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 researchers in France to submit striking images from their work; a jury picks around twenty laureates from nearly two hundred entries. It is joint work with Rémi Coulon, Sabetta Matsumoto and Henry Segerman, and shows the view inside a finite-volume hyperbolic 3-manifold.

The prize was announced in September 2022 and carried in CNRS le Journal. The CNRS then included the image in a traveling exhibition of twenty panels, produced in both French and English, which it also made available to download from its médiathèque.

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.