Growth of Face-Homogeneous Tessellations

TL;DR

This paper investigates face-homogeneous tessellations in hyperbolic geometry, identifying those with the slowest exponential growth rate, specifically the golden mean, and explores the differences between monomorphic and polymorphic sequences.

Contribution

It determines the minimal growth rate of monomorphic face-homogeneous hyperbolic tessellations and analyzes the growth rates of polymorphic sequences, expanding understanding of tessellation classifications.

Findings

Minimal growth rate is the golden mean, b3=(1+1/2)

Sequences [4,6,14] and [3,4,7,4] achieve this minimal growth

Polymorphic tessellations grow faster than the minimal rate

Abstract

A tessellation of the plane is face-homogeneous if for some integer $k\geq3$ there exists a cyclic sequence $\sigma=[p_0,p_1,\ldots,p_{k-1}]$ of integers $\geq3$ such that, for every face $f$ of the tessellation, the valences of the vertices incident with $f$ are given by the terms of $\sigma$ in either clockwise or counter-clockwise order. When a given cyclic sequence $\sigma$ is realizable in this way, it may determine a unique tessellation (up to isomorphism), in which case $\sigma$ is called monomorphic, or it may be the valence sequence of two or more non-isomorphic tessellations (polymorphic). A tessellation which whose faces are uniformly bounded in the Euclidean plane is called a Euclidean tessellation; a non-Euclidean tessellation whose faces are uniformly bounded in the hyperbolic plane is called hyperbolic. Hyperbolic tessellations are well-known to have exponential growth. We seek the face-homogeneous hyperbolic tessellation(s) of slowest growth and show that the least growth rate of monomorphic face-homogeneous tessellations is the "golden mean," $\gamma=(1+\sqrt{5})/2$, attained by the sequences $[4,6,14]$ and $[3,4,7,4]$. A polymorphic sequence may yield non-isomorphic tessellations with different growth rates. However, all such tessellations found thus far grow at rates greater than $\gamma$.