There is a regular (n, m) tessellation of
H2 for all integers n and m such that 1/n+1/m<1/2
(*)
This makes use of the hyperbolic parallel axiom in its equivalent form
of the sum of the angles of a triangle is less than
180 degrees.
An n-gon can be simply divided into (n-2) triangles with a common vertex.
If a is the interior angle of the n-gon we have
na < (n-2).180
(1)
To fit exactly m of these about a point we must have
ma = 2.180
(2)
Eliminate a from (1) and (2) to get (n+m)/nm
< 1/2 or the required result
1/n + 1/m < 1/2
(*)
The equation (2) also gives us the angles of the regular triangles that
tesselate H2 . For instance for n = 7, 8, and 9 we have the
regular tringles having interior angles 51.4, 45 and 40 degrees respectively.
You can check these figures on the Cabri figure for regular triangle tesselations
tri.tess.fig.
