How hyperbolic tilings are programmed
The programs to make hyperbolic tilings are written in
MLgraph library. Symmetry groups of tilings are computed formally using a definition
by generators and relations and a canonical rewriting system to decide
equality. Hyperbolic isometries are defined as homographies over
complex numbers.Tiles are defined essentially as lists of points and isometries are applied
In the paper Tilings as a programming exercise ,
I describe a generic implementation of tilings using Objective Caml modules to parameterize
over Geometries, Symmetry groups and Patterns.
The fish picture was first programmed by
for a student programming project at ENS .
had described the geometry of this picture in
The non-euclidean symmetry of Escher's picture Circle limit III,
Leonardo 12, pp 19-25, Pergamon Press, 1979 .
Philippe Lechenadec has given a canonical rewriting system for
hyperbolic groups in
Canonical forms in finitely presented algebras, Pitman, 1986 .
Adrien Douady explained how to define the hyperbolic rotations numerically.
The kangaroo picture was designed for the 1996 poster of
the game Kangourou des Mathématiques by
Guy Cousineau, Claudie Missenard and Raoul Raba.