Coincidences in numbers of graph vertices corresponding to regular planar hyperbolic mosaics

The aim of this paper is to determine the elements which are in two pairs of sequences linked to the regular mosaics $\{4,5\}$ and $\{p,q\}$ on the hyperbolic plane. The problem leads to the solution of diophantine equations of certain types.Comment: 10 pages, 2 figures, Annales Mathematicae et Informaticae 43 (2014

The growing ratios of hyperbolic regular mosaics with bounded cells

In 3- and 4-dimensional hyperbolic spaces there are four, respectively five, regular mosaics with bounded cells. A belt can be created around an arbitrary base vertex of a mosaic. The construction can be iterated and a growing ratio can be determined by using the number of the cells of the considered belts. In this article we determine these growing ratios for each mosaic in a generalized way.Comment: 17 pages, 3 figure

Fibonacci words in hyperbolic Pascal triangles

The hyperbolic Pascal triangle ${\cal HPT}_{4,q}$ $(q\ge5)$ is a new mathematical construction, which is a geometrical generalization of Pascal's arithmetical triangle. In the present study we show that a natural pattern of rows of ${\cal HPT}_{4,5}$ is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the hyperbolic Pascal triangles. The geometrical properties of a ${\cal HPT}_{4,q}$ imply a graph structure between the finite Fibonacci words.Comment: 10 pages, 4 figures, Acta Univ. Sapientiae, Mathematica, 201