What do you get if you glue 120 buckyballs together in 4D and fill in the gaps with truncated tetrahedra?
You get a uniform polytope with 120 buckyballs and 600 truncated tetrahedra, called the Truncated-icosahedral hexacosihecatonicosachoron. This cool polychoron has 1200 triangular ridges, 720 pentagons, and a whopping 2400 hexagons. It contains 7200 edges. :-)
OK, OK, so all this is already known on George Olshevsky's uniform polytope page, but it's still cool 'cos I independently discovered this just earlier today. I was playing around with truncating the regular polychora, and found some interesting uniform polychora. Here's a list of the ones I found:
* The rectified 5-cell consists of 5 tetrahedra and 5 octahedra.
* The bitruncate (or mesotruncate) of the 5-cell is an interesting polychoron with 10 truncated tetrahedral cells. This is interesting 'cos all the cells are identical, and the hexagonal faces of the truncated tetrahedra are joined in complementary orientation, so that every edge is only touching one triangle.
* The rectified 8-cell consists of 16 tetrahedra and 8 cuboctahedra.
* The bitruncate (or mesotruncate) of the 8-cell is the same as the bitruncate of the 16-cell, and consists of 16 truncated tetrahedra and 8 truncated octahedra. Like the bitruncated 5-cell, the hexagonal faces of the cells are joined in complementary orientation.
* The rectified 16-cell is, of course, the 24-cell; and the rectified 24-cell consists of 24 cubes and 24 cuboctahedra.
* What do you get if you take 48 cubes, cut off their corners, and fold them up into 4D? You get the bitruncate (mesotruncate) of the 24-cell, which is made of 48 truncated cubes. Like the other mesotruncates, the octagonal faces of the 48 cells are joined in complementary orientation so that only one triangular face is attached to each edge. Again, I found this particularly interesting 'cos all 48 cells are identical.
* The coolest part is with the 120-cell and the 600-cell. Rectifying the 120-cell yields a polychoron with 600 tetrahedra and 120 icosidodecahedra; and rectifying the 600-cell gives a polychoron with 600 octahedra and 120 icosahedra. If you make a deeper cut, you get the bitruncate, made of 120 buckyballs and 600 truncated tetrahedra.