Pentagonal bipyramids lead to the smallest flexible embedded polyhedron

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.