# A sufficient condition for a polyhedron to be rigid

## TL;DR

This paper establishes a new sufficient condition based on edge length relations that guarantees a polyhedron's rigidity, and explores properties of dihedral angles during flexing.

## Contribution

It introduces a novel criterion involving rational linear combinations of edge lengths to determine polyhedral rigidity and analyzes dihedral angle invariants during flexing.

## Key findings

- Polyhedra are rigid if each edge length cannot be expressed as a rational linear combination of the others.
- Flexible polyhedra have certain linear combinations of dihedral angles that remain constant during flexing.
- Coefficients of these invariant linear combinations are integers and do not change during flex.

## Abstract

We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed continuously by changing its dihedral angles only. We prove that the polyhedron is not flexible if for each of its edges the following holds true: the length of this edge is not a linear combination with rational coefficients of the lengths of the remaining edges. We prove also that if a polyhedron is flexible, then some linear combinations of its dihedral angles remain constant during the flex. In this case, the coefficients of such a linear combination do not alter during the flex, are integers, and do not equal to zero simultaneously.