  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 $\gamma$.
