## Escher, Conway, Kali, etc.

One can express the symmetry types of, say, Escher's "Circle Limit" prints using Conway's orbifold notation, best known in the context of symmetries of Euclidean plane patterns.

For example, Circle Limit III has symmetry type $433$ (with Euler characteristic $-1/12$).

Where can I find an explicit algorithm that produces generators for some appropriate subgroup of the isometries of the PoincarĂ© model of the hyperbolic plane given a suitable Conway notation? Only certain notation give rise to rigid orbifolds, so I'd also like to know how to read off the number of moduli from the Conway notation. Absent rigidity, I'd really like a parametric family of generating sets realizing all the distinct forms of the underlying orbifold.

The popular program Kali facilitates drawing symmetric Euclidean patterns? Does anyone distribute some appropriate hyperbolic counterpart?

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Yes! I had to rummage around, but D. Huson has such a program; he now works at Uni Bielefeld in bioinformatics, so it's somewhat hard to find.

http://www-ab.informatik.uni-tuebingen.de/software/2dtiler/welcome.html

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Bioinformaticians in Bielefeld are somewhat hard to find? :) – Mariano SuĂˇrez-Alvarez Jan 3 2011 at 3:16
No; what I should say is that there was a bad link on Geometry Junkyard to this, and so I had to hunt for Huson on the web, and finally found him doing bioinformatics at Bielefeld. – Robert Haraway Jan 3 2011 at 3:49

Can't speak for the Conway -> generators, but for drawing, there is this http://www.plunk.org/~hatch/HyperbolicApplet/

I am not sure why "rummaging" was necessary for the D. Huson program, since there is a link to it on the "OrbifoldNotation" wikipedia page.

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 It was probably before that link was up, or before I had heard of orbifold notation. – Robert Haraway Jan 3 2011 at 3:50 Thanks! Toyed with the program a bit, but it doesn't seem to do any smart uniformization. So sometimes it works and sometimes it doesn't. – David Feldman Jan 3 2011 at 6:39