Growth of Edge-Homogeneous Tessellations

Graves, Stephen J.; Pisanski, Tomaž; Watkins, Marley W.

doi:10.1137/070707026

Cited by 6 publications

(29 citation statements)

References 5 publications

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“…Although it was shown in [6] (see pages 3-4) that, for any connected planar map with bounded covalences, the above definition of growth rate is equivalent to the growth rate with respect to the standard graph-theoretic metric, we need to show that said growth rate is independent of the root of the Bilinski diagram in question. Proof.…”

Section: Growth Formulasmentioning

confidence: 99%

“…When M = [m i,j ] is such a matrix, the entry m i,j is the number of faces of type f j that are offspring of a face of type f i . We require the following result from [6]. …”

Section: Accretionmentioning

confidence: 99%

“…A general sufficient condition for polymorphism is given. The difficulties posed by polymorphism are illustrated by an example; the polymorphic sequence [4,4,6,8] is considered in some depth in Subsection 4.2. In particular, we see by this example that two different tessellations having the same (polymorphic) valence sequence may well have different growth rates.…”

Section: Introductionmentioning

confidence: 99%

“…Subsection 2.5 presents a rigorous theoretical treatment of growth rate with respect to regional distance in a Bilinski diagram. Subsection 2.6 concludes the Preliminaries with a review of the completely resolved case of edge-homogeneous tessellations, summarizing results from [8] and [6].…”

Section: Introductionmentioning

confidence: 99%

“…In [6] they determined the growth rate of any edge-homogeneous tessellation as a function of its edge-symbol and proved that the least growth rate for an exponentially-growing, edge-homogeneous tessellation is 1 2 (3 + √ 5) ≈ 2.618. The goal of this article is to obtain an analogous result for face-homogeneous tessellations.…”

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.

show abstract

Section: Growth Formulasmentioning

confidence: 99%

“…When M = [m i,j ] is such a matrix, the entry m i,j is the number of faces of type f j that are offspring of a face of type f i . We require the following result from [6]. …”

Section: Accretionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Growth of face-homogeneous tessellations

Graves

Watkins

2017

Ars Math. Contemp.

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Tessellations with arbitrary growth rates

Graves

2010

Discrete Mathematics

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Edge-transitive maps of low genus

et al. 2011

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Graver and Watkins classified edge-transitive maps on closed surfaces into fourteen types. In this note we study these types for maps in orientable and non-orientable surfaces of small genus, including the Euclidean and hyperbolic plane. We revisit both finite and infinite one-ended edge-transitive maps. For the finite ones we give precise description that should enable their enumeration for a given number of edges. Edge-transitive maps on surfaces with small genera are classified in the paper.

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Growth of Edge-Homogeneous Tessellations

Cited by 6 publications

References 5 publications

Growth of face-homogeneous tessellations

Growth of face-homogeneous tessellations

Tessellations with arbitrary growth rates

Edge-transitive maps of low genus

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