Circle Packings from Tilings of the Plane

TL;DR

This paper introduces a new class of fractal circle packings derived from plane tilings, extending previous polyhedral packings, and explores their symmetry groups and mathematical properties.

Contribution

It generalizes existing circle packings using tilings, proves existence and uniqueness via infinite Koebe-Andreev-Thurston theorems, and analyzes their symmetry groups.

Findings

Established existence and uniqueness of new fractal circle packings

Provided structure theorems for symmetry groups of these packings

Illustrated number-theoretic and group-theoretic significance with examples

Abstract

We introduce a new class of fractal circle packings in the plane, generalizing the polyhedral packings defined by Kontorovich and Nakamura. The existence and uniqueness of these packings are guaranteed by infinite versions of the Koebe-Andreev-Thurston theorem. We prove structure theorems giving a complete description of the symmetry groups for these packings. And we give several examples to illustrate their number-theoretic and group-theoretic significance.