Klein-Arnold tensegrities

TL;DR

This paper introduces new classes of infinite, periodic tensegrities linked to algebraic multidimensional continued fractions, revealing a novel connection between rigidity theory and continued fraction geometry, with frameworks exhibiting rational self-stresses.

Contribution

It establishes a new connection between geometric rigidity and continued fractions, introduces classes of tensegrities with rational self-stresses, and develops a projective Maxwell-Cremona lifting principle.

Findings

New classes of infinite, periodic tensegrities derived from algebraic continued fractions.

Frameworks possess rational self-stress coefficients.

A projective Maxwell-Cremona lifting principle is formulated.

Abstract

In this paper, we introduce new classes of infinite and combinatorially periodic tensegrities, derived from algebraic multidimensional continued fractions in the sense of F. Klein. We describe the stress coefficients on edges through integer invariants of these continued fractions, as initiated by V.I. Arnold, thereby creating a novel connection between geometric rigidity theory and the geometry of continued fractions. Remarkably, the new classes of tensegrities possess rational self-stress coefficients. To establish the self-stressability of the frameworks, we present a projective version of the classical Maxwell-Cremona lifting principle, a result of independent interest.