Pretty things: hyperbolic planar tesselations by Don Hatch. Presented in the conformal Poincaré disc mapping, which is the most common; it’s analogous to stereographic projection of a sphere. Another favorite mapping is the half-plane, which has no analogue that I can think of.
But I’ve never seen a conformal ‘Mercator’ mapping, preserving one line. Instead of a circle, the infinite hyperbolic plane would become an infintely long but finitely wide strip; Escher’s Circle Limit, transformed through such a projection, would make a nifty frieze (or corridor carpet).
Sadly I’ve yet to find enough information (clear enough for my lazy mind) on doing stuff in hyperbolic space.
In 2008(?) I succeeded in making hyperbolic strip designs. I showed them to Vladimir Bulatov, who showed me prior art (which did not surprise me) and wonderfully extended the concept. Now it’s popularly known as the Bulatov Band.