattern matches that of diamond. Similarly, besides the cubic diamond there's a hexagonal form called lonsdaleite made by meteor impacts. It's impossible to find pure samples (for obvious reasons), but simulations of it show that a pure form would be harder than diamond!
http://en.wikipedia.org/wiki/Lonsdaleite
Wikipedia says 58% harder. So, someday we should have a contest to see who can scratch whom: pure lonsdaleite or Wurtzite boron nitride!Dec 2, 2012
I'm not sure how to express this (and I'm really not sure anything is there, really, to express) but the rhombicosidodecahedron and truncated icosahedron look a lot like the rhombitrihexagonal tiling and hexagonal tiling, respectively, just with 12 hexes replaced with pentagons, and that ends up curling them up off the plane.Dec 3, 2012- +Jerry Federspiel - there is definitely something precise to express here! In my series, and this post here, I explain Coxeter diagrams. When you understand this notation you can see the rhombicosidodecahedron is
•—5—o—3—•
while the rhombitrihexagonal tiling is
•—6—o—3—•
and so on - each member I listed of the 'dodecahedron/icosahedron family' has a corresponding planar tiling obtained by replacing the 5 in the Coxeter diagram by a 6. There's a lot of math lurking here, if one wants.Dec 3, 2012 
So back in 1977 after botching a quals question on the simplicity of A5, I took some old APL output I'd never managed to draw, describing the Cayley graph of A5. Lo and behold, it was the edges of this solid! I stayed up and made one, which I still have in my office.Dec 3, 2012- Sorry about that qual question! But your comment is very interesting. The fancier solid called the truncated icosidodecahedron, with 120 vertices, can be seen as the Cayley graph of the group of all rotation and reflection symmetries of the dodecahedron: A5 × Z/2. This is part of a more general pattern that I'll explain soon in this series. But I hadn't noticed that the rhombicosidodecahedron, with the 60 vertices, can be seen as the Cayley graph of the group of all rotation symmetries of the dodecahedron, namely A5. I suspect this is part of a general pattern too! I'll try to figure this out...Dec 3, 2012