Tessellations with arbitrary growth rates

Graves, Stephen J.

doi:10.1016/j.disc.2010.04.023

Cited by 2 publications

(7 citation statements)

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“…In this contribution we have explored some properties of regular hyperbolic tilings and identified all cases where the Golden Section appears as the fraction of peripheral cells in the bulk of a tiling. We found that there are only two such tilings, the dual pair (7, 3) and (3,7). In all other possible cases the fraction of periphery in the total exceeds ϕ.…”

Section: Discussionmentioning

confidence: 81%

The Golden Section's heptagonal connections

Došlić¹

2021

Proceedings of the 3rd Croatian Combinatorial Days

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It is well known that the Golden Section plays an important role in the geometry of several polygons and polyhedra; the best known example is the length of a diagonal in the regular pentagon with unit side. In this contribution we show how the Golden Section appears as the solution of an enumerative problem connected with heptagons, more precisely, with heptagonal tilings of the hyperbolic plane. The results are then generalized by investigating whether it also appears in other types of hyperbolic tilings.

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Section: Discussionmentioning

confidence: 81%

The Golden Section's heptagonal connections

Došlić¹

2021

Proceedings of the 3rd Croatian Combinatorial Days

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“…The minimal polymorphic valence sequence under the partial order on cyclic sequences, namely [4,4,4,5], is unfortunately not amenable to study via our methods. In fact, there is no well-defined transition matrix between coronas, and this problem is shared by all valence sequences of the form [4, 4, 4, q] for q > 4.…”

Section: Two Non-isomorphic Tessellations With the Same Valence Sequencementioning

confidence: 99%

“…In the previous example, we constructed the sequence in the process of transforming T 1 with valence sequence [4,5,4,5] into T 2 with valence sequence [4,6,6,4,5]; however, the process of creating {T j : j ∈ N 0 } is identical in any case where T 1 and T 2 are facehomogeneous and uniformly concentric with monomorphic valence sequences σ 1 and σ 2 , respectively, where σ 1 < σ 2 . Thus by Lemma 2.18, we obtain the following result.…”

Section: Monomorphic Uniformly Concentric Sequencesmentioning

confidence: 99%

“…With respect to the ordering of cyclic sequences, the least polymorphic valence sequence with positive angle excess is [4,4,4,5]; that is to say, every cyclic sequence σ such that σ < [4,4,4,5] is either not realizable as a tessellation, is realizable only by a finite map or a Euclidean tessellation, or is monomorphic. While all valence sequence of length 3 are monomorphic, k-covalent polymorphic valence sequences abound for k ≥ 4.…”

Section: Polymorphic Valence Sequences 41 a Sufficient Condition Formentioning

confidence: 99%

“…In this schema, all Euclidean tessellations have growth rate equal to 1, and hyperbolic tessellations have growth rate strictly greater than 1. The first author has shown by construction in [5] that, given any > 0, there exists a hyperbolic tessellation with growth rate exactly 1 + . In general, these latter tessellations have few if any combinatorial or geometric symmetries.…”

Section: Introductionmentioning

confidence: 99%

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Growth of face-homogeneous tessellations

Graves

Watkins

2017

Ars Math. Contemp.

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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.

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Tessellations with arbitrary growth rates

Cited by 2 publications

References 3 publications

The Golden Section's heptagonal connections

The Golden Section's heptagonal connections

Growth of face-homogeneous tessellations

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