### Growth of face-homogeneous tessellations

#### Abstract

A tessellation of the plane is

*face-homogeneous*if for some integer*k*≥ 3 there exists a cyclic sequence*σ*= [*p*_{0},*p*_{1}, …,*p*_{k − 1}] of integers ≥ 3 such that, for every face*f*of the tessellation, the valences of the vertices incident with*f*are given by the terms of*σ*in either clockwise or counter-clockwise order. When a given cyclic sequence*σ*is realizable in this way, it may determine a unique tessellation (up to isomorphism), in which case*σ*is called*monomorphic*, or it may be the valence sequence of two or more non-isomorphic tessellations (*polymorphic*). A tessellation whose faces are uniformly bounded in the hyperbolic plane but not uniformly bounded in the Euclidean plane is called a*hyperbolic tessellation*. Such tessellations are well-known to have exponential growth. We seek the face-homogeneous hyperbolic tessellation(s) of slowest growth rate and show that the least growth rate of such monomorphic tessellations is the “golden mean,” γ = (1+√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*γ*.#### Keywords

Face-homogeneous, tessellation, growth rate, valence sequence, exponential growth, transition matrix, Bilinski diagram, hyperbolic plane

DOI: https://doi.org/10.26493/1855-3974.862.bb5

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications