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    Growth of Face-Homogeneous Tessellations

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    A tessellation of the plane is face-homogeneous if for some integer kβ‰₯3k\geq3 there exists a cyclic sequence Οƒ=[p0,p1,…,pkβˆ’1]\sigma=[p_0,p_1,\ldots,p_{k-1}] of integers β‰₯3\geq3 such that, for every face ff of the tessellation, the valences of the vertices incident with ff are given by the terms of Οƒ\sigma in either clockwise or counter-clockwise order. When a given cyclic sequence Οƒ\sigma is realizable in this way, it may determine a unique tessellation (up to isomorphism), in which case Οƒ\sigma 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," Ξ³=(1+5)/2\gamma=(1+\sqrt{5})/2, attained by the sequences [4,6,14][4,6,14] and [3,4,7,4][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 Ξ³\gamma.Comment: Article 32 pages, appendix 44 pages. Article to appear without appendix in Ars Mathematica Contemporane
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