La preuve par l'image

Bleu Hyperbolique won the grand prix du jury in La preuve par l’image, the CNRS’s annual competition inviting French researchers to submit striking images from their work — a jury selects around twenty laureates from nearly two hundred entries for a touring exhibition. The image, a view inside a finite-volume hyperbolic 3-manifold, is joint work with Rémi Coulon, Sabetta Matsumoto, and Henry Segerman.

The Mathematics

This is a tiling of hyperbolic 3-space arising from an orbifold with two cusps, whose fundamental domain is an ideal cube in ℍ³. We have drawn the edges of the cube and punctured each ideal vertex with a horoball — the circular holes visible where the edges meet.

Technique

We wrote a ray tracer that traces geodesics in the hyperboloid model of ℍ³ inside Minkowski 4-space. Restricting to a single fundamental domain, we apply the required side-pairing transformation each time a ray leaves one face, re-entering through another.