Zero-sum cycles in flexible polyhedra
Metric Geometry
2022-03-21 v2 Algebraic Geometry
Combinatorics
Abstract
We show that if a polyhedron in the three-dimensional affine space with triangular faces is flexible, i.e., 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. 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 analogue, which is trivial to solve.
Cite
@article{arxiv.2009.14041,
title = {Zero-sum cycles in flexible polyhedra},
author = {Matteo Gallet and Georg Grasegger and Jan Legerský and Josef Schicho},
journal= {arXiv preprint arXiv:2009.14041},
year = {2022}
}