The regular icosahedron hides many regular pentagons, surrounding each of its vertices. Cut it through the planes of two parallel pentagons, and you decompose it into two pyramids and a pentagonal antiprism. But what if you add a second antiprism, and glue the result back together?
The result is a 17-vertex triangulated planar graph called the Errera graph. Its main claim to fame is as one of the counterexamples to Kempe's false proof of the four color theorem, but it also describes the shape of clusters of gold atoms. Its dual graph (a planar graph with 12 pentagonal faces and 6 hexagonal faces) is one of the fullerenes, although too small to be a stable one.
This construction from an icosahedron shows that the Errera graph can be realized as a non-convex deltahedron, a polyhedron in which all faces are equilateral triangles. It can also be made convex, but at the expense of making its faces non-regular. You can also think of the same construction as forming the Errera graph from two icosahedra by slicing a vertex off each one and gluing them together on the resulting pentagonal faces.
I had somehow acquired the impression that the Errera graph got its name as an unusual spelling of "error", from the fact that it was used to point out Kempe's error. But that impression itself turns out to be erroneous. It's actually someone's name: Belgian mathematician Alfred Errera, who published it in his 1921 doctoral dissertation.
March 28 2017, 00:43:33 UTC
Hmm, I just learned a new word. So an "antiprism" is a solid in which sides and vertices lie opposite one another along the axis?
March 28 2017, 01:41:20 UTC
Yep. See https://en.wikipedia.org/wiki/Antiprism