Exotic spherical flexible octahedra and counterexamples to the Modified Bellows Conjecture
Alexander A. Gaifullin
TL;DR
This work identifies and geometrically constructs exotic flexible octahedra in $\mathbb{S}^3$, a new family beyond Bricard’s classical types. It analyzes their configuration spaces, proving the existence of a unique essential 1D component $\Gamma$ with two real parts $\Gamma_{+}$ and $\Gamma_{-}$ and provides an explicit volume formula that varies along the flexion. The key result is that the oriented volume of these exotic octahedra is nonconstant during flexion, even after antipodal vertex replacements, making them counterexamples to the Modified Bellows Conjecture in $\mathbb{S}^3$. The paper also discusses higher-dimensional generalizations and the real-valued volume branch, outlining avenues for counterexamples to the Modified Bellows Conjecture in $\mathbb{S}^n$ for $n\ge 4$ and posing questions about volume multivaluedness on configuration spaces.
Abstract
In 2014 the author showed that in the three-dimensional spherical space, alongside with 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.