2017/07/11 by Graves, Stephen J., Watkins, Mark E.
#05B45 #05C10 #05C12 (Secondary) #05C63 (Primary) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1707.03443
A tessellation of the plane is face-homogeneous if for some integer k≥3 there exists a cyclic sequence σ=[p0,p1,…,pk-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 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," 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.