We show that if a polyhedron in the three-dimensional affine space with triangular faces is flexible, that is, can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and (Formula presented.). We do this via elementary combinatorial considerations, made possible by a well-known compactification of the three-dimensional affine space as a quadric in the four-dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one-dimensional analog, which is trivial to solve.
Zero-sum cycles in flexible polyhedra
Gallet M.;Schicho J.
2022-01-01
Abstract
We show that if a polyhedron in the three-dimensional affine space with triangular faces is flexible, that is, can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and (Formula presented.). We do this via elementary combinatorial considerations, made possible by a well-known compactification of the three-dimensional affine space as a quadric in the four-dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one-dimensional analog, which is trivial to solve.File | Dimensione | Formato | |
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Bulletin of London Math Soc - 2022 - Gallet - Zero‐sum cycles in flexible polyhedra.pdf
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