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 runner rug).
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.
The inversion of a half-plane map of Hⁿ is a conformal disc map.
The inversion of a stereographic map of Sⁿ is also a stereographic map.