Wherever I go on the internet, I keep seeing the same types of hyperbolic tessellations -- regular {p,q} things, sometimes truncated, rectified or rhombated -- snubs, on the best days. But there is a range of much more interesting tilings that are still uniform, yet have strange connections.
I have been interesting in them ever since I found out that a simple vertex configuration (a,a,a,b) -- three polygons of one type and one of another type -- can lead to up to four globally distinct uniform tilings. I heard about Mr. Conway's orbifold notation, but I never heard of anyone actually using his theory to classify the tilings and seriously explore them, to determine which tilings are possible for which vertex configurations. Did that ever happen?
The only software I ever found that is capable of displaying arbitrary tilings was this little applet, over 13 years old now:
http://www.plunk.org/~hatch/HyperbolicApplet/
I learned how to transform my own notation into what the program accepts, to explore tessellations that are not easily connectible to Wythoffian constructions like these little marvels:
"(3,5,5,5,5,5) (0 1)2 3(5)"
"(3,5,5,5,5,5) (0 1)(2 4)(3 5)"
"(3,5,5,5,5,5) (0 1)(2 4)[3 5]"
"(3,5,5,5,5,5) (0 1)2 4(5)"
"(3,5,5,5,5,5) (0 1)(2)(3)(4)(5)"
These are five fundamentally distinct uniform tilings with one triangle and five pentagons at every vertex, and by downloading the applet and running it with these parameters (like HyperbolicApplet.jar symbol="(3,5,5,5,5,5) (0 1)2 3(5)"), I can explore them and plainly see their differences. Isn't that fascinating? But I was never able to drum up much interest -- math is just a hobby for me, after all. As far as I can understand, Conway's method studies symmetries of vertices, while mine is more of a combinatorical approach that literally builds the tilings from fundamental pieces -- and it's probably much more complicated then it needs to be.
I'm just wondering whether there's any interest in this.