Hyperbolic Tessellations of a Pentagon in Mathematica

Question

I want to implement this in Mathematica:

I realize I can use ParametricPlot to get the lines, and then use the Mesh option to fill in the colors. Can anyone tell me what a general formula for the equations of these lines are? Does it generalize to a regular n-gon?

Answer 1

I happen to have some code lying around that will do something close to what you want and you can view that code here: http://facstaff.unca.edu/mcmcclur/mathematicaGraphics/PTiling/.

A couple of comments are in order. The ideas behind the code are all described in Saul Stahl's excellent book, The Poincare Half-Plane - specifically, the chapter on the Poincare disk. I wrote the code to illustrate some ideas for a geometry class that I was teaching in 1999 so it must have been for version 3 or 4. I have done nothing to try to optimize the code for any subsequent version. Regardless, if you define the function PTiling on that page and then execute PTiling[5, 2*5 - 4, 3], you should (after several minutes) get something like the following:

Obviously, we have just a black and white picture illustrating the boundaries of the pentagons that you want but, hopefully, this is a good start. I think that one could use portions of disks, rather than circles, to get closer to what you want.