The HyperbolicHelper script creates hyperbolic polygons. The polygons may then be tessellated.
The Inversion3ptFast script does tessellation of one or more polygons. It uses Fast htm code, and also will invert points.
It still needs "Separate" and "re-Join" and Color preservation and Planar, to make tessellated surfaces easier.
The Escher circle limit art forms tessellations.
The InversionCenterFast script does tessellation using a center point and a radius point. It is Fast and has option to invert points.

The mathematical nomenclature for the tessellation is {p,q}, where p is the number of sides of a polygon, (sideCt),
and q is the number of polygons which meet at each polygon vertex, (vertexCt).
Note that a p-sided polygon may be subdivided into p regions, or 2*p hyperbolic triangles, etcetera.
Also note that there are some constraints on the minimal values for p and q, as per the table in reference #1.
p and q are positive integers, and (p - 2)(q - 2) > 4.
Lower combinations of p and q do not work for this script. (They only work for planes and spheres.)
Reference #1: http://www.d.umn.edu/~ddunham/md07tlk.pdf Creating Regular Repeating Hyperbolic Patterns.
https://cms.math.ca/cmb/v40/coxeter6871.pdf The trigonometry.
http://moniker.name/worldmaking/?p=385 Hyperbolic Geometry: Poincaré Disc Tessellation How-to.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.133.8746&rep=rep1&type=pdf Compass and Straightedge in the Poincaré Disk.

- Brian

http://www.ma.utexas.edu/users/gilbert/M333L/chp5vers4.pdf