Isoperimetric Formulas for Hyperbolic Animals

TL;DR

This paper derives isoperimetric formulas for hyperbolic animals, extending Euclidean results by linking hyperbolic tessellations, Sturmian words, and discrete ball parameters, revealing new complexities.

Contribution

It introduces hyperbolic animal isoperimetric formulas and connects Sturmian words to hyperbolic tessellation structures, a novel approach in geometric combinatorics.

Findings

Derived isoperimetric formulas for hyperbolic animals.

Established a connection between Sturmian words and hyperbolic tessellations.

Revealed new complexities in hyperbolic animals not seen in Euclidean cases.

Abstract

An animal is a planar shape formed by attaching congruent regular polygons along their edges. In 1976, Harary and Harborth gave closed isoperimetric formulas for Euclidean animals. Here, we provide analogous formulas for hyperbolic animals. We do this by proving a connection between Sturmian words and the parameters of a discrete analogue of balls in the graph determined by hyperbolic tessellations. This reveals a complexity in hyperbolic animals that is not present in Euclidean animals.