Visualizing curvature on the Lorenz manifold

The Lorenz manifold is an intriguing two-dimensional surface that illustrates chaotic dynamics in the well-known Lorenz system. While it is not possible to find the Lorenz manifold as an explicit analytic solution, we have developed a method for calculating a numerical approximation that builds the surface up as successive geodesic level sets. The resulting mesh approximation can be read as crochet instructions, which means that we are able to generate a three-dimensional model of the Lorenz manifold. We mount the crocheted Lorenz manifold using a stiff rod as the z-axis, and bendable wires at the outer rim and the two solutions that are perpendicular to the z-axis. The crocheted model inspired us to consider the geometrical properties of the Lorenz manifold. Specifically, we introduce a simple method to determine and visualize local curvature of a smooth surface. The colour coding according to curvature reveals a striking pattern of regions of positive and negative curvature on the Lorenz manifold.