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 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.
@article{862,
title = {Growth of face-homogeneous tessellations},
journal = {ARS MATHEMATICA CONTEMPORANEA},
volume = {14},
year = {2017},
doi = {10.26493/1855-3974.862.bb5},
language = {EN},
url = {http://dml.mathdoc.fr/item/862}
}
Graves, Stephen J.; Watkins, Mark E. Growth of face-homogeneous tessellations. ARS MATHEMATICA CONTEMPORANEA, Tome 14 (2017) . doi : 10.26493/1855-3974.862.bb5. http://gdmltest.u-ga.fr/item/862/