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 .