Rational tensegrities through the lens of toric 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. We introduce a link between self-stresses and Chow rings on toric varieties. More precisely, for a given rational tensegrity framework $F$, we construct a glued toric surface $X_F$. We show that the abelian group of tensegrities on $F$ is isomorphic to a subgroup of the Chow group $A^1(X_F;Q)$. In the case of planar frameworks, we show how to explicitly carry out the computation of tensegrities via classical tools in toric geometry.