Normally, we can use complex numbers to describe a plane, with their norm related to the Euclidean metric. There are two lattice systems (Gaussian and Eisenstein integers) that permit integers with unique factorizations and such.
My question is: is there an analogue of these systems in hyperbolic plane?
One challenge is that hyperbolic plane doesn't have a consistent set of cardinal directions. If I'd want to build a Gaussian integer analogue based on {5,4} tesselation, the way to get to a particular lattice point would have to be described locally, as a series of turns. It's not obvious how addition would work without global orientation, not to mention multiplication.
Is there a way to transform a hyperbolic tessellation into a consistent 2-dimensional number system?