Zero-sum cycles in flexible polyhedra

TL;DR

This paper proves that flexible polyhedra with triangular faces in 3D space always contain a cycle of edges whose weighted lengths sum to zero, revealing a fundamental geometric property of such structures.

Contribution

It introduces a novel combinatorial approach using compactification and degeneration techniques to analyze flexibility in polyhedra.

Findings

Existence of zero-sum edge cycles in flexible polyhedra

Use of compactification in affine space for geometric analysis

Reduction of a 3D problem to a trivial 1D case

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.