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Klein-Arnold tensegrities

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arxiv 2410.12729 v1 pith:O6NM7EFN submitted 2024-10-16 math.CO math.MG

Oleg Karpenkov , Fatemeh Mohammadi , Christian Müller , Bernd Schulze This is my paper · ORCID
classification math.COmath.MG
keywords continuedfractionstensegritiesclassescoefficientsalgebraicarnoldclassical
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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.

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  1. Liftings of surfaces in the plane

    math.CO 2025-05 conditional novelty 6.0 of 10

    A self-stress on a framework of surfaces yields a monodromy-free piecewise-linear lifting whenever the self-stress space has dimension greater than 3 times the first Betti number of the surface.

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