FRACTAL TILINGS FROM SUBSTITUTION TILINGS

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950009
Author(s):  
XINCHANG WANG ◽  
PEICHANG OUYANG ◽  
KWOKWAI CHUNG ◽  
XIAOGEN ZHAN ◽  
HUA YI ◽  
...  
Keyword(s):  
Short Paper ◽  
Self Similarity ◽  
Substitution Rule ◽  
Substitution Tiling ◽  
Simple Strategy ◽  
Substitution Tilings

A fractal tiling or [Formula: see text]-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of [Formula: see text]-tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule.

scholarly journals On substitution tilings of the plane with n-fold rotational symmetry

2015 ◽  
Vol Vol. 17 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Gregory R. Maloney
Keyword(s):  
Rotational Symmetry ◽  
Substitution Tiling ◽  
Computer Assistance ◽  
Discrete Algorithms ◽  
Special Cases ◽  
International Audience ◽  
Substitution Tilings

Discrete Algorithms International audience A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with elevenfold symmetry appearing in the literature.

scholarly journals Fractal Tilings Based on Successive Adjacent Substitution Rule

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Peichang Ouyang ◽  
Xiaosong Tang ◽  
Kwokwai Chung ◽  
Tao Yu
Keyword(s):  
Self Similarity ◽  
Substitution Rule ◽  
Visual Appeal ◽  
Penrose Tilings ◽  
General Method

A fractal tiling or f-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. f-tilings have complicated structures and strong visual appeal. However, so far, the discovered f-tilings are very limited since constructing such f-tilings needs special talent. Based on the idea of hierarchically subdividing adjacent tiles, this paper presents a general method to generate f-tilings. Penrose tilings are utilized as illustrators to show how to achieve it in detail. This method can be extended to treat a large number of tilings that can be constructed by substitution rule (such as chair and sphinx tilings and Amman tilings). Thus, the proposed method can be used to create a great many of f-tilings.

scholarly journals Cut-and-Project Schemes for Pisot Family Substitution Tilings

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 511 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Shigeki Akiyama ◽  
Yasushi Nagai
Keyword(s):  
Point Spectrum ◽  
The Other ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Internal Space ◽  
Sufficient Condition ◽  
Abstract Space ◽  
Dynamical Spectrum ◽  
Substitution Tiling ◽  
Substitution Tilings

We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m ∈ N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.

scholarly journals Cut-and-Project Schemes for Pisot Family Substitution Tilings

2018 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Shigeki Akiyama ◽  
Yasushi Nagai
Keyword(s):  
Point Spectrum ◽  
The Other ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Internal Space ◽  
Sufficient Condition ◽  
Abstract Space ◽  
Dynamical Spectrum ◽  
Substitution Tiling ◽  
Substitution Tilings

We consider Pisot family substitution tilings in $\R^d$ whose dynamical spectrum is pure point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family property and the other from the pure point spectrum respectively. The first CPS has an internal space $\R^m$ for some integer $m \in \N$ defined from the Pisot family property, and the second CPS has an internal space $H$ which is an abstract space defined from the property of the pure point spectrum. However it is not known how these two CPS's are related. Here we provide a sufficient condition to make a connection between the two CPS's. In the case of Pisot unimodular substitution tiling in $\R$, the two CPS's turn out to be same due to [5, Remark 18.5].

ACTIVE TILE SELF-ASSEMBLY, PART 2: SELF-SIMILAR STRUCTURES AND STRUCTURAL RECURSION

2014 ◽  
Vol 25 (02) ◽  
pp. 165-194 ◽  
Author(s):  
NATAŠA JONOSKA ◽  
DARIA KARPENKO
Keyword(s):  
Self Assembly ◽  
The Self ◽  
Assembly System ◽  
Mathematical Framework ◽  
Assembly Model ◽  
Self Similarity ◽  
Substitution Tiling ◽  
Structural Recursion ◽  
Self Similar

We introduce a mathematical framework to describe self-similarity and structural recursion within the active tile self-assembly model, thereby providing a connection between substitution tiling and algorithmic self-assembly. We show that one such structurally recursive assembly system can simulate the dynamics of the self-similar substitution tiling known as the L-shape tiling.

scholarly journals Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings

2020 ◽  
Vol 76 (5) ◽  
pp. 600-610
Author(s):  
Dong-il Lee ◽  
Shigeki Akiyama ◽  
Jeong-Yup Lee
Keyword(s):  
Discrete Spectrum ◽  
Additional Condition ◽  
Internal Space ◽  
Representative Point ◽  
Primitive Substitution ◽  
Substitution Tiling ◽  
Regular Model ◽  
Model Sets ◽  
Project Scheme ◽  
Substitution Tilings

Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

On the frequency module of the hull of a primitive substitution tiling

2022 ◽  
Vol 78 (1) ◽  
Author(s):  
April Lynne D. Say-awen ◽  
Dirk Frettlöh ◽  
Ma. Louise Antonette N. De Las Peñas
Keyword(s):  
Statistical Properties ◽  
Absolute Frequency ◽  
Aperiodic Tilings ◽  
Primitive Substitution ◽  
Substitution Tiling ◽  
Tiling Spaces ◽  
The Family ◽  
Frequency Module ◽  
Substitution Tilings ◽  
The Absolute

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 {\bb Z}-module, where {\bb Z} 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.

Pinwheel tiling fractal graph- a notion to edge cordial and cordial labeling

2016 ◽  
Vol 5 (2) ◽  
pp. 84
Author(s):  
Sathakathulla A A
Keyword(s):  
Self Similarity ◽  
Geometric Figure ◽  
Substitution Tiling ◽  
Self Similar

<p>A fractal is a complex geometric figure that continues to display self-similarity when viewed on all scales. Tile substitution is the process of repeatedly subdividing shapes according to certain rules. These rules are also sometimes referred to as inflation and deflation rule. One notable example of a substitution tiling is the so-called Pinwheel tiling of the plane. Many examples of self-similar tiling are made of fractiles: tiles with fractal boundaries. . The pinwheel tiling was the first example of this sort. There are many as such as family of tiling fractal curves, but for my study, I have considered this Pinwheel and its two intriguing Pinwheel properties of tiling fractals. These fractals have been considered as a graph and the same has been viewed under the scope of cordial and edge cordial labeling to apply this concept for further study in Engineering and science applications.</p>

scholarly journals Squirals and beyond: substitution tilings with singular continuous spectrum

2013 ◽  
Vol 34 (4) ◽  
pp. 1077-1102 ◽  
Author(s):  
MICHAEL BAAKE ◽  
UWE GRIMM
Keyword(s):  
Dynamical System ◽  
Mixed Type ◽  
Constructive Proof ◽  
Singular Continuous Spectrum ◽  
Dynamical Spectrum ◽  
Substitution Rule ◽  
Continuous Components ◽  
Lattice Dynamical System ◽  
Substitution Tilings ◽  
Inflation Rule

AbstractThe squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of${ \mathbb{Z} }^{2} $. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalization to bijective block substitutions of trivial height on${ \mathbb{Z} }^{d} $.

The promises and perfidy of cryomicroscopy and analysis

1994 ◽  
Vol 52 ◽  
pp. 382-383
Author(s):  
Patrick Echlin
Keyword(s):  
Low Temperature ◽  
High Energy ◽  
Short Paper ◽  
List Type ◽  
Time Resolved ◽  
Energy Beam ◽  
Cellular Analysis ◽  
Dissolved Ions ◽  
Embedding Medium ◽  
Low Temperature Microscopy

The unusual title of this short paper and its accompanying tutorial is deliberate, because the intent is to investigate the effectiveness of low temperature microscopy and analysis as one of the more significant elements of the less interventionist procedures we can use to prepare, examine and analyse hydrated and organic materials in high energy beam instruments. The promises offered by all these procedures are well rehearsed and the litany of petitions and responses may be enunciated in the following mantra.Vitrified water can form the perfect embedding medium for bio-organic samples.Frozen samples provide an important, but not exclusive, milieu for the in situ sub-cellular analysis of the dissolved ions and electrolytes whose activities are central to living processes.The rapid conversion of liquids to solids provides a means of arresting dynamic processes and permits resolution of the time resolved interactions between water and suspended and dissolved materials.The low temperature environment necessary for cryomicroscopy and analysis, diminish, but alas do not prevent, the deleterious side effects of ionizing radiation.Sample contamination is virtually eliminated.

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