An almost trivial observation about the icosahedron

TL;DR

This paper shows that the incidence structure of the icosahedron's vertex neighborhoods admits only two projectively equivalent realizations, linking geometric configurations to the pentagram map and revealing their rigidity.

Contribution

It provides a novel interpretation of the icosahedron's incidence structure through the pentagram map, demonstrating its rigidity and classifying all realizations.

Findings

Exactly two realizations up to projective equivalence.

Realizations correspond to the vertex sets of the icosahedron, great dodecahedron, and small stellated dodecahedron.

The configuration's rigidity is explained via a quadratic constraint on pentagon and pentagram shapes.

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.