How to Crochet a Space-Filling Pancake: the Math, the Art and What Next

Hinke M. Osinga and Bernd Krauskopf
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture (2014)
Pages 19–26

Abstract

The chaotic behavior of the famous Lorenz system is organized by an amazing surface: the Lorenz manifold. Initial conditions on different sides of this surface will behave differently after some time, which means that the Lorenz manifold encodes how the chaotic dynamics manifests itself throughout the whole three-dimensional phase space. To convey its geometric properties, imagine a pancake that grows and grows to fill the entire space without creasing or developing self-intersections. Our mathematical research into the Lorenz manifold naturally led to artistic expression, when we realized that our computational method to find it generates crochet instructions. The crocheted Lorenz manifold not only received media attention, but also gave new insight into the geometry of chaos. Here we survey the developments that followed. They include our involvement with the mathematical art community and our collaboration with artist Benjamin Storch that led to a steel sculpture. In turn, our involvement with art contributed to the development of new ideas for studying and visualizing the Lorenz manifold, for example, by considering its intersection curves on a suitably chosen sphere.

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