**Authors:** Maurice Margenstern

**Keywords:** Hyperbolic tessellations, algorithmic approach.

### Abstract

In this paper we revisit Poincare's theorem in the light of the splitting method which was introduced by the author in [3]. This led to the definition of combinatoric tilings.

We show that all tessellations which are constructed on a triangle with interior angles

and

with

are combinatoric, except when p=2 and q=3. At the price of a small extension of the definition of a combinatoric tiling, which we call quasi-combinatoric, we show that all tessellations with the above numbers p, q and r are quasi-combinatoric for all possible values of p, q and r, the case when p=2 and q=3 being included. As a consequence, see [3, 8], there is a bijection of the tiling being restricted to an angular sector S

_{0} and a tree which we call the spanning tree of the splitting. According , there is also a polynomial P

_{p,q,r} which allows us to compute the number of triangles which are associated with the nodes of the n

^{th} level in the tree: this will be examined in the second part of the paper. We also show that the tessellations which are constructed on an isosceles triangle with interior angles

, p odd, for the vertex angle and

for the basis angles with

are quasi-combinatoric and indeed combinatoric for p>=5. However, all the tessellations which are constructed on an equilateral triangle with interior angle

are combinatoric tilings.

LITA, EA 3097,

Universite de Metz,

Ile du Saulcy,

57045 Metz Cedex, France

E-mail:

### Fulltext

–

0.21 Mb