Counting Polygons and Fishes in Uniform Tessellations of the Hyperbolic Plane

Abstract

SummaryIn this article we consider uniform tessellations of the hyperbolic plane. Two counts are performed. The first, considers the number of polygons in layers of the hyptagrid and is shown to be related to Fibonacci numbers. The second, considers the number of fishes in layers of a superimposed octagrid on Escher’s circle limit III. To excute these counts we solve two linear recurrence relations, homogeneous and non-homogeneous. The initial conditions are set up by performing tessellations using a software in hyperbolic geometry.MSC: 51M10 AcknowledgmentsThe author is indebted to the Editor and an anonymous referee for their valuable comments that substantially improved the exposition of the paper. Special thanks are also due to Douglas Dunham for the permission to use his recreation of Escher’s Circle Limit III.Notes1 Note that the online version of this article has color diagrams.Additional informationNotes on contributorsElias AbboudELIAS ABBOUD (MR Author ID: 249090) received his D.Sc from the Technion-Haifa, Israel. Since 1992, he has taught Mathematics at Beit Berl College. Between the years 2010–2017. he served as the Math Chair in the Arab Academic Institution within the Faculty of Education of Beit Berl College. Since 2001, he also works partially at the Academic Arab College of Education-Haifa.