G4G's Virtual Celebration of Mind | February

CoM | Computer Graphics in Curved Space

Visiting the Thurston Geometries: Computer Graphics in Curved Space

A beautiful observation of classical physics is that “light travels in straight lines” is only an approximation to reality. More precisely, light always takes a geodesic – a path between two points minimizing its time of travel.

While this is often used to explain physical phenomena mathematically (from refraction to mirages to gravitational lensing), we may turn the tables and also use this observation to experience mathematical phenomena physically.

That is, given an abstract three-dimensional mathematical space, we may realistically simulate “what it would be like” to be there, by tracing the geodesics connecting our eyes to objects around us. In joint work with Remi Coulon, Sabetta Matsumoto, and Henry Segerman, we have produced real-time computer and VR simulations of the eight Thurston Geometries: in this talk, I will take you on a brief adventure to each of these worlds and talk a little about how we made it happen.

Steve Trettel is a Szego Assistant Professor of Mathematics at Stanford University, and his research focuses primarily on low dimensional topology and differential geometry. Steve grew up in Minnesota and completed his undergraduate at the University of Minnesota before moving to California to pursue a Ph.D. at UC Santa Barbara. Steve is passionate about mathematical illustration and particularly enjoys tackling difficult visualization problems with a computer.