Farey Bryophylla

TL;DR

This paper introduces conformal bryophylla as a new class of fractal-like sets on the boundary of hyperbolic 3-space, extending Farey tessellation concepts to higher dimensions and classifying their properties.

Contribution

It presents the first steps towards generalizing Farey tessellation to three-dimensional hyperbolic space through conformal bryophylla and classifies these sets.

Findings

Classified all conformal bryophylla.

Analyzed properties of their limiting sets.

Established parallels with Farey tessellation.

Abstract

The construction of the Farey tessellation in the hyperbolic plane starts with a finitely generated group of symmetries of an ideal triangle, i.e. a triangle with all vertices on the boundary. It induces a remarkable fractal structure on the boundary of the hyperbolic plane, encoding every element by the continued fraction related to the structure of the tessellation. The problem of finding a generalisation of this construction to the higher dimensional hyperbolic spaces has remained open for many years. In this paper we make the first steps towards a generalisation in the three-dimensional case. We introduce conformal bryophylla, a class of subsets of the boundary of the hyperbolic 3-space which possess fractal properties similar to the Farey tessellation. We classify all conformal bryophylla and study the properties of their limiting sets.