Is the $r/R$ ratio for any polyhedron always the same as the $r/R$ ratio of the dual of that polyhedron?
Given any polyhedron, we can find the biggest sphere that fits inside it (its insphere) and the smallest sphere outside it (its circumsphere). I'm interested in $r/R$, which is the ratio of the radius of the insphere to the radius of the circumsphere.
Motivation: the worst-case linear distortion of a polyhedral global map projection is roughly (at least) $r/R$. The $r/R$ ratio is one of many ways to measure how close a polyhedron is to a sphere. Some of those measures say the dodecahedron is the closest Platonic solid to the sphere. Others say the icosahedron is the closest Platonic solid to the sphere. Polyhedrons with larger r/R ratio are better, in the sense that they (usually) produce maps with less linear distortion. I'm surprised to discover that that ratio seem to be equal between the dodecahedron and the icosahedron (both ~ $0.7946$), and a (somewhat smaller) ratio is the same between the cube and the octahedron (both ~ $0.5773$). (I'm not so surprised that an (even smaller) ratio is the same between the remaining Platonic solid and its dual polyhedron: $1/3$). Can this surprising-to-me fact be generalized to some larger group of polyhedrons? What other polyhedron have a $r/R$ ratio the same as the $r/R$ ratio of its dual polyhedron? Do any polyhedron have a $r/R$ ratio different from the $r/R$ ratio of its dual polyhedron?