Zero‐sum cycles in flexible polyhedra. Issue 1 (4th March 2022) - Gallet, Matteo; Grasegger, Georg; Legerský, Jan; Schicho, Josef
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 − 1 -1 . 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.