IMI/Publicaţii/CSJM/Ediţii/CSJM v.10, n.3 (30), 2002/

The splitting method and Poincare's theorem: (I) - the geometric part

Authors: Maurice Margenstern
Keywords: Hyperbolic tessellations, algorithmic approach.


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 S0 and a tree which we call the spanning tree of the splitting. According , there is also a polynomial Pp,q,r which allows us to compute the number of triangles which are associated with the nodes of the nth 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


Adobe PDF document0.21 Mb