Nonlinear algebra with tensegrity structures

TL;DR

This paper explores tensegrity structures using nonlinear algebra, providing explicit examples, computational tools, and insights into the algebraic geometry of equilibrium solutions, making complex concepts accessible to undergraduates.

Contribution

It introduces a nonlinear algebraic framework for tensegrity analysis, including explicit examples and computational resources, extending linear equilibrium equations to polynomial systems.

Findings

Minors of structured matrices determine the singular locus.

Singular points exhibit interesting phenomena in tensegrity 3-prism.

Polynomial systems and algebraic varieties are essential in understanding tensegrity solutions.

Abstract

In this paper, we discuss tensegrity from the perspective of nonlinear algebra in a manner accessible to undergraduates. We compute explicit examples and include the SAGE and Julia code so that readers can continue their own experiments and computations. The entire framework is a natural extension of linear equations of equilibrium, but to describe the space of solutions will require (nonlinear) polynomials. In our examples, minors of a structured matrix determine the singular locus of the algebraic variety of interest. At these singular points, more interesting phenomena can occur, which we investigate in the context of the tensegrity 3-prism, our running example. Tools from algebraic geometry, commutative algebra, semidefinite programming, and numerical algebraic geometry will be used. Although at first it is all linear algebra, the examples will motivate the study of systems of polynomial equations. In particular, we will see the importance of varieties cut out by determinants of matrices.