Triangle Symmetry

The Mathematics

A triangle orbifold is a tiling of a space by repeated reflections of a single triangle. We begin with a hyperbolic structure on the (3,3,4) triangle orbifold, which unrolls into a beautiful projective tiling of the Klein disk. Within real projective geometry this tiling can be deformed by deforming hyperbolic geometry itself — and this artwork depicts that one-parameter family of deformations.

Technique

I computed the exact one-parameter family of convex real projective structures on this orbifold, then wrote a projective tiling generator that takes a fundamental domain and reflects it thousands of times across its symmetry group. Doing this at fifteen points spaced along the family, I assembled the results into the final image.