# Talk:Euler Characteristic Exploration

This seems to be a bit redundant with Platonic Solids Exploration, although it's more directed. It would be nice if this one went faster and further, maybe using the Platonic Solids Exploration as a starting point but then moving into more interesting polyhedra, and possibly doing some disk tessellations and/or the torus.

Any thoughts? I'm kind of assuming this exploration is being used regularly. If not, I'd like to reclaim the title for something with more depth.

So, actually, I stuck a bit of stuff at the end under the title 'Going Further'. But I cover Platonic Solids Exploration, so I'll actually only use the 'going futher' stuff. But this probably isn't the right solution. What if we renamed the original one 'Euler Characteristic of Platonic Solids Exploration'? Bryan 10:06, 14 November 2011 (CST)

No comments in two years, so I'm going to split off Euler Characteristic of Platonic Solids Exploration. Bryan (talk) 10:07, 11 November 2013 (CST)

## Solutions

1. Rhombic dodecahedron: Faces are 120-90-120-90 with defect 60, so 60/720=1/12, so F=12, E=24, V=14.
2. Cuboctahedron: F=14, E=24, V=12 (it's dual to the rhombic dodecahedron)
3. Concentric Rinds: Faces are 45-60-90 with defect 15, so 15/720 = 48, so F=48, E=72, V=26.
4. Deltoidal icositetrahedron: Faces are 90-90-90-120 with defect 30, so 30/720 = 1/24, so F=24, E=48, V=26
5. Square lattice: F=12, E=31, V=20, $\chi = 1$.
6. Fullerene molecule: F=14, E=39, V=26, $\chi = 1$.
7. Torus: F=60, E=120, V=60, $\chi = 0$.