Tessellations in the Hyperbolic Plane

Abstract

In this paper a study of hyperbolic tessellations is presented, with emphasis on regular tessellations. The tessellations have been present in our daily lives for a long time, since 5000 a.C, in bee hives and pot decorations, for example, but in the 16th century, with the studies of Johannes Kepler, German mathematician and physicist, these pavimentations left to have a purely aesthetic character and come to be seen as a mathematical study, with several applications. When studying regular tessellations in the Euclidean plane, it is noticed that there are a limited number of them, more precisely, there are only three of them in this plane. Thus there arises an expansion about the study of regular tessellations, now on the hyperbolic plane, since in this plane there are infinite possibilities, further enriching the study of tessellations. Then,
after a general study on hyperbolic geometry, two well-known Euclidean models of this geometry were considered: the upper half-plane and the Poincar ́e disc. These models provide a better visualization of the hyperbolic plane.