Nonlinear algebra via tensegrity structures

Alexander Heaton

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Submission date: 05. Sep. 2019
Bibtex
MSC-Numbers: 74-01, 13-01
Keywords and phrases: tensegrity, rigidity
Link to arXiv: See the arXiv entry of this preprint.

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 code so that readers can continue their own experiments and computations. The entire framework is a natural extension of linear equations of equilibrium, however, to describe the space of solutions will require (nonlinear) polynomials. The maximal minors of a certain matrix A will cut out the algebraic variety of prestress solutions from the configuration space of a given structure. Imposing an additional list of inequalities defines the semi-algebraic set of tensegrity solutions for a specific choice of cables and bars. These inequalities will come from the entries of a vector b in the left nullspace of A. Tools from algebraic geometry and commutative algebra, such as primary decomposition, can be used to single out certain spaces of configurations. Although at first it is all linear algebra, the examples will motivate the study of systems of polynomial equations, including the algebraic geometry and commutative algebra used to solve them. In particular, we will see the importance of varieties cut out by determinants of matrices.