In his work he mainly focuses on the pentagrid {5,4} and heptagrid {7,3}:
In what ways are these tilings special? How do they compare to hyperbolic tilings in general? I am wondering what insights from these two cases are unique to these two cases, and what insights can transfer to all other possible tilings in the hyperbolic plane.
I would like to build a visualization system according to his work, but am trying to figure out the coordinate system first, and it appears he has figured out a coordinate system for these two cases but it won't transfer to the general case and I'm not sure why. So wondering if you could summarize what results from these two cases transfer to all tilings in the plane, or which ones are special to these two cases.
Most likely, these examples were selected because of their relative simplicity in terms of the number of sides per polygon and the number of polygons meeting at each vertex. With three identical polygons per vertex each one has to have at least seven sides; with four polygons per vertex the minimum number of sides is five. These are the examples chosen in the work.
There are some other special properties for each of these two cases. The $\{7,3\}$ tiling (and the dual $\{3,7\}$ tiling) can be fitted onto the Klein quartic, one of the most widely studied and appealing hyperbolic manifolds.
The $\{5,4\}$ tiling, again along with its dual, represents a particularly simple case whose Poincare-disc projection may be constructed via unmarked straightedge and compasses (and thus is definable by strictly Euclidean geometry). Exercise for the reader: construct the central pentagon in this projection.