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An almost trivial observation about the icosahedron

Jürgen Richter-Gebert

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.

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 satisfying a homothety relation , and show that this condition forces 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.
Paper Structure (6 sections, 4 theorems, 11 equations, 6 figures)

This paper contains 6 sections, 4 theorems, 11 equations, 6 figures.

Table of Contents

  1. How rigid is the icosahedron?
  2. The configuration
  3. Kepler–Poinsot polyhedra
  4. A guest appearance of the pentagram map
  5. The nitty gritty calculations
  6. Incidence Theorems

Key Result

Theorem 1

Consider any set of 12 points $\{ \overline{0},\ldots, \overline{5},\underline{0},\ldots, \underline{5} \}$ in $\mathbb{R}^3$ satisfying non-degeneracy $ND$ requirements specified below. If for each $(a,b,c,d,e)\in D$ the points $a,b,c,d,e$ are coplanar then up to projective equivalence there are on

Figures (6)

  • Figure 1: The edge graph of the great dodecahedron (left) $\mathcal{G}$ is identical to the edge graph of the icosahedron. The small stellated dodecahedron $\mathcal{G^*}$ (right) also has the same edge graph, however with a geometrically different embedding. The embedding can be derived from $\mathcal{G}$ by a suitable permutation of the vertices.
  • Figure 2: The transition from the great dodecahedron (left) via the dodecadodecahedron (middle) to the small stellated dodecahedron (right)
  • Figure 3: The projection of $(\overline{1}^*,\ldots,\overline{5}^*)$ to $(\underline{1},\ldots,\underline{5})$.
  • Figure 4: Pentagram maps between the two pentagons $(\overline{1},\overline{2},\overline{3},\overline{4},\overline{5})$ and $(\underline{1},\underline{2},\underline{3},\underline{4},\underline{5})$.
  • Figure 5: A geometric situation in which the doubly iterated pentagram map produces a homothetic copy of the original pentagon.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4