TL;DR
This paper establishes necessary metric conditions for the flexibility of octahedra, providing a new description of Bricard's octahedra that aids in solving problems in metric geometry and exploring volume polynomial properties.
Contribution
It introduces a novel metric-based characterization of flexible octahedra, enhancing understanding and analysis of their geometric properties.
Findings
Derived necessary metric conditions for octahedron flexibility
Provided a new description of Bricard's octahedra
Supported exploration of volume polynomial coefficients
Abstract
For flexibility of an octahedron we find necessary metric conditions in terms of edge lengths. These conditions yield a new description of Bricard's octahedra, suitable for solving some problems in metric geometry of octahedra, in particular, for searching the proof of I.\,Hh~Sabitov hypothesis that all non-leading coefficients of the volume polynomial for an octahedron of third type are zero.
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