In search for a perfect shape of polyhedra: Buffon transformation (Q330018)

Summary: For an arbitrary polygon generate a new one by joining the centres of consecutive edges. Iteration of this procedure leads to a shape which is affine equivalent to a regular polygon. This regularisation effect is usually ascribed to Count Buffon (1707--1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine \(B\)-regular polyhedra. The main result is the proof of existence of star-shaped affine \(B\)-regular polyhedra with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdière and Lovász.