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.

翻译：长期以来，人们认为Steffen多面体是具有不自交柔性多面体中顶点数最少的。Maksimov指出，将一个面细分为三份的五角双锥是唯一已知顶点数更少且是否存在不自交柔性尚不明确的多面体。由于将面细分为三份不会改变其可动性，我们重点研究柔性五角双锥。当双锥发生柔性变形时，两个锥体相对顶点间的距离会发生变化；将双锥位置与该距离相关联可得到一个代数映射，该映射决定了有理函数域的非平凡扩张。基于Nelson提出的构造方法，我们根据该域扩张的伽罗瓦群对柔性五角双锥进行分类，并为每个类别提供了实例。令人惊讶的是，我们的某个构造产生了可扩展为具有8个顶点的嵌入柔性多面体的柔性五角双锥，从而解决了这一悬而未决的问题。