Inductive constructions for frameworks on a two-dimensional fixed torus

2012/03/29 by Elissa Ross, Ross, Elissa · 3 citations
Engineering · #52C25 #Advanced Antenna and Metasurface Technologies #Advanced Materials and Mechanics #FOS: Mathematics #Metric Geometry (math.MG) #Structural Analysis and Optimization

paper · pdf · doi:10.48550/arxiv.1203.6561

openalex publication_date 2012/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An infinite periodic framework in the plane can be represented as a framework on a torus, using a \mathbb Z²-labelled gain graph. We find necessary and sufficient conditions for the generic minimal rigidity of frameworks on the two-dimensional fixed torus \mathcal T₀². It is also shown that every minimally rigid periodic orbit framework on \mathcal T₀² can be constructed from smaller frameworks through a series of inductive constructions. These are fixed torus adapted versions of the results of Laman and Henneberg respectively for finite frameworks in the plane. The proofs involve the development of inductive constructions for \mathbb Z²-labelled graphs.

Citations

• Infinite Bar-Joint Frameworks, Crystals and Operator Theory
• Crystal frameworks, symmetry and affinely periodic flexes
• Generic Global Rigidity
• Periodic Rigidity on a Variable Torus Using Inductive Constructions

Cited by

• Uniquely realizable crystalline structures
• Non-Euclidean Crystallographic Rigidity
• Symmetry-forced rigidity of frameworks on surfaces

Related

• The rigidity of periodic frameworks as graphs on a fixed torus
• Periodic Rigidity on a Variable Torus Using Inductive Constructions
• One brick at a time: a survey of inductive constructions in rigidity theory
• Rigidity of Frameworks Supported on Surfaces
• A characterisation of generically rigid frameworks on surfaces of revolution