Branched Cover from (4 4 3/2) Schwarz Triangle - Greg Egan

Branched Cover from (4 4 3/2) Schwarz Triangle

A Schwarz triangle is a spherical triangle that can be used to generate a tiling of a branched covering of the sphere by repeatedly reflecting this triangle across its edges. Sometimes we get an actual tiling of the sphere, but in general we get a branched covering, because the same point can lie in the interior of several triangles, and there may be branch points at the corners of the triangles.

Tübingen Tiling - Greg Egan

Tübingen Tiling

A systematic way to generate quasiperiodic tilings of the plane is to take a lattice in higher dimensions and slice it at a funny angle. Greg Egan has created an applet that generates quasiperiodic tilings by projecting selected triangles from an $n$-dimensional lattice called the $\mathrm{A}_n$ lattice onto a plane. This particular picture comes from the $\mathrm{A}_4$ lattice.