Rational tensegrities through the lens of toric geometry

TL;DR

This paper explores multidimensional rational tensegrity frameworks using toric geometry, establishing a connection between tensegrities and algebraic geometry tools, and providing explicit computational methods for planar cases.

Contribution

It introduces a novel approach to analyze multidimensional tensegrities through toric geometry, linking tensegrity groups to Chow groups and offering explicit computational techniques.

Findings

Tensegrity groups are isomorphic to subgroups of Chow groups of associated toric surfaces.

Explicit methods for computing tensegrities in planar frameworks are developed.

The framework extends classical tensegrity models to higher dimensions using algebraic geometry.

Abstract

A classical tensegrity model consists of an embedded graph in a vector space with rigid bars representing edges, and an assignment of a stress to every edge such that at every vertex of the graph the stresses sum up to zero. The tensegrity frameworks have been recently extended from the two dimensional graph case to the multidimensional setting. We study the multidimensional tensegrities using tools from toric geometry. For a given rational tensegrity framework $\mathcal{F}$, we construct a glued toric surface $X_\mathcal{F}$. We show that the abelian group of tensegrities on $\mathcal{F}$ is isomorphic to a subgroup of the Chow group $A^1(X_\mathcal{F};\QQ)$. In the case of planar frameworks, we show how to explicitly carry out the computation of tensegrities via classical tools in toric geometry.