Visualizing curvature on the Lorenz manifold

Abstract

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 an explicit analytic expression for the Lorenz manifold, 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. The crocheted model directly motivated us to investigate the curvature properties of the Lorenz manifold by means of determining an approximation to the Gaussian curvature from our geodesic mesh representation. We then translate this information to colour, which leads to a new visualization of the geometry of the Lorenz manifold. This colouring enhances the aesthetics of the visualization by revealing a striking pattern of regions of positive and negative Gaussian curvature.