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Pentagonal bipyramids lead to the smallest flexible embedded polyhedron

Matteo Gallet, Georg Grasegger, Jan Legerský, Josef Schicho

Abstract

Steffen's polyhedron was believed to have the least number of vertices among polyhedra that can flex without self-intersections. Maksimov clarified that the pentagonal bipyramid with one face subdivided into three is the only polyhedron with fewer vertices for which the existence of a self-intersection-free flex was open. Since subdividing a face into three does not change the mobility, we focus on flexible pentagonal bipyramids. When a bipyramid flexes, the distance between the two opposite vertices of the two pyramids changes; associating the position of the bipyramid to this distance yields an algebraic map that determines a nontrivial extension of rational function fields. We classify flexible pentagonal bipyramids with respect to the Galois group of this field extension and provide examples for each class, building on a construction proposed by Nelson. Surprisingly, one of our constructions yields a flexible pentagonal bipyramid that can be extended to an embedded flexible polyhedron with 8 vertices. The latter hence solves the open question.

Pentagonal bipyramids lead to the smallest flexible embedded polyhedron

Abstract

Steffen's polyhedron was believed to have the least number of vertices among polyhedra that can flex without self-intersections. Maksimov clarified that the pentagonal bipyramid with one face subdivided into three is the only polyhedron with fewer vertices for which the existence of a self-intersection-free flex was open. Since subdividing a face into three does not change the mobility, we focus on flexible pentagonal bipyramids. When a bipyramid flexes, the distance between the two opposite vertices of the two pyramids changes; associating the position of the bipyramid to this distance yields an algebraic map that determines a nontrivial extension of rational function fields. We classify flexible pentagonal bipyramids with respect to the Galois group of this field extension and provide examples for each class, building on a construction proposed by Nelson. Surprisingly, one of our constructions yields a flexible pentagonal bipyramid that can be extended to an embedded flexible polyhedron with 8 vertices. The latter hence solves the open question.

Paper Structure

This paper contains 9 sections, 27 theorems, 47 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Basics of Galois theory
  4. Relevant graphs and subgraphs
  5. Motions
  6. Flexible octahedra and the Nelson construction
  7. Galois theory for motions
  8. A flexible embedded polyhedron with 8 vertices
  9. Equations for the oriented Cayley-Menger variety

Key Result

Theorem 1

There are only two possibilities for the Galois group of a non-degenerate motion of a pentagonal bipyramid. Both these possibilities arise as Galois groups of some particular flexible pentagonal bipyramids obtained via Nelson constructions.

Figures (8)

  • Figure 1: An example of a flexible embedded polyhedron with 8 vertices.
  • Figure 2: The graph of a pentagonal bipyramid (left) and of the almost tetrahedron $\mathbb{T}_{{1},{2}}$ (right).
  • Figure 3: The idea of the Nelson construction.
  • Figure 4: Instances of the motion of the pentagonal bipyramid based on Nelson construction of two flexible octahedra with Type III.
  • Figure 5: Instances of the motion of the pentagonal bipyramid based on Nelson construction of two flexible octahedra with Type I.
  • ...and 3 more figures

Theorems & Definitions (64)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 3.1
  • ...and 54 more