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Symmetry
Volume 9
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Symmetry, Volume 9, Issue 2 (February 2017) – 9 articles

Cover Story (view full-size image): We introduced the class of Cyclotomic Aperiodic Substitution Tilings (CASTs) which covers a wide range of new and well known substitution tilings, e.g. those discovered by R. Penrose, R. Ammann, F.P.M. Beenker and L. Danzer. We investigated general properties, detailed substitution matrices and minimal inflation multipliers, as well as practical use cases to identify new specimens with individual dihedral symmetry Dn or D2n.
The figure shows an aperiodic rhomb substitution tiling (based on the substitution rules in Figure 14, example n = 6) with individual dihedral symmetry D12 and minimal inflation multiplier under the given boundary conditions. While for “odd n”, the identification of substitution rules is a rather straight forward exercise, the “even n” case tends to be more complex. View the paper
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256 KiB  
Article
Symmetry Analysis and Conservation Laws of the Zoomeron Equation
by Tanki Motsepa, Chaudry Masood Khalique and Maria Luz Gandarias
Symmetry 2017, 9(2), 27; https://doi.org/10.3390/sym9020027 - 21 Feb 2017
Cited by 37 | Viewed by 4867
Abstract
In this work, we study the (2 + 1)-dimensional Zoomeron equation which is an extension of the famous (1 + 1)-dimensional Zoomeron equation that has been studied extensively in the literature. Using classical Lie point symmetries admitted by the equation, for the first [...] Read more.
In this work, we study the (2 + 1)-dimensional Zoomeron equation which is an extension of the famous (1 + 1)-dimensional Zoomeron equation that has been studied extensively in the literature. Using classical Lie point symmetries admitted by the equation, for the first time we develop an optimal system of one-dimensional subalgebras. Based on this optimal system, we obtain symmetry reductions and new group-invariant solutions. Again for the first time, we construct the conservation laws of the underlying equation using the multiplier method. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models)
1601 KiB  
Article
A Symmetry Particle Method towards Implicit Non‐Newtonian Fluids
by Yalan Zhang, Xiaojuan Ban, Xiaokun Wang and Xing Liu
Symmetry 2017, 9(2), 26; https://doi.org/10.3390/sym9020026 - 17 Feb 2017
Cited by 8 | Viewed by 6249
Abstract
In this paper, a symmetry particle method, the smoothed particle hydrodynamics (SPH) method, is extended to deal with non‐Newtonian fluids. First, the viscous liquid is modeled by a non‐Newtonian fluid flow and the variable viscosity under shear stress is determined by the Carreau‐Yasuda [...] Read more.
In this paper, a symmetry particle method, the smoothed particle hydrodynamics (SPH) method, is extended to deal with non‐Newtonian fluids. First, the viscous liquid is modeled by a non‐Newtonian fluid flow and the variable viscosity under shear stress is determined by the Carreau‐Yasuda model. Then a pressure correction method is proposed, by correcting density error with individual stiffness parameters for each particle, to ensure the incompressibility of fluid. Finally, an implicit method is used to improve efficiency and stability. It is found that the nonNewtonian behavior can be well displayed in all cases, and the proposed SPH algorithm is stable and efficient. Full article
(This article belongs to the Special Issue Symmetry in Cooperative Applications II)
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Figure 1

Figure 1
<p>Viscous forces model. V: velocity.</p> Full article ">Figure 2
<p>Illustration for smoothed particle hydrodynamics (SPHs) method. It is a symmetry particle method and the physical attributions of each particle is weighted in summation by its neighbors.</p> Full article ">Figure 3
<p>A falling water column with our method. (<b>a</b>) is Newtonian fluid; (<b>b</b>) is the non-Newtonian fluid.</p> Full article ">Figure 4
<p>The free fall of a water column. (<b>a</b>) is Newtonian fluid (our method); (<b>b</b>) is the non-Newtonian fluid (Andrade et al. [<a href="#B25-symmetry-09-00026" class="html-bibr">25</a>]) with <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo> </mo> <mi>s</mi> </mrow> </semantics> </math>; (<b>c</b>) is the non-Newtonian fluid (Andrade et al. [<a href="#B25-symmetry-09-00026" class="html-bibr">25</a>]) with <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>t</mi> <mo>=</mo> <mn>3.0</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo> </mo> <mi>s</mi> </mrow> </semantics> </math>; (<b>d</b>) is the non-Newtonian fluid (our method) with <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>t</mi> <mo>=</mo> <mn>3.0</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo> </mo> <mi>s</mi> </mrow> </semantics> </math>.</p> Full article ">Figure 5
<p>A water block falling in a pool. (<b>a</b>) is Newtonian fluid; (<b>b</b>) is the non-Newtonian fluid.</p> Full article ">
3385 KiB  
Article
Deformable Object Matching Algorithm Using Fast Agglomerative Binary Search Tree Clustering
by Jaehyup Jeong, Insu Won, Hunjun Yang, Bowon Lee and Dongseok Jeong
Symmetry 2017, 9(2), 25; https://doi.org/10.3390/sym9020025 - 10 Feb 2017
Cited by 5 | Viewed by 5024
Abstract
Deformable objects have changeable shapes and they require a different method of matching algorithm compared to rigid objects. This paper proposes a fast and robust deformable object matching algorithm. First, robust feature points are selected using a statistical characteristic to obtain the feature [...] Read more.
Deformable objects have changeable shapes and they require a different method of matching algorithm compared to rigid objects. This paper proposes a fast and robust deformable object matching algorithm. First, robust feature points are selected using a statistical characteristic to obtain the feature points with the extraction method. Next, matching pairs are composed by the feature point matching of two images using the matching method. Rapid clustering is performed using the BST (Binary Search Tree) method by obtaining the geometric similarity between the matching pairs. Finally, the matching of the two images is determined after verifying the suitability of the composed cluster. An experiment with five different image sets with deformable objects confirmed the superior robustness and independence of the proposed algorithm while demonstrating up to 60 times faster matching speed compared to the conventional deformable object matching algorithms. Full article
(This article belongs to the Special Issue Symmetry in Complex Networks II)
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Figure 1

Figure 1
<p>Flowchart of the proposed algorithm.</p> Full article ">Figure 2
<p>Example of the feature points in an image: (<b>a</b>) feature points using only SIFT; and (<b>b</b>) the feature points using feature selection.</p> Full article ">Figure 3
<p>Example of matching pairs that overlap or not.</p> Full article ">Figure 4
<p>Comparison example of a transform matrix (<span class="html-italic">T<sub>i</sub></span>): (<b>a</b>) rigid object in the images; and (<b>b</b>) deformable object in the images.</p> Full article ">Figure 5
<p>Example of a symmetric similarity matrix (<span class="html-italic">N<sub>M</sub></span> = 5).</p> Full article ">Figure 6
<p>Example of binary search tree (<span class="html-italic">t</span> = 5). The circles in blue indicate the nodes in BT<span class="html-italic"><sub>t</sub></span> and the oval in purple indicate two candidate node {<span class="html-italic">i</span> = 8, <span class="html-italic">j</span> = 35}. (<b>a</b>) Node 8 is searched in BT<sub>0</sub> (red dotted arrow and circle); (<b>b</b>) Node 35 is inserted as a new leaf node in BT<sub>0</sub> (red solid arrow and red number in the circle).</p> Full article ">Figure 7
<p>Examples of mismatching results without using cluster verification.</p> Full article ">Figure 8
<p>Examples of reference and query (deformable) images: (<b>a</b>) clothes; (<b>b</b>) snack packs; (<b>c</b>) SMVS (using TPS); (<b>d</b>) IN-Natural (using TPS); and (<b>e</b>) Oxbulid (using TPS).</p> Full article ">Figure 9
<p>Accuracy of the proposed and other algorithms.</p> Full article ">Figure 10
<p>Recall vs. Precision curve of the proposed and other algorithms.</p> Full article ">Figure 11
<p>Examples of matching results using proposed algorithm.</p> Full article ">Figure 11 Cont.
<p>Examples of matching results using proposed algorithm.</p> Full article ">
1308 KiB  
Article
Single Image Super-Resolution by Non-Linear Sparse Representation and Support Vector Regression
by Yungang Zhang and Jieming Ma
Symmetry 2017, 9(2), 24; https://doi.org/10.3390/sym9020024 - 10 Feb 2017
Cited by 8 | Viewed by 4769
Abstract
Sparse representations are widely used tools in image super-resolution (SR) tasks. In the sparsity-based SR methods, linear sparse representations are often used for image description. However, the non-linear data distributions in images might not be well represented by linear sparse models. Moreover, many [...] Read more.
Sparse representations are widely used tools in image super-resolution (SR) tasks. In the sparsity-based SR methods, linear sparse representations are often used for image description. However, the non-linear data distributions in images might not be well represented by linear sparse models. Moreover, many sparsity-based SR methods require the image patch self-similarity assumption; however, the assumption may not always hold. In this paper, we propose a novel method for single image super-resolution (SISR). Unlike most prior sparsity-based SR methods, the proposed method uses non-linear sparse representation to enhance the description of the non-linear information in images, and the proposed framework does not need to assume the self-similarity of image patches. Based on the minimum reconstruction errors, support vector regression (SVR) is applied for predicting the SR image. The proposed method was evaluated on various benchmark images, and promising results were obtained. Full article
(This article belongs to the Special Issue Symmetry in Systems Design and Analysis)
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Figure 1

Figure 1
<p>Main framework of the proposed single image super-resolution (SR) method.</p> Full article ">Figure 2
<p>Example SR results and their PSNR values on ‘pepper’. Top row: ground truth HR image, bicubic interpolation (PSNR: 29.88), LLE [<a href="#B38-symmetry-09-00024" class="html-bibr">38</a>] (PSNR: 26.85). Bottom row: Yang et al. [<a href="#B26-symmetry-09-00024" class="html-bibr">26</a>] (PSNR: 26.12), Glasner et al. [<a href="#B4-symmetry-09-00024" class="html-bibr">4</a>] (PSNR: 24.87), Self-learning [<a href="#B30-symmetry-09-00024" class="html-bibr">30</a>] (PSNR: 29.96), and ours (PSNR: 30.48).</p> Full article ">Figure 3
<p>Example SR results and their PSNR values on ‘baby’. Top row: ground truth HR image, bicubic interpolation (PSNR: 32.48), LLE [<a href="#B38-symmetry-09-00024" class="html-bibr">38</a>] (PSNR: 31.65). Bottom row: Yang et al. [<a href="#B26-symmetry-09-00024" class="html-bibr">26</a>] (PSNR: 32.78), Glasner et al. [<a href="#B4-symmetry-09-00024" class="html-bibr">4</a>] (PSNR: 32.13), Self-learning [<a href="#B30-symmetry-09-00024" class="html-bibr">30</a>] (PSNR: 33.76), and ours (PSNR: 34.64).</p> Full article ">Figure 4
<p>Example SR results and their PSNR values on ‘flag’. Top row: ground truth HR image, bicubic interpolation (PSNR: 27.08), LLE [<a href="#B38-symmetry-09-00024" class="html-bibr">38</a>] (PSNR: 25.99). Bottom row: Yang et al. [<a href="#B26-symmetry-09-00024" class="html-bibr">26</a>] (PSNR: 24.23), Glasner et al. [<a href="#B4-symmetry-09-00024" class="html-bibr">4</a>] (PSNR: 24.41), Self-learning [<a href="#B30-symmetry-09-00024" class="html-bibr">30</a>] (PSNR: 27.48), and ours (PSNR: 27.67).</p> Full article ">Figure 5
<p>Example SR results and their PSNR values on ‘man’. Top row: ground truth HR image, bicubic interpolation (PSNR: 27.45), LLE [<a href="#B38-symmetry-09-00024" class="html-bibr">38</a>] (PSNR: 27.83). Bottom row: Yang et al. [<a href="#B26-symmetry-09-00024" class="html-bibr">26</a>] (PSNR: 24.05), Glasner et al. [<a href="#B4-symmetry-09-00024" class="html-bibr">4</a>] (PSNR: 27.92), Self-learning [<a href="#B30-symmetry-09-00024" class="html-bibr">30</a>] (PSNR: 28.32), and ours (PSNR: 28.88).</p> Full article ">Figure 6
<p>Performance of different super-resolution methods on denoising are compared in this experiment. Top row: ground truth image (Lena), noisy image (<math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>25</mn> </mrow> </semantics> </math>), LLE [<a href="#B38-symmetry-09-00024" class="html-bibr">38</a>] (PSNR: 21.34). Bottom row: Yang et al. [<a href="#B26-symmetry-09-00024" class="html-bibr">26</a>] (PSNR: 23.67), Glasner et al. [<a href="#B4-symmetry-09-00024" class="html-bibr">4</a>] (PSNR: 23.56), Self-learning [<a href="#B30-symmetry-09-00024" class="html-bibr">30</a>] (PSNR: 24.22), and ours (PSNR: 25.45).</p> Full article ">
158 KiB  
Erratum
Erratum: Rauh, A. Coherent States of Harmonic and Reversed Harmonic Oscillator. Symmetry, 2016, 8, 46
by Alexander Rauh
Symmetry 2017, 9(2), 23; https://doi.org/10.3390/sym9020023 - 9 Feb 2017
Viewed by 2833
Abstract
n/a Full article
20627 KiB  
Article
A Study on Immersion of Hand Interaction for Mobile Platform Virtual Reality Contents
by Seunghun Han and Jinmo Kim
Symmetry 2017, 9(2), 22; https://doi.org/10.3390/sym9020022 - 5 Feb 2017
Cited by 30 | Viewed by 6921
Abstract
This study proposes gaze-based hand interaction, which is helpful for improving the user’s immersion in the production process of virtual reality content for the mobile platform, and analyzes efficiency through an experiment using a questionnaire. First, three-dimensional interactive content is produced for use [...] Read more.
This study proposes gaze-based hand interaction, which is helpful for improving the user’s immersion in the production process of virtual reality content for the mobile platform, and analyzes efficiency through an experiment using a questionnaire. First, three-dimensional interactive content is produced for use in the proposed interaction experiment while presenting an experiential environment that gives users a high sense of immersion in the mobile virtual reality environment. This is designed to induce the tension and concentration of users in line with the immersive virtual reality environment. Additionally, a hand interaction method based on gaze—which is mainly used for the entry of mobile virtual reality content—is proposed as a design method for immersive mobile virtual reality environment. The user satisfaction level of the immersive environment provided by the proposed gaze-based hand interaction is analyzed through experiments in comparison with the general method that uses gaze only. Furthermore, detailed analysis is conducted by dividing the effects of the proposed interaction method on user’s psychology into positive factors such as immersion and interest and negative factors such as virtual reality (VR) sickness and dizziness. In this process, a new direction is proposed for improving the immersion of users in the production of mobile platform virtual reality content. Full article
(This article belongs to the Special Issue Symmetry in Complex Networks II)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>The flow of the proposed interactive content: (<b>a</b>) start the content; (<b>b</b>) select cards; (<b>c</b>) result of card exchange; (<b>d</b>) card game finish condition and event after finish.</p> Full article ">Figure 2
<p>Overview of the proposed gaze-based hand interaction. HMD: head-mounted display.</p> Full article ">Figure 3
<p>Interaction process of the proposed content using gaze: (<b>a</b>) card browsing; (<b>b</b>) card selection using gaze.</p> Full article ">Figure 4
<p>Construction of mobile virtual reality input environment using a Leap Motion device.</p> Full article ">Figure 5
<p>Interaction of content through the proposed gaze-based hand interaction: (<b>a</b>) card selection process; (<b>b</b>) action event process when the game finishes.</p> Full article ">Figure 6
<p>Implementation result of the proposed mobile virtual reality content including our hand interaction: (<b>a</b>) content starting screen; (<b>b</b>) content initial card setting screen; (<b>c</b>) card selection interaction using gaze and hand; (<b>d</b>) content finish condition; (<b>e</b>) generation of a virtual object for event in the event of content’s finish; (<b>f</b>) event object control through hand interaction; (<b>g</b>) delivery of information by converting the reaction speed of users into their scores.</p> Full article ">Figure 7
<p>Analysis results for the suitability and satisfaction of the proposed virtual reality content (left to right: suitability level of five items, score distribution between 1 and 5, satisfaction factors consisting of three items).</p> Full article ">Figure 8
<p>Comparison experiment result of the gaze-based hand interaction and the method using gaze only: (<b>a</b>) results of the experimental group who experienced H followed by G; (<b>b</b>) results of the experimental group who experienced G followed by H; (<b>c</b>) results of the experimental group who experienced G only; and (<b>d</b>) results of the experimental group who experienced H only.</p> Full article ">Figure 9
<p>Detailed analysis results for psychological factors of the proposed gaze-based hand interaction: (<b>a</b>) satisfaction distribution of the proposed hand interaction; (<b>b</b>) distribution of positive factors and score analysis results; (<b>c</b>) distribution of the positive psychological factors of participants who experienced the gaze only; (<b>d</b>) distribution of the positive psychological factors of participants who experienced the gaze-based hand interaction only.</p> Full article ">
2270 KiB  
Article
Aesthetic Patterns with Symmetries of the Regular Polyhedron
by Peichang Ouyang, Liying Wang, Tao Yu and Xuan Huang
Symmetry 2017, 9(2), 21; https://doi.org/10.3390/sym9020021 - 3 Feb 2017
Cited by 5 | Viewed by 8253
Abstract
A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With the resulting algorithm, a flexible form of invariant mappings is achieved to generate aesthetic patterns with symmetries of the regular polyhedra. Full article
(This article belongs to the Special Issue Polyhedral Structures)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>The five regular polyhedra.</p> Full article ">Figure 2
<p>(<b>a</b>) The blue spherical right triangle <math display="inline"> <semantics> <msub> <mo>▵</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> </msub> </semantics> </math> surrounded by planes <math display="inline"> <semantics> <mrow> <msubsup> <mo>Π</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> <mi>α</mi> </msubsup> <mo>,</mo> <mspace width="4pt"/> <msubsup> <mo>Π</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> <mi>β</mi> </msubsup> </mrow> </semantics> </math>, and <math display="inline"> <semantics> <msubsup> <mo>Π</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> <mi>γ</mi> </msubsup> </semantics> </math> forms a fundamental region associated with group <math display="inline"> <semantics> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> </semantics> </math>; (<b>b</b>) Let <math display="inline"> <semantics> <mrow> <mi>Q</mi> <mo>∈</mo> <msub> <mo>▵</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>∉</mo> <msub> <mo>▵</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> </msub> </mrow> </semantics> </math> be two points on the different sides of <math display="inline"> <semantics> <msubsup> <mo>Π</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> <mi>β</mi> </msubsup> </semantics> </math>. Then, <math display="inline"> <semantics> <mrow> <msub> <mi>β</mi> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> and <span class="html-italic">Q</span> lie on the same side of <math display="inline"> <semantics> <msubsup> <mo>Π</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>]</mo> </mrow> <mi>β</mi> </msubsup> </semantics> </math>, and the distance between them is smaller than <math display="inline"> <semantics> <msub> <mi>P</mi> <mn>0</mn> </msub> </semantics> </math> and <span class="html-italic">Q</span>; and (<b>c</b>) A schematic illustration that shows how Theorem 1 transforms <math display="inline"> <semantics> <mrow> <msup> <mi>u</mi> <mn>1</mn> </msup> <mo>∉</mo> <msub> <mo>▵</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </msub> </mrow> </semantics> </math> into <math display="inline"> <semantics> <msub> <mo>▵</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </msub> </semantics> </math> symmetrically. In this case, <math display="inline"> <semantics> <msup> <mi>u</mi> <mn>1</mn> </msup> </semantics> </math> is first transformed by <math display="inline"> <semantics> <msub> <mi>γ</mi> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </msub> </semantics> </math> so that <math display="inline"> <semantics> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>=</mo> <msub> <mi>γ</mi> <mrow> <mo>[</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>]</mo> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>u</mi> <mn>1</mn> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> goes into red tile. Then, <math display="inline"> <semantics> <msup> <mi>u</mi> <mn>2</mn> </msup> </semantics> </math> is transformed by <math display="inline"> <semantics> <msub> <mi>β</mi> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </msub> </semantics> </math> so that <math display="inline"> <semantics> <mrow> <msup> <mi>u</mi> <mn>3</mn> </msup> <mo>=</mo> <msub> <mi>β</mi> <mrow> <mo>[</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>]</mo> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> goes into green tile. At last, <math display="inline"> <semantics> <msup> <mi>u</mi> <mn>3</mn> </msup> </semantics> </math> is transformed by <math display="inline"> <semantics> <msub> <mi>α</mi> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </msub> </semantics> </math> so that <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>α</mi> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>u</mi> <mn>3</mn> </msup> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mo>▵</mo> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 3
<p>Two spherical patterns with [3,3] symmetries.</p> Full article ">Figure 4
<p>Two spherical patterns with [3,4] symmetries.</p> Full article ">Figure 5
<p>Two spherical patterns with [3,5] symmetries.</p> Full article ">
168 KiB  
Editorial
Acknowledgement to Reviewers of Symmetry in 2016
by Symmetry Editorial Office
Symmetry 2017, 9(2), 20; https://doi.org/10.3390/sym9020020 - 26 Jan 2017
Viewed by 3856
Abstract
The editors of Symmetry would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2016.[...] Full article
2984 KiB  
Article
Cyclotomic Aperiodic Substitution Tilings
by Stefan Pautze
Symmetry 2017, 9(2), 19; https://doi.org/10.3390/sym9020019 - 25 Jan 2017
Cited by 6 | Viewed by 8615
Abstract
The class of Cyclotomic Aperiodic Substitution Tilings (CASTs) is introduced. Its vertices are supported on the 2 n -th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers [...] Read more.
The class of Cyclotomic Aperiodic Substitution Tilings (CASTs) is introduced. Its vertices are supported on the 2 n -th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry D n or D 2 n , i.e., the tiling contains an infinite number of patches of any size with dihedral symmetry D n or D 2 n only by iteration of substitution rules on a single tile. Full article
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Figure 1

Figure 1
<p>The diagonals <math display="inline"> <semantics> <msub> <mi>μ</mi> <mrow> <mn>11</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> </semantics> </math> of a regular hendecagon with side length 1 can be written as a sum of 22-nd roots of unity as described in Equation (9).</p> Full article ">Figure 2
<p>Cyclotomic Aperiodic Substitution Tilings (CAST) for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math> with minimal inflation multiplier. The black tips of the prototiles mark their respective chirality.</p> Full article ">Figure 3
<p>CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math> with minimal inflation multiplier as described in [<a href="#B33-symmetry-09-00019" class="html-bibr">33</a>] (Figure 1 and Section 3, 2nd matrix).</p> Full article ">Figure 4
<p>Generalized Lançon-Billard tiling.</p> Full article ">Figure 5
<p>CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math> with inflation multiplier <math display="inline"> <semantics> <msub> <mi>μ</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </semantics> </math>.</p> Full article ">Figure 6
<p>CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>11</mn> </mrow> </semantics> </math> with inflation multiplier <math display="inline"> <semantics> <msub> <mi>μ</mi> <mrow> <mn>11</mn> <mo>,</mo> <mn>5</mn> </mrow> </msub> </semantics> </math>. One vertex star within prototile <math display="inline"> <semantics> <msub> <mi>P</mi> <mn>4</mn> </msub> </semantics> </math> has been chosen to illustrate the individual dihedral symmetry <math display="inline"> <semantics> <msub> <mi>D</mi> <mn>11</mn> </msub> </semantics> </math>.</p> Full article ">Figure 7
<p>The ”Edges” of a substitution rules are defined as the boundaries of the supertile (dashed line) and the rhombs bisected by it along one of their diagonals. The figure illustrates how the rhombs can be placed accordingly for even edge configuration (<b>a</b>); and odd edge configuration (<b>b</b>). The inner angles of the rhombs are integer multiples of <math display="inline"> <semantics> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </semantics> </math> and are denoted by small numbers near the tips (Example <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 8
<p>Substitution rule for rhomb <math display="inline"> <semantics> <msub> <mi>R</mi> <mn>1</mn> </msub> </semantics> </math> or <math display="inline"> <semantics> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </semantics> </math>: The three possible configurations of the tip are shown in (<b>a</b>), (<b>b</b>) and (<b>c</b>) (Example <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 9
<p>Substitution rule for rhomb <math display="inline"> <semantics> <msub> <mi>R</mi> <mn>1</mn> </msub> </semantics> </math> or <math display="inline"> <semantics> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </semantics> </math>: Rhomb <math display="inline"> <semantics> <msub> <mi>R</mi> <mi>k</mi> </msub> </semantics> </math> on the edge and its relatives at the corresponding edge (black) and the opposite edge (blue) (Example <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 10
<p>Substitution rule for rhomb <math display="inline"> <semantics> <msub> <mi>R</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>±</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>/</mo> <mn>2</mn> </mrow> </msub> </semantics> </math>: The Kannan-Soroker-Kenyon (KSK) criterion requires the existence of rhomb <math display="inline"> <semantics> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> </semantics> </math> on the edge as shown in (<b>c</b>) and (<b>d</b>). However, this is not the case for rhomb <math display="inline"> <semantics> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </semantics> </math> as shown in (<b>a</b>) and (<b>b</b>) (Case 1b, example <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 11
<p>Substitution rule for rhomb <math display="inline"> <semantics> <msub> <mi>R</mi> <mn>1</mn> </msub> </semantics> </math> or <math display="inline"> <semantics> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </semantics> </math>: The possible configurations of the edges orientations are shown in (<b>a</b>), (<b>b</b>), (<b>c</b>) and (<b>d</b>) (Example <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 12
<p>Substitution rule for rhomb <math display="inline"> <semantics> <msub> <mi>R</mi> <mn>1</mn> </msub> </semantics> </math> or <math display="inline"> <semantics> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </semantics> </math>: KSK criterion for corresponding edges with different orientations with <math display="inline"> <semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics> </math> are shown in (<b>a</b>), (<b>b</b>), (<b>c</b>) and <math display="inline"> <semantics> <mrow> <msub> <mi>α</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>α</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>;</mo> <mspace width="0.277778em"/> <mi>k</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </semantics> </math> are shown in (<b>d</b>), (<b>e</b>) and (<b>f</b>).</p> Full article ">Figure 13
<p>Orientations of edges (and orientations of lines of symmetry) for rhomb prototiles in case 1b (Example <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>11</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 14
<p>Rhombic CAST examples for case 1a (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics> </math>) and case 1b (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 15
<p>Rhombic CAST examples for case 2a (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics> </math>) and case 2b (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>). The shown example for case 2b was slightly modified to reduce the number of prototiles to <math display="inline"> <semantics> <mfenced separators="" open="&#x230A;" close="&#x230B;"> <mi>n</mi> <mo>/</mo> <mn>2</mn> </mfenced> </semantics> </math> as in case 1b. In detail, the edges of the rhomb prototiles have orientation as shown in <a href="#symmetry-09-00019-f013" class="html-fig">Figure 13</a>.</p> Full article ">Figure 16
<p>Rhombic CAST examples for case 3a (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics> </math>) and case 3b (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 16 Cont.
<p>Rhombic CAST examples for case 3a (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics> </math>) and case 3b (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 17
<p>Rhombic CAST examples for case 4a (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics> </math>) and case 4b (<math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 18
<p>CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 19
<p>CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>, derived from the Goodman-Strauss tiling in [<a href="#B39-symmetry-09-00019" class="html-bibr">39</a>].</p> Full article ">Figure 20
<p>CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math>, derived from the generalized Lançon-Billard tiling in <a href="#symmetry-09-00019-f004" class="html-fig">Figure 4</a>.</p> Full article ">Figure 21
<p>Extended Girih CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 22
<p>Girih CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 23
<p>Another Girih CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics> </math>, derived from [<a href="#B56-symmetry-09-00019" class="html-bibr">56</a>] (Figures 14 and 15) and patterns shown in the Topkapi Scroll, in detail [<a href="#B53-symmetry-09-00019" class="html-bibr">53</a>] (Panels 28, 31, 32, 34).</p> Full article ">Figure 24
<p>Extended Girih CAST for the case <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>.</p> Full article ">
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