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Computer Aided Geometric Design


The problem of detecting auxetic behavior, originating in materials science and mathematical crystallography, refers to the property of a flexible periodic bar-and-joint framework to widen, rather than shrink, when stretched in some direction. The only known algorithmic solution for detecting infinitesimal auxeticity is based on the rather heavy machinery of fixed-dimension semi-definite programming. In this paper we present a new, simpler algorithmic approach which is applicable to a natural family of 3D periodic bar-and-joint frameworks with 3 degrees-of-freedom. This class includes most zeolite structures, which are important for applications in computational materials science. We show that the existence of auxetic deformations is related to properties of an associated elliptic curve. A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve.


Periodic framework, Auxetic deformation, Elliptic curve, Aronhold invariants, Zeolites



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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.


Peer reviewed accepted manuscript.



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