This work considers the scaling properties characterizing the hyperuniformity (or anti‐hyperuniformity) of long‐wavelength fluctuations in a broad class of one‐dimensional substitution tilings. A ...simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit‐periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between −1 and 3. Limit‐periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law.
This work examines the long‐wavelength scaling properties of self‐similar substitution tilings, placing them in their hyperuniformity classes. Quasiperiodic, non‐PV (Pisot–Vijayaraghavan number) and limit‐periodic examples are analyzed. Novel behavior is demonstrated for certain limit‐periodic cases.
k‐Isocoronal tilings Taganap, Eduard; De Las Peñas, Ma. Louise Antonette
Acta crystallographica. Section A, Foundations and advances,
January 2019, 2019-Jan-01, 2019-01-01, 20190101, Volume:
75, Issue:
1
Journal Article
Peer reviewed
Open access
In this article, a framework is presented that allows the systematic derivation of planar edge‐to‐edge k‐isocoronal tilings from tile‐s‐transitive tilings, s ≤ k. A tiling is k‐isocoronal if its ...vertex coronae form k orbits or k transitivity classes under the action of its symmetry group. The vertex corona of a vertex x of is used to refer to the tiles that are incident to x. The k‐isocoronal tilings include the vertex‐k‐transitive tilings (k‐isogonal) and k‐uniform tilings. In a vertex‐k‐transitive tiling, the vertices form k transitivity classes under its symmetry group. If this tiling consists of regular polygons then it is k‐uniform. This article also presents the classification of isocoronal tilings in the Euclidean plane.
This article presents a method to determine planar edge‐to‐edge k‐isocoronal tilings – tilings whose vertex coronae form k orbits or k transitivity classes under the action of the symmetry group.
A brief introductory review is provided of the theory of tilings of 3‐periodic nets and related periodic surfaces. Tilings have a transitivity p q r s indicating the vertex, edge, face and tile ...transitivity. Proper, natural and minimal‐transitivity tilings of nets are described. Essential rings are used for finding the minimal‐transitivity tiling for a given net. Tiling theory is used to find all edge‐ and face‐transitive tilings (q = r = 1) and to find seven, one, one and 12 examples of tilings with transitivity 1 1 1 1, 1 1 1 2, 2 1 1 1 and 2 1 1 2, respectively. These are all minimal‐transitivity tilings. This work identifies the 3‐periodic surfaces defined by the nets of the tiling and its dual and indicates how 3‐periodic nets arise from tilings of those surfaces.
After a brief review of tilings of 3‐periodic nets, the use of essential rings is proposed to identify transitive tilings.
We study substitution tilings that are also discrete plane tilings, that is, satisfy a relaxed version of cut-and-projection. We prove that the Sub Rosa substitution tilings with a 2
n
-fold ...rotational symmetry for odd
n
>
5
defined by Kari and Rissanen are not discrete planes—and therefore not cut-and-project tilings either. We then define new Planar Rosa substitution tilings with a 2
n
-fold rotational symmetry for any odd
n
, and show that these satisfy the discrete plane condition. The tilings we consider are edge-to-edge rhombus tilings. We give an explicit construction for the 10-fold case, and provide a construction method for the general case of any odd
n
. Our methods are to lift the tilings and substitutions to
R
n
using the lift operator first defined by Levitov, and to study the planarity of substitution tilings in
R
n
using mainly linear algebra, properties of circulant matrices, and trigonometric sums. For the construction of the Planar Rosa substitutions we additionally use the Kenyon criterion and a result on De Bruijn multigrid dual tilings.
On the Structure of Ammann A2 Tilings Durand, Bruno; Shen, Alexander; Vereshchagin, Nikolay
Discrete & computational geometry,
04/2020, Volume:
63, Issue:
3
Journal Article
Peer reviewed
Open access
We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in the sense of Solomyak (Discret Comput Geom ...20:265–279,
1998
). By the same techniques we show that Ammann A2 tilings are not robust in the sense of Durand et al. (J Comput Syst Sci 78(3):731–764,
2012
).
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same
vertex-type
, which is a cyclic tuple of integers that ...determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.
A coloring of a planar semiregular tiling {\cal T} is an assignment of a unique color to each tile of {\cal T}. If
G
is the symmetry group of {\cal T}, the coloring is said to be perfect if every ...element of
G
induces a permutation on the finite set of colors. If {\cal T} is
k
-valent, then a coloring of {\cal T} with
k
colors is said to be precise if no two tiles of {\cal T} sharing the same vertex have the same color. In this work, perfect precise colorings are obtained for some families of
k
-valent semiregular tilings in the plane, where
k
≤ 6.
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More ...specifically, we generalize results on Delaney–Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney–Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
On the frequency module of the hull of a primitive substitution tiling Say-awen, April Lynne D.; Frettlöh, Dirk; De Las Peñas, Ma. Louise Antonette N.
Acta crystallographica. Section A, Foundations and advances,
January 2022, 2022-Jan-01, 2022-01-01, 20220101, Volume:
78, Issue:
1
Journal Article
Peer reviewed
Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive ...substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal ‐module, where is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.
As part of the study of aperiodic tilings and tiling spaces, the frequency module of the hull of a primitive substitution tiling is computed.
Primitive substitution tilings with rotational symmetries Say-awen, April Lynne D.; De Las Peñas, Ma. Louise Antonette N.; Frettlöh, Dirk
Acta crystallographica. Section A, Foundations and advances,
July 2018, 2018-Jul-01, 2018-07-01, 20180701, Volume:
74, Issue:
4
Journal Article
Peer reviewed
This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are given of the substitutions σ6 and σ7 ...which give rise to aperiodic primitive substitution tilings with dense tile orientations and which are invariant under six‐ and sevenfold rotations, respectively; the derivation of the symmetry orders of their hulls is also presented.
The idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution, is introduced. Two substitutions giving rise to six‐ and sevenfold rotation‐invariant tilings are also presented.
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