Exotic Spherical Flexible Octahedra and Counterexamples to the Modified Bellows Conjecture
Abstract
Abstract In 2014, the author showed that in the three-dimensional spherical space, alongside the three classical types of flexible octahedra constructed by Bricard, there exists a new type of flexible octahedra, which was called exotic. In the present paper, we give a geometric construction for exotic flexible octahedra, describe their configuration spaces, and calculate their volumes. We show that the volume of an exotic flexible octahedron is nonconstant during the flexion, and, moreover, the volume remains nonconstant if we replace any set of vertices of the octahedron with their antipodes. So exotic flexible octahedra are counterexamples to the Modified Bellows Conjecture proposed by the author in 2015.
Fields of Study & Subjects
Structural Analysis and Optimization
Point processes and geometric inequalities
Computational Geometry and Mesh Generation
Publication details
- Journal / source
- Functional Analysis and Its Applications
- Publisher
- Pleiades Publishing Ltd
- Volume
- 60
- Issue
- 2
- Pages
- 107–128
- Publication date
- 1 June 2026
- Language
- EN