# 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.

# Pentagon-Decagon Branched Covering

Two regular pentagons and a regular decagon fit snugly at a point: their interior angles sum to 360°. Despite this, you cannot tile the plane with regular pentagons and decagons. However, there is a branched covering of the plane tiled with pentagons and decagons, which map to regular pentagons and decagons on the plane. Here Greg Egan has drawn a portion of this branched covering.

# Pentagon-Decagon Packing

Two regular pentagons and a regular decagon meet snugly at a vertex: their interior angles sum to 360°. However, they can’t tile the plane. However, they come fairly close, as shown in this picture by Greg Egan.

# Pattern-Equivariant Homology of a Penrose Tiling

The Penrose kite and dart are a pair of tiles that can be used to create aperiodic tilings of the plane. This image illustrates a ‘pattern-equivariant 1-chain’, a tool used by James J. Walton to study the topology of the kite and dart tiling, and other aperiodic tilings.

# 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.