Many Paths to the Beach

The Mathematics

This is an inside view of a three-manifold that is asymptotically ordinary space (ℝ³) on one end and a flat 3-torus on the other, smoothly joined by an S²×I tube. The viewer stands within the 3-torus and gazes down the tube toward ℝ³, where a Southern California beach lies on the far side. Many copies of the tube and the beach are visible at once — one for every homotopy class of paths out to infinity.

Technique

I wrote a custom Riemannian ray tracer for computing geodesics in three-manifolds of this topology — a compact part, an S²×I part, and an asymptotically flat part, their metrics glued together by smooth bump functions. A spherical photograph is placed in the asymptotically flat end, and rays are traced from a point in the 3-torus until they reach that scene.