For two thousand years geometry was synonymous with the perfectly flat expanse imagined by Euclid. But nineteenth‑century investigations into the parallel postulate lifted a veil from our eyes, revealing the richer realms charted by Gauss and Riemann. In this talk we’ll take an “insider’s tour” of those curved landscapes.

We begin by thinking carefully about what it means to see, and use this to step inside new geometries, by tracing light rays along their geodesics. Modern computing affords us the ability to make this thought experiment a reality, with interactive ray-traced demos and experiments.  Using these, we will explore curved spaces important to modern mathematics, and physics, including the curved spacetime we live in.