1 research outputs found
Growth of Face-Homogeneous Tessellations
A tessellation of the plane is face-homogeneous if for some integer
there exists a cyclic sequence of integers
such that, for every face of the tessellation, the valences of the
vertices incident with are given by the terms of in either
clockwise or counter-clockwise order. When a given cyclic sequence is
realizable in this way, it may determine a unique tessellation (up to
isomorphism), in which case 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," , attained by the sequences and
. A polymorphic sequence may yield non-isomorphic tessellations with
different growth rates. However, all such tessellations found thus far grow at
rates greater than .Comment: Article 32 pages, appendix 44 pages. Article to appear without
appendix in Ars Mathematica Contemporane