Initial text in the uploaded worksheet describes an attempt to partially tile the Poincare disk and asks to be shown how a known complex transformation can be used to accomplish this.

 Poincare_tiling.mw

The worksheet below contains the desired hyperbolic action and asks for the specific transformation that performs it.

Hyperbolic_Transformation.mw

I have corrected the above worksheet so that the Poincare disk is now centred at [0,0].

I hope this helps you answer my question

In the worksheet below is an equation for non-Euclidean lines. How can Maple plot these?

NonEuclideanLines.mw

This worksheet creates geodesics in the Poincare disk by transformation of a series of circles of diminishing radii in the complex plane.

The intersections of the geodesics are meant to create the first few pentagonal uniform tilings in the Poincare disk.

I do not know the mathematically correct way to create such a display, so the radii of the circles are only a trial and error approximation.

What Maple code will provide the radii of the complex circles which produce an accurate uniform pentagonal tiling?

Is there a better overall strategy for producing uniform tilings of the Poincare disk? 

HyperbolicTiling.mw

This worksheet attempts to implement figures in Section 3 of the book "Indra's pearls" by Mumford, Series and Wright using the  command complexplot.

The worksheet seems to display Figure 3.1 on page 63 accurately but fails to correctly display Figure 3.4 on page 67.

Can complexplot be coded to accurately display this figure?

Indras_pearls_spirals.mw