robinhouston on Nostr: That’s interesting, because self-duality is almost a sort of symmetry. Let me ...

That’s interesting, because self-duality is almost a sort of symmetry.

Let me explain. Given a polyhedron P, which for these purposes we can think of as a graph of vertices and edges drawn on the surface of a sphere, we can construct another graph, a bipartite one, that has two types of vertex: a vertex for every of P, and also a vertex for every face of P. Whenever a vertex is on the boundary of a face, we connect the corresponding vertices of our new graph.

It would be nice if there were a standard name for this derived graph, but every paper I’ve read that uses it gives it a different name. Archdeacon and Richter call it the radial graph; Felsner and Rote call it the angle graph; Ziegler calls it the quad graph.

Of those names, I think I like ‘quad graph’ the best, because it describes an important feature of this graph, which is that all of its faces are quadrilaterals. So for now let’s call it that.