August 20, 2019
2017 – 2018
While searching for an equation that could provide a model for the unpredictable behavior of the weather, mathematician Edward Lorenz came up with the concept of the “butterfly effect”— the notion that an action as small as the flap of a butterfly’s wing can dramatically influence future events. Lorenz found that, no matter how close to each other the initial meteorological conditions were, the model predictions would diverge at an exponential rate. His discovery led to the development of chaos theory — the branch of mathematics that deals with complex systems whose behaviors are highly sensitive to slight changes.
Recent Yale College graduate Scott Weady, whose field is applied mathematics, wondered about the possibilities of using virtual reality (VR) to tell the story of chaos in the Lorenz system of equations.
“The Lorenz equations are a great tool for studying how the states of systems change over time,” Weady says. Using mathematical models of atmospheric temperature variations (atmospheric convection), Weady rendered the equation solutions in VR, using Unity, the most common software for virtual reality development. The rendering produced an intricate structure known as a “strange attractor.” Virtual reality allows users to interact with this mathematical structure in an intuitive way, exploring the geometric space and various phenomena common to the study of dynamical systems such as bifurcations (sudden qualitative changes in a system’s behavior), instabilities, and sensitivity to initial conditions.
“What I want to do with VR,” Weady says, “is develop models that will allow someone to see the differential equations in action by sharing a three-dimensional space with the strange attractor and seeing how it responds to variations in parameters and initial conditions.”
Weady thinks that VR has significant pedagogical potential to illustrate complex mathematical concepts in an intuitive and interactive way, and that the same techniques could be used to explore other mathematical models in areas including climate science, neuroscience, and fluid mechanics. “You don’t need to know any math to be intrigued by these VR models,” he says. “They illustrate the overall concepts and are just cool in themselves!”
Currently, Weady is working on developing the educational possibilities of the VR application, with the goals of making applied mathematics concepts accessible to those without formal training, and getting younger students excited about pursuing advanced mathematics.