Non-Euclidean Crystallographic Rigidity

2025/10/09 by Esson, Jack, Kastis, Eleftherios, Schulze, Bernd
#52A21 (secondary) #52C25 (primary) 05C22 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.08128

Abstract

This paper establishes combinatorial characterisations of forced-symmetric and forced-periodic rigidity (under a fixed lattice) of bar-joint frameworks in non-Euclidean normed planes. In ℓ_q-planes for q∈(1,∞)\backslash\2\, we prove characterisations for periodic rigidity and finite reflectionally-symmetric rigidity. We also characterise symmetric rigidity in this space with respect to the orientation-reversing wallpaper group ℤ²\rtimesC_s, otherwise known as pm in crystallography. In the ℓ₁ and ℓ_∞-planes, we provide characterisations for periodic rigidity and ℤ²\rtimesC_s-symmetric rigidity. All of these characterisations are proved by inductive constructions involving Henneberg-type graph operations.

Citations

• Rigid graphs in cylindrical normed spaces
• Quotient graphs of symmetrically rigid frameworks
• Symbol functions for symmetric frameworks
• Generic symmetry-forced infinitesimal rigidity: translations and rotations
• Rigidity of symmetric frameworks in normed spaces
• Motions of grid-like reflection frameworks
• Finite and infinitesimal rigidity with polyhedral norms
• Symmetry-forced rigidity of frameworks on surfaces
• Infinitesimal rigidity for non-Euclidean bar-joint frameworks
• Frameworks with forced symmetry I: Reflections and rotations
• Matroids of Gain Graphs in Applied Discrete Geometry
• Periodic Rigidity on a Variable Torus Using Inductive Constructions
• Inductive constructions for frameworks on a two-dimensional fixed torus
• The rigidity of periodic frameworks as graphs on a fixed torus
• The orbit rigidity matrix of a symmetric framework
• Pebble Game Algorithms and Sparse Graphs

Related

• Orientation-Reversing Crystallographic Rigidity
• Generic rigidity of frameworks with orientation-preserving crystallographic symmetry
• The almost periodic rigidity of crystallographic bar-joint frameworks
• Generic rigidity with forced symmetry and sparse colored graphs
• Rigidity of symmetric frameworks in normed spaces