Zero‐sum cycles in flexible polyhedra. Issue 1 (4th March 2022)

• Record Type: Journal Article
• Authors: Gallet, Matteo; Grasegger, Georg; Legerský, Jan; Schicho, Josef
• Is Part Of: Bulletin of the London Mathematical Society. Volume 54:Issue 1(2022)
• Page Start: 112
• Page End: 125
• Publication Date: 2022-03-04
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12562
• Languages: English
• ISSNs: 0024-6093

Abstract

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.