Constructing isostatic frameworks for the $\ell^\infty$ plane

TL;DR

This paper introduces a new constructive method to realize 2-tree decompositions as minimally rigid bar-joint frameworks in the $ ext{L}^ ext{1}$ and $ ext{L}^ ext{infinity}$ planes, advancing understanding of rigidity in normed spaces.

Contribution

It presents a novel coloured multi-graph construction technique for realizing 2-tree decompositions as isostatic frameworks in $ ext{L}^ ext{1}$ and $ ext{L}^ ext{infinity}$ spaces, including symmetry considerations.

Findings

Every 2-tree decomposition can be realized as an isostatic framework in the $ ext{L}^ ext{1}$ or $ ext{L}^ ext{infinity}$ plane.

The method adapts to incorporate symmetry in frameworks.

Open problems in rigidity and symmetry in normed spaces are discussed.

Abstract

We use a new coloured multi-graph constructive method to prove that every 2-tree decomposition can be realised in the plane as a bar-joint framework which is minimally rigid (isostatic) with respect to $\ell^1$ or $\ell^\infty$ distance constraints. We show how to adapt this technique to incorporate symmetry and indicate several related open problems on rigidity, redundant rigidity and forced symmetric rigidity in normed spaces.