Vertices of self-similar tiles

Deng, Da-Wen; Ngai, Sze-Man

doi:10.1215/ijm/1258138223

Cited by 7 publications

(5 citation statements)

References 13 publications

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“…Now we shall see that most examples from fractal geometry have a simple automatic structure. Self-affine tiles will not be mentioned since their automata are well described in [1,13,28,29,30,31,42,47]. Sierpinski gasket and tetrahedron.…”

Section: More Examplesmentioning

confidence: 99%

“…In this case [9,6,36], as well as in the case of self-affine tilings [1,13,28,29,30,31,42,47], a topology-generating automaton can be determined directly from the IFS. It was termed neighbor map in [9] since the states are neighbor maps of the form h = f −1 u f v .…”

Section: Propositionmentioning

confidence: 99%

“…This work was extended in numerous papers on canonical number systems (cf. [2] and [18, Section 2.4]) and on the topology of self-affine tiles in the plane [1,13,28,29,30,31,42] and in space [47]. Roughly speaking, self-affine tiles are the unit intervals of canonical number systems.…”

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confidence: 99%

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Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Bandt

Mekhontsev

2022

Fractal Fract

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Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.

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Section: More Examplesmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

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Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Bandt

Mekhontsev

2022

Fractal Fract

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“…1. If A is similar to a similarity, then by using the method of Strichartz and Wang 17 (see also 6, 9), the Hausdorff dimension of the intersections can be computed. If A is not similar to a similarity, we do not know how to compute the Hausdorff dimension of the intersections.…”

Section: Comments and Questionsmentioning

confidence: 99%

Structure of planar integral self‐affine tilings

Deng

Jiang

Ngai

2011

Mathematische Nachrichten

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For a self-affine tile in R 2 generated by an expanding matrix A ∈ M2 (Z) and an integral consecutive collinear digit set D, Leung and Lau [Trans. Amer. Math. Soc. 359, 3337-3355 (2007).] provided a necessary and sufficient algebraic condition for it to be disklike. They also characterized the neighborhood structure of all disklike tiles in terms of the algebraic data A and D. In this paper, we completely characterize the neighborhood structure of those non-disklike tiles. While disklike tiles can only have either six or eight edge or vertex neighbors, non-disklike tiles have much richer neighborhood structure. In particular, other than a finite set, a Cantor set, or a set containing a nontrivial continuum, neighbors can intersect in a union of a Cantor set and a countable set.

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“…Deng and Ngai [3] calculated the dimensions of subsets of the boundary of the Levy dragon which are the common intersection of at least a given number of neighbors in the tiling of the plane by copies of the Levy dragon. Ngai and Tang [8] proved that the closure of each interior component of the Levy dragon is a topological disk.…”

Section: Introductionmentioning

confidence: 99%

The Finite Number of Interior Component Shapes of the Levy Dragon

Alster¹

2009

Discrete Comput Geom

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The Levy dragon is a connected self-similar tile with disconnected interior. It was previously known that there are at least 16 different shapes of its interior components. Using simple properties of an infinite sequence of curves which converge into the Levy dragon, it is proved that the number of different shapes of the interior components is finite. A detailed description of the buildup of those shapes as unions of various contractions of three convex polygonal shapes is given, and the number of shapes is determined.

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Vertices of self-similar tiles

Cited by 7 publications

References 13 publications

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Structure of planar integral self‐affine tilings

The Finite Number of Interior Component Shapes of the Levy Dragon

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