TITLE: Description of the order-5 square tiling of the hyperbolic plane as a graph
QUESTION [6 upvotes]: What is a description of the graph representing the faces of the Order-5 square tiling. Alternatively, this can be seen as the graph of vertices of the its dual, the Order 4 pentagon tiling.
Preferably, it should be a description useful for use with a computer (all that is needed is a way to find the neighbors of any node). The reason why is got interested in using hyperbolic geometry in games, so as a short cut to all the calculus needed I decided I'd make a bunch of rooms corresponding to a tessellation of the hyperbolic plane (and approximate each room as Euclidean).
Also, these description should be discrete (i.e. not rely on real numbers (which would need to be approximated with floating numbers on a computer))?

REPLY [3 votes]: You can represent this as the $\Delta(2,4,5)$ Von Dyck group.
In particular, you can generate the group by

Rotate left 90 degrees ($L$)
Rotate right 90 degrees ($R$)
Move forward one cell, and then turn around ($F$)

Given a string of such symbols, we can simplify to a normal form with the following operations:

Replace $L^2$ with $R^2$
Remove $F^2$
Replace $R^3$ with $L$
Replace $(FL)^2F$ with $(RF)^2R$
Replace $FLFR^2(FL)^2$ with $RF(RFR)^2F$
Replace $(FLFR^2)^2$ with $RF(RFR)^2FL$

(generated with sagemath).
Any two representations of the same element in the Von Dyck group will normalize to the same result under these operations.
Finally, note that each element represents both a square, and a direction. Therefore, we will say that two elements represent the same square by ignoring $R$ and $L$ at the beginning of the string.
Two cells are adjacent if you can get from one to the other with zero to three $R$ operations and exactly one $F$ operation.