Rigidity of polytopes with edge length and coplanarity constraints

TL;DR

This paper explores a new form of polytope rigidity where edge lengths and face planarity are preserved, but face shapes can change, leading to insights on generic rigidity in higher dimensions.

Contribution

It introduces a novel rigidity concept allowing face shape changes, constructs flexible polytopes, and proves a conjecture on generic rigidity in three dimensions.

Findings

Regular cubes are flexible under this notion.

Flexibility is rare among polytopes.

Convex polytopes are generically rigid in 3D.

Abstract

We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present techniques for constructing flexible polytopes and find that flexibility seems to be an exceptional property. Based on this observation, we introduce a notion of generic realizations for polytopes and conjecture that convex polytopes are generically rigid in dimension $d\geq 3$. We prove this conjecture in dimension $d=3$. Motivated by our findings we also pose several questions that are intended to inspire future research into this notion of polytope rigidity.