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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.

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 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,
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