An almost trivial observation about the icosahedron

Abstract

We consider the incidence structure formed by the twelve pentagons given by the vertex neighborhoods of the icosahedron. Interpreting this structure purely in terms of coplanarity conditions, we show that -- up to projective equivalence -- it admits exactly two realizations. Both realizations coincide with the vertex set of the regular icosahedron and interpreted as cell complex they correspond to the great dodecahedron and the small stellated dodecahedron. The key step is to reinterpret the configuration via the pentagram map. We prove that any realization gives rise to a pentagon $X$ satisfying a homothety relation $P^2(X)\sim X$, and show that this condition forces $X$ to be an affine image of either a regular pentagon or a regular pentagram. This reduces the problem to a quadratic constraint and explains the rigidity of the configuration.