# Necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex

## TL;DR

This paper introduces new equations based on Dehn invariants and the rigidity matrix that establish necessary conditions for extending a first-order flex of a polyhedron into a full flex, advancing the understanding of polyhedral flexibility.

## Contribution

It derives novel equations involving Dehn invariants and the rigidity matrix that serve as necessary conditions for flex extension, a significant theoretical advancement.

## Key findings

- New equations satisfied by first-order flexes of polyhedra
- Identification of Dehn invariants as sources of these equations
- Provision of necessary conditions for flex extension

## Abstract

We derive fundamentally new equations that are satisfied by first-order flexes of a flexible polyhedron. Moreover, we indicate two sources of such new equations. These sources are the Dehn invariants and rigidity matrix. The equations derived provide us with fundamentally new necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex.