Flexible pentagonal bipyramids and their Galois groups

When a bipyramid flexes, the distance between the two opposite vertices of the two pyramids changes. Therefore, there is a map that associates each realization of the bipyramid to the distance between the two opposite vertices. From an algebraic point of view, this determines a field extension between the field of univariate rational functions and the field of rational functions on the configuration curve of the bipyramid. In this talk, we present a classification of flexible pentagonal bipyramids with respect to the Galois group of this field extension.

As a consequence of the result, we construct an embedded flexible polyhedron with 8 vertices. This answers negatively a long-standing question whether the flexible embedded polyhedron by Steffen, which has 9 vertices, is an embedded flexible polyhedron with the least number of vertices.