Pentagon Hyperbolic Tessellations in Mathematics

Question

I want to implement this in Mathematica:

I understand that I can use ParametricPlot to get the rows, and then use the Mesh parameter to fill the colors. Can someone tell me what is the general formula for the equations of these lines? Does it generalize a regular regular n-gon?

Answer 1

I seem to have a code that will do something close to what you want, and you can view this code here: http://facstaff.unca.edu/mcmcclur/mathematicaGraphics/PTiling/ .

A few comments are fine. The ideas that underlie the code are described in Saul Steel’s remarkable book, The Half-Point of Poincare — in particular, the chapter on the Poincare disk. I wrote code to illustrate some ideas for the geometry class that I taught in 1999, so it should have been for version 3 or 4. I did nothing to try to optimize the code for any subsequent version. Regardless if you define the PTiling function on this page and then do the PTiling[5, 2*5 - 4, 3] , you should (in a few minutes) get something like the following:

Obviously, we only have a black and white picture illustrating the borders of the pentagons you want, but hopefully this is a good start. I think you can use parts of the disks rather than circles to get closer to what you want.