On the Infinigons of the Hyperbolic Plane, A combinatorial approach

In this paper, we pay a new visit to an object of hyperbolic geometry which, perhaps, did not draw on itself all the attention it may deserve. The paper gives a simple definition of this object, infinigons, which was implicit in general considerations about filings of the hyperbolic plane, and which was not definied in its all possible extensions. From the simple construction of the infinigons and using the ideas of the splitting method being introduced by the author in the case of tilings being based on the replication of regular polygons, we give an algorithmic construction of the infinigrids. On the way, we give a simple geometrical characterisation of the infinigons in terms of pencils of lines.