Pentagonal bipyramids lead to the smallest flexible embedded polyhedron

TL;DR

This paper classifies flexible pentagonal bipyramids using algebraic methods and constructs an embedded flexible polyhedron with 8 vertices, solving an open problem in the field.

Contribution

It introduces a classification of flexible pentagonal bipyramids via Galois groups and constructs the first embedded flexible polyhedron with 8 vertices.

Findings

Classification of flexible pentagonal bipyramids based on Galois groups

Construction of an embedded flexible polyhedron with 8 vertices

Solution to the open problem of minimal vertices for flexible polyhedra

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.