[Paper Review] A non FLC regular pentagonal tiling of the plane

Maria Ramirez-Solano|arXiv (Cornell University)|Mar 8, 2013

Quasicrystal Structures and Properties19 references3 citations

TL;DR

This paper demonstrates that a conformally regular pentagonal tiling of the plane, constructed via a substitution rule with complex scaling, is not finite local complexity (FLC) under conformal isomorphisms. Despite being a conformal substitution tiling with well-defined combinatorial structure, the tiling contains an infinite number of distinct tile shapes under conformal equivalence, violating FLC and precluding compactness in the hull under this group action.

ABSTRACT

In this paper we describe the pentagonal tiling of the plane defined in the article "A regular pentagonal tiling of the plane" by P. L. Bowers and K. Stephenson as a conformal substitution tiling and summarize many of its properties given in the mentioned article. We show furthermore why such tiling is not FLC with respect to the set of conformal isomorphisms.

Motivation & Objective

• To characterize the conformally regular pentagonal tiling of the plane as a conformal substitution tiling with complex scaling.
• To analyze the tiling's properties using the framework of topological dynamics and hull construction.
• To investigate whether the tiling satisfies finite local complexity (FLC) under the group of conformal isomorphisms.
• To demonstrate that FLC fails under conformal isomorphisms due to an infinite number of non-conformally equivalent tile shapes.
• To clarify the limitations of FLC in non-isometric settings, particularly in conformal geometry, and to propose alternative notions of complexity.

Proposed method

• Constructs the tiling as a conformal substitution tiling using a complex scaling factor λ = (−324)^{1/5}, derived from the work of Bowers and Stephenson.
• Applies the conformal map Φ to transform the tiling into a geometric realization in the complex plane, with tiles mapped to regions via analytic continuation.
• Uses the substitution rule ω = Φ⁻¹ ∘ α ∘ Φ, where α(z) = λz, to generate successive approximations of supertiles.
• Analyzes convergence of tile boundaries using the Hausdorff metric, showing that normalized tiles converge to a limit shape scaled by b ≈ 1.3.
• Applies the metric d on the orbit set O(T) and constructs the continuous hull Ω_T as the completion of O(T), enabling topological dynamical system analysis.
• Employs a contradiction argument in Theorem 6.2 to show that FLC fails under conformal isomorphisms due to an infinite number of non-similar tiles of bounded diameter.

Experimental results

Research questions

• RQ1 Can the conformally regular pentagonal tiling be described as a conformal substitution tiling with a finite set of prototiles and complex scaling?
• RQ2 Does this tiling satisfy finite local complexity (FLC) with respect to the group of conformal isomorphisms?
• RQ3 What is the geometric and topological behavior of the tiling’s hull under conformal equivalence, and how does it differ from classical FLC?
• RQ4 Why does the standard notion of FLC fail in the conformal setting, and what alternative complexity criteria might apply?
• RQ5 Can combinatorial FLC be preserved under isomorphisms between subcomplexes, even when geometric FLC fails?

Key findings

• The tiling is a conformal substitution tiling generated by a complex scaling factor λ = (−324)^{1/5}, with each prototile replaced by conformally scaled copies.
• The central tile τ₀ is mapped via Φ to a region τ₀ in the complex plane, and successive approximations tₙ = Φ₁(Kₙ) converge to τ₀ in Hausdorff metric.
• Normalized tiles from the sequence (τⱼⁿ) converge in Hausdorff metric to b·b̄τ₀, where b ≈ 1.3, indicating a limit shape under scaling.
• The tiling contains an infinite number of tiles with diameter less than b·diam(τ₀), b ≈ 1.3, and no two are conformally equivalent.
• The tiling fails to satisfy FLC under the group of conformal isomorphisms, as there are infinitely many non-conformally equivalent tiles of bounded size.
• The hull Ω_T is not compact under the standard metric when FLC fails, and the group action loses key topological properties like minimality and transitivity.