In 1877, Richard Dedekind discovered one of the most famous pictures in mathematics : the black&white tessellation of the upper half-plane in hyperbolic triangles. Recall that the group
and as minus the identity matrix acts trivially, it is really an action of the modular group
At some points (such as
At other points (such as
Now, for the edges. There are three types of edges : even edges connecting a cusp and an even vertex (they form the
The geodesics (the semi-circles and the vertical lines) are made of edges and they come in two types : even lines are complete geodesics which are unions of two even edges (such as the semi-circle
If we write rational numbers in reduced form
This notation was set up to define the notion of a special polygon which is a connected polygonal region
- Even edges in
come in pairs and each such pair forms an even line. - Odd edges in
come in pairs and each pair meets at an odd vertex where they make an internal angle of . - Any odd edge e is side-paired to a different odd edge f which makes on internal angle
with e. - If e and f are even edges in
forming an even line, then either e is side-paired to f or else e,f form a free side and is side-paired to a different such free side. are among the vertices of .
The sides of P are : the odd edges on the boundary, the free sides and the even edges on non-free sides. The vertices of P are the intersections of adjacent sides.
For example, the region inside the thick edges is a special polygon. Its boundary consists of 8 even edges (two on the 4 complete geodesics : the vertical lines at 0 and 1 and the semi-circles
We have several option for the side-pairing, the only forded pairing being the two odd edges which have to be paired. For the even edges we can either consider 0,2 or 4 of the geodesics as free sides and pair these, or we can have 0,2 or 4 non-free sides and then we have to pair up the two even edges making such a non-free side.
The number of sides of the special polygon depends on the number of free sides chosen. For 0 free sides, there are 10 sides and vertices. For 2 free sides, there are 8 sides and vertices and for 4 free sides we have 6 sides and vertices.
Special polygons are a combinatorial gadget to describe the subgroups of finite index in the modular group
Some technical issues : if some of the latex-pictures don’t show up nicely it often helps to resize the browser-window and resize it back. The drawing of the special polygon was made using the LaTeX-package MFPIC which is an easy to use interface to MetaPost.
Reference
Ravi S. Kulkarni “An arithmetic-geometric method in the study of the subgroups of the modular group” Amer. J. Math. 113 (1991) 1053-1133