2020/09/30 by Matteo Gallet, Georg Grasegger, Jan Legerský +1 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Materials and Mechanics #Affine space #Affine transformation #Combinatorics #Compactification (mathematics) #Computational Geometry and Mesh Generation #Euclidean space #Mathematics #Polyhedron #Pure mathematics #Quadric #Structural Analysis and Optimization #Topology (electrical circuits) #Zero (linguistics) #math.AG #math.CO #math.MG #msc:52B10 #msc:52C25 #msc:70B15
paper · pdf · doi:10.1112/blms.12562
published in Bulletin of the London Mathematical Society 54(1), 112-125 (Wiley)
openalex publication_date 2022/02/01 · arxiv created 2022/03/18 · arxiv updated 2022/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.