Rigidity of polytopes with edge length and coplanarity constraints

TL;DR

The paper studies rigidity of polytopes under fixed edge lengths and face planarity constraints, showing the regular cube is flexible under this notion and introducing a notion of generic realizations. It proves that 3D polyhedra are generically rigid, using a contraction-sequence strategy together with Tutte embeddings and the Maxwell-Cremona correspondence, and it develops a framework to define genericity on irreducible components via Zariski convexity. The work connects to classical results on rigidity (Cauchy, Dehn, Gluck) and extends them to general combinatorial types, while outlining a program for higher dimensions, second-order rigidity, and edge-length perturbations. It also raises numerous open questions about the extent of flexibility in non-convex or non-Euclidean settings and the role of affine transformations in polytope rigidity. Overall, the paper provides a rigorous 3D confirmation of generic rigidity for polytopes and lays out a robust toolkit for exploring rigidity in broader geometric contexts.

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.

Figures (10)

• Figure 1: Examples of continuously flexing 3-dimensional polytopes.
• Figure 2: Flexible polygons.
• Figure 4: A 3-polytope that is affinely flexible because it has only five edge direction. It is also a Minkowski sum of the shaded faces.
• Figure 5: The cuboctahedron with two stacked faces is not a Minkow-ski sum, but is still flexible. This is because the stacking restricts only one of the three degrees of freedom that come from the three relative rotations of the Minkowski summands of the cuboctahedron.
• Figure 6: Each flex of the "cube with pyramid roof" is dissectable since it can be split into a flexing cube and a rigid pyramid using the shown hyperplane.

Theorems & Definitions (34)

• Conjecture 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 3.2
• Conjecture 5.1
• Theorem 5.2
• Definition 5.3
• Remark 5.4
• Example 5.5
• Definition 5.6