Hyperbolic Tilings with Truly Hyperbolic Crochet Motifs

Joshua Holden and Lana Holden
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture (2014)
Pages 405–408 Short Papers

Abstract

Until now, most methods for making a hyperbolic plane from crochet or similar fabrics have fallen into one of two categories. In one type, the work has constant negative curvature but does not naturally lend itself to a polygonal tiling. In the other, polygonal tiles are attached in such a way that the final product approximates a hyperbolic plane on the large scale but does not have truly constant curvature. We show how crochet can be used to create polygonal tiles that have constant negative curvature in themselves and can therefore be joined into a large region of a hyperbolic plane without significant stretching. Formulas from hyperbolic trigonometry are used to show how, in theory, any regular tiling of the hyperbolic plane can be produced in this way.

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