On the Dedekind tessellation

TL;DR

This paper reinterprets the Dedekind tessellation by characterizing circles algebraically through modular group actions, linking geometry, group theory, and number theory, and providing algorithms for visualization.

Contribution

It introduces a novel algebraic characterization of circles in the Dedekind tessellation using Lorentz transformations, expanding the understanding beyond ideal triangles.

Findings

Complete algebraic description of circles in the tessellation

Algorithms for computer visualization of the tessellation

New insights into the interplay of geometry, group theory, and number theory

Abstract

The Dedekind tessellation -- the regular tessellation of the upper half-plane by the Mobius action of the modular group -- is usually viewed as a system of ideal triangles. We change the focus from triangles to circles and give their complete algebraic characterization with the help of a representation of the modular group acting by Lorentz transformations on Minkowski space. This interesting example of the interplay of geometry, group theory and number theory leads also to convenient algorithms for computer drawing of the Dedekind tessellation.