Growth of Face-Homogeneous Tessellations

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 $\gamma$.