I stumbled onto these hyperbolic planar tesselations as I was researching different tesselation strategies for a project I’m developing. I asked my friend Alex Mollere, a Phd. candidate in applied mathematics at UT Austin, to explain:

… that’s the poincare disc model for the hyperbolic plane, a model for the space satisfying the postulates of the non-euclidean hyperbolic geometry in which infinitely many different lines may pass through a point P not on a line l without intersecting l, i.e., there is more than one line parallel to another given line. lines (i.e. the analog of “straight lines” in Euclidean plane) are the arcs of circles intersecting the poincare disc such that the angle between the boundary of the disc and the segment of the circle intersecting it is 90 degrees. The tesselations (just like in the Euclidean plane) are formed by segments of these lines. the dual tesselation is formed by the line segments intersecting the line segments of the original tesselation at 90 degrees, as you truncate the corners of the polygons forming the original regular tesselation, it gradually becomes its dual.