Hindawi Complexity Volume 2018, Article ID 7315695, 8 pages https://doi.org/10.1155/2018/7315695

Research Article The Visualization of Spherical Patterns with Symmetries of the Wallpaper Group

Shihuan Liu ,1,2 Ming Leng,1 and Peichang Ouyang 1

1 School of Mathematics & Physics, Jinggangshan University, Ji’an 343009, China 2Sichuan Province Key Lab of Signal and Information Processing, Southwest Jiaotong University, Chengdu 611756, China

Correspondence should be addressed to Peichang Ouyang; g [email protected]

Received 17 October 2017; Accepted 1 January 2018; Published 12 February 2018

Academic Editor: Michele Scarpiniti

Copyright © 2018 Shihuan Liu et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By constructing invariant mappings associated with wallpaper groups, this paper presents a simple and efcient method to generate colorful wallpaper patterns. Although the constructed mappings have simple form and only two parameters, combined with the color scheme of orbit trap algorithm, such mappings can create a great variety of aesthetic wallpaper patterns. Te resulting wallpaper patterns are further projected by central projection onto the sphere. Tis creates the interesting spherical patterns that possess infnite symmetries in a fnite space.

1. Introduction Fourier series to achieve it [8]. Chung and Chan [9] and Lu et al. [10] later presented similar ideas to create colorful Wallpaper groups (or plane crystallographic groups) are wallpaper patterns. Recently, Douglas and John discovered a mathematical classifcation of two-dimensional repetitive very simple approach to yield interesting wallpaper patterns patterns. Te frst systematic proof that there were only 17 of fractal characteristic [11]. possible wallpaper patterns was carried out by Fedorov in Te key idea behind [7–10] is equivariant mapping, 1891 [1] and later derived independently by Polya´ in 1924 [2]. which is not easy to achieve, since such mapping must be Wallpaper groups are characterized by translations in two commutable with respect to symmetry group. In this paper, independent directions, which give rise to a lattice. Patterns we present a simple invariant method to create wallpaper with wallpaper symmetry can be widely found in architecture patterns. It has independent mapping form and only two and decorative art [3–5]. It is surprising that the three- parameters. Combined with the color scheme of orbit trap dimensional 230 crystallographic groups were enumerated algorithm, our approach can be conveniently utilized to yield before the planar wallpaper groups. rich wallpaper patterns. Te art of M. C. Escher features the rigorous mathe- Escher’s Circle Limits I–IV are unusual and visually matical structure and elegant artistic charm, which might attractive because they realized infnity in a fnite unit disc. betheoneandonlyinthehistoryofart.Aferhisjourney Inspired by his arts, we use central projection to project to the Alhambra, La Mezquita, and Cordoba, he created wallpaper patterns onto the fnite sphere. Tis obtains the many mathematically inspired arts and became a master in aesthetic patterns of infnite symmetry structure in the creating wallpaper arts [6]. With the development of modern fnite sphere space. Such patterns look beautiful. Combined computers, there is considerable research on the automatic with simulation and printing technologies, these computer- generation of wallpaper patterns. In [7], Field and Golubitsky generated patterns could be utilized in wallpaper, textiles, frst proposed the conception of equivariant mappings. Tey ceramics, carpet, stained glass windows, and so on, producing constructed equivariant mapping to generated chaotic cyclic, both economic and aesthetic benefts. dihedral, and wallpaper attractors. Carter et al. developed an Te remainder of this paper is organized as follows. In easier method that used equivariant truncated 2-dimensional Section 2, we frst introduce some basic conceptions and the 2 Complexity

� (�) � � Table 1: Te concrete invariant mapping ��,� forms associated with 17 wallpaper groups. In the fourth column, the subscripts and identify the lattice kind (� � represents square lattice, while �� represents diamond lattice) and wallpaper group type, respectively.

Wallpaper Point Extra symmetry set Invariant mapping group group

�� ,�1 (�) = ∑ �� [� (�)] p1 �1 None �� � �∈�1

�� ,� (�) = ∑ �� [� (�)] p2 �2 None �� 2 � �∈�2

�� ,�� (�) = ∑ �� [� (�)]+ ∑ �� [(�1�) (�)] �� �1 �1 (�, �) = (�, −�) �� � � �∈�1 �∈�1 2 �1(�, �) = (�, −�), ��� � �� ,��� (�) = ∑ �� [� (�)]+∑ { ∑ �� [(���) (�)]} 2 � (�, �) = (−�, �) �� � � 2 �∈�2 �=1 �∈�2

�� ,�� (�) = ∑ �� [� (�)]+ ∑ �� [(�1�) (�)] �� �1 �1 (�, �) = (�+�,−�) �� � � �∈�1 �∈�1 2 �1(�, �) = (� + �, −�), ��� � �� ,��� (�) = ∑ �� [� (�)] + ∑ { ∑ �� [(���) (�)]} 2 � (�, �) = (−�, �) �� � � 2 �∈�2 �=1 �∈�2 2 �1(�, �) = (�, −�), �� � �� ,�� (�) = ∑ �� [� (�)]+∑ { ∑ �� [(���) (�)]} 1 � (�, �) = (� + �, � − �) �� � � 2 �∈�1 �=1 �∈�1 � (�, �) = (�, −�) 1 , 4 �2(�, �) = (� − �, � + �) ��� � �� ,��� (�) = ∑ �� [� (�)]+∑ { ∑ �� [(���) (�)]} 2 � (�,�)=(�+�,�−�) �� � � 3 , �∈�2 �=1 �∈�2 �4(�, �) = (−�, �)

�� ,�4 (�) = ∑ �� [� (�)] p4 �4 None �� � �∈�4

�� ,�4� (�) = ∑ �� [� (�)]+ ∑ �� [(�1�) (�)] �4��4 �1 (�, �) = (�+�,−�) �� � � �∈�4 �∈�4

�� ,�4� (�) = ∑ �� [� (�)]+ ∑ �� [(�1�) (�)] �4��4 �1 (�, �) = (�, −�) �� � �� �∈�4 �∈�4 2 �1(�,�)=(�+�,�−�), ��� � �� ,��� (�) = ∑ �� [� (�)]+∑ { ∑ �� [(���) (�)]} 2 � (�, �) = (� − �, � + �) �� � � 2 �∈�2 �=1 �∈�2

�� ,�3 (�) = ∑ �� [� (�)] p3 �3 None �� � �∈�3

�� ,�3�1 (�) = ∑ �� [� (�)]+ ∑ �� [(�1�) (�)] p3m1 �3 �1 (�, �) = (−�, �) �� � � �∈�3 �∈�3

�� ,�31� (�) = ∑ �� [� (�)]+ ∑ �� [(�1�) (�)] p31m �3 �1 (�, �) = (�, −�) �� � � �∈�3 �∈�3

�� ,�6 (�) = ∑ �� [� (�)] p6 �6 None �� � �∈�6

�� ,�6� (�) = ∑ �� [� (�)]+ ∑ �� [(�1�) (�)] �6��6 �1 (�, �) = (�, −�) �� � � �∈�6 �∈�6 lattices with respect to wallpaper groups. To create patterns forms a lattice [12, 13]. Since a lattice is a fnitely generated free 2 with symmetries of the wallpaper group, we will explicitly abelian group, it is isomorphic to � and fully spans the real 2 construct invariant mappings associated with 17 wallpaper vector space � [14]. A lattice may be viewed as a regular tiling groups (the concrete mapping forms are summarized in of a space by a primitive cell. Lattices have many signifcant Table 1) in Section 3. In Section 4, we describe how to applications in pure mathematics, particularly in connection create colorful wallpaper patterns. Finally, we show some to Lie algebras, number theory, and group theory [15]. spherical wallpaper patterns obtained by central projection In this section, we mainly introduce the lattices associated in Section 5. with wallpaper group. Firstly, we introduce some basic conceptions. Te symmetry group of an object is the set of all isometries 2. The Lattice of Wallpaper Groups under which the object is invariant with composition as the group operation. A point group (sometimes called rosette In geometry and group theory, a lattice in 2-dimensional group) is a group of isometries that keep at least one point 2 2 Euclidean plane � is essentially a subgroup of � .Or, fxed. 2 2 equivalently, for any basis vectors of � ,thesubgroupofall Point groups in � come in two infnite families: dihedral linear combinations with integer coefcients of the vectors group �� whichisthesymmetrygroupofaregularpolygon Complexity 3

� and cyclic group � that only comprises rotation transforma- �� {∑ � (�⋅V) + ∑ (�⋅V)} � � =(cos(2�/�) − sin(2�/�) ) �=(−1 0 ) 1 2 tions of �.Let � sin(2�/�) cos(2�/�) and 01. V∈� V∈� � =( )=�� (�) , Ten their matrix group can be represented as �� ={�,�= � �� {∑ � (�⋅V) + ∑ (�⋅V)} 1, 2, 3, . . . , �} � =� ∪{��� , �=1,2,3,...,�} 3 4 and � � � . V∈� V∈� A wallpaper group is a type of topologically discrete group 2 in � which contains two linearly independent translations. (2) 2 � ∗ ∗ A lattice in is the symmetry group of discrete translational where �=�� +�� ,�,�∈�. symmetry in two independent directions. A tiling with this ∗ lattice of translational symmetry cannot have more but may Proof. Since � is the dual lattice of �, ∀V ∈�,wehave�⋅V = � ∗ ∗ ∗ ∗ have less symmetry than the lattice itself. Let be a lattice in (�� +�� )⋅V =�(� ⋅ V)+�(� ⋅ V)=2��for certain �2 �∗ � ∀�∈� .Alattice is called the dual lattice of if, and �∈�.Tusweget��{∑V∈� ��((� + �) ⋅ V)+∑V∈�((� + �) ⋅ V)} = ∀V ∈�∗ �⋅V � V ,theinnerproduct is an integer, where and ��{∑V∈� ��(� ⋅V)+∑V∈�(� ⋅V)},since�� and �� are functions of �2 � �2 �2 � are vectors in .Let be a mapping from to and let period 2�(�,�=1,2,3,4). Consequently, the mapping ��(�) �2 � be a symmetry group in ; is called an invariant mapping constructed by �� (� = 1,2,3,4)satisfes (2). Tis completes 2 with respect to � if, ∀�∈�and ∀� ∈ �, �(�) = �(��). the proof. By the crystallographic restriction theorem, there are 2 � (�) only 5 lattice types in � [16]. Although wallpaper groups Essentially Lemma 1 says that � is a double period 2� have totally 17 types, their lattices can be simplifed into two mapping (of period ) along the independent translational �∗ lattices: square and diamond lattices. For convenience, we directions of . require that the inner product of the mutual dual lattice of a Teorem 2. � 2×2 wallpaper group be an integer multiple of 2�.Troughoutthe Let be a fnite group realized by matrices �2 � paper,forsquarelattice,wechooselattice�� = {(1, 0), (0, 1)} acting on by multiplication on the right and let be an ∗ �2 �2 with dual lattice �� = {2�(1, 0), 2�(0, 1)}; for diamond lattice, arbitrary mapping from to .Tenmapping √ we choose lattice �� = {(1, 0), (1/2)(−1, 3)} with dual lattice � � = ∑ �[� � ], �∈�2, �∗ ={(2�/√3)(√3, −1), 2�(0, −2/√3)} �,� ( ) ( ) � . �∈� (3) In this paper, we use standard crystallographic notations �1 of wallpaper groups [16, 17]. Among 17 wallpaper groups, , is an invariant mapping with respect to �. �2, ��, ���, ��, ���, ��, ���, �4, ���, �4�,and�4� �3, �3�1, �31�, �6, �6� possess square lattice, while and Proof. For �∈�, by closure of the group operation, we see possess diamond lattice. that �� runs through � as � does. Terefore we have

∗ 3. Invariant Mapping with respect to ��,� [� (�)] = ∑ �[�(��)]= ∑ �[� (�)] �∈� �∗∈� Wallpaper Groups (4) =� (�) , In this section, we explicitly construct invariant mappings �,� associated with wallpaper groups. To this end, we frst prove �∗ =��∈� � (�) the following lemma. where .Tismeansthat �,� is an invariant mapping with respect to �. Lemma 1. Suppose that �� (� = 1,2,3,4) are sine or cosine ∗ functions, � is a wallpaper group with lattice �={�,�}, � = Combining Lemma 1 and Teorem 2, we immediately ∗ ∗ {� ,� } is the dual lattice of �,and� and � are real numbers. derive the following theorem. Ten mapping Teorem 3. Let � in Teorem 2 have the form �� as in Lemma 1. Suppose that � is a cyclic group �� or dihedral �� {∑� (�⋅V) + ∑ (�⋅V)} ∗ 1 2 group �� with lattice �; � is the dual lattice associated with � (�) =( V∈� V∈� ), � � (�) �2 �2 � . Assume that ��,� is a mapping from to of the ��3 {∑�4 (�⋅V) + ∑ (�⋅V)} (1) following form: V∈� V∈� � (�) = ∑ � [� (�)], �∈�2. 2 ��,� � ∀�∈�, �∈� (5) ∗ is invariant with respect to � ,or��(�) has translation ∗ Ten �� ,�(�) is an invariant mapping with respect to both � invariance of � ;thatis, ∗ � and � . �� {∑ � �+� ⋅ V + ∑ �+� ⋅ V } 1 2 (( ) ) (( ) ) Wallpaper groups possess globally translation symmetry ( V∈� V∈� ) along two independent directions as well as locally point �� {∑ � ((�+�) ⋅ V) + ∑ ((�+�) ⋅ V)} group symmetry. For the wallpaper groups that only have 3 4 � (�) V∈� V∈� symmetries of a certain point group, mapping ��,� in 4 Complexity

BEGIN start x =0 end x = 6∗3.1415926 start y =0 end y = 6∗3.1415926 //Set pi = 3.1415926 step x =(endx –startx)/X res //Xres is the resolution in X direction step y =(endy –starty)/Y res //YresistheresolutioninY direction FOR i =0TOX res DO FOR j =0TOY res DO x =startx + i ∗ step x y =starty + j ∗ step y FOR k = 1 TO MaxIter//MaxIter is the number of iterations, the default set is 100 ∗ � (�) � / Given a invariant mapping ��,� associated with a wallpaper group as iteration function and initial point (x, y), function Iteration (x, y) iterates MaxIter times. Te iterated sequences are stored in the array Sequence∗/. Sequence [�] =Iteration(x, y) END FOR /∗Inputting Sequence, the color scheme OrbitTrap outputs the color [r, g, b]∗/ [r, g, b] = OrbitTrap (Sequence) Set color [r, g, b] to point (x, y) END FOR END FOR END

Algorithm 1: CreatingWallpaperPattern()// algorithm for creating patterns with the wallpaper symmetry.

Teorem 3 can be used to create wallpaper patterns well. 4. Colorful Wallpaper Patterns from However, except for the symmetries of a point group, some Invariant Mappings wallpaper groups may possess other symmetries. For exam- ple,exceptforsymmetriesofdihedralgroup�3,wallpaper Invariant mapping method is a common approach used in group �31� still has a refection along horizontal direction, creating symmetric patterns [18–23]. Color scheme is an 2 say symmetry �1(�, �) = (�, −�) ((�, �) ∈ � ); besides algorithm that is used to determine the color of a point. Given � �2 symmetry, wallpaper group ��� contains perpendicular a color scheme and domain , by iterating invariant mapping � (�) �∈� � refections; that is, �1(�, �) = (�, −�) and �2(�, �) = (−�, �). ��,� , , one can determine the color of .Coloring For a wallpaper group � with extra symmetry set Δ= points in � pointwise, one can obtain a colorful pattern in � � {�1,�2,�3,...,��}, based on mapping (5), we add proper with symmetries of the wallpaper group . Figures 1-2 � (�) are four wallpaper patterns obtained in this manner. Tese terms so that the resulting mapping ��,� is also invariant with respect to �. Tis is summarized in Teorem 4. patternswerecreatedbyVC++6.0onaPC(SVGA).In Algorithm1,weprovidethepseudocodesothattheinterested Teorem 4. Let � in Teorem 2 have the form �� in Lemma 1. reader can create their own colorful wallpaper patterns. Suppose that � is a wallpaper group with symmetry group � Te color scheme used in this paper is called orbit trap and extra symmetry set Δ={�1,�2,�3,...,��}. Assume that � algorithm [10]. We refer the reader to [10, 20] for more details ∗ is the lattice of � and � is the dual lattice associated with �. about the algorithm (the algorithm is named as function � (�) �2 �2 OrbitTrap() in Algorithm 1). It has many parameters to adjust Let ��,� be a mapping from to of the following form: color, which could enhance the visual appeal of patterns � (�) = ∑ � [� (�)] ��,� � efectively. Compared with the complex equivariant mapping �∈� constructed in [7–10], our invariant mappings possess not (6) only simple form but also sensitive dynamical system prop- � { } erty, which can be used to produce infnite wallpaper patterns + ∑ ∑ � [(� �) (�)] ,�∈�2. { � � } easily. For example, Figures 1(a) and 2(b) were created �=1 �∈�,��∈Δ { } by mappings �� ,�4�(�) and �� ,�3�1(�),respectively,in �� �� � � Ten �� ,�(�) is an invariant mapping with respect to both � which the specifc mappings �� and �� were ∗ � and � . 2.12 cos { ∑ cos (�⋅V) + ∑ (�⋅V)} We refer the reader to [7, 8, 17] for more detailed descrip- V∈� V∈� � =( � � ), tion about the extra symmetry set of wallpaper groups. By �� (7) Teorems 3-4, we list the invariant mappings associated with 1.03 cos { ∑ sin (�⋅V) + ∑ (�⋅V)} 17 wallpaper groups in Table 1. V∈�� V∈�� Complexity 5

(a) (b)

Figure 1: Colorful wallpaper patterns with the �4� (a) and �6� (b) symmetry.

(a) (b)

Figure 2: Colorful wallpaper patterns with the ��� (a) and �3�1 (b) symmetry.

2 3 2 2 2 Let � ={(�,�,�)∈� |� +� +� =1}be the unit sphere 1.1 { ∑ (�⋅V) + ∑ (�⋅V)} 3 cos sin in � ,let�=�be a projection plane, where � is a negative V∈�� V∈�� 3 3 �� =( ). (8) constant. Assume that �(�, �, �) ∈ � ;then�(�, �, �) ∈ � � � 3 −0.52 sin { ∑ cos (�⋅V) + ∑ (�⋅V)} and � (−�, −�, −�) ∈ � are a pair of antipodal points. For any 3 V∈�� V∈�� point �(�, �, �) ∈ � , there exist a unique line � through the � origin (0, 0, 0) and � (and � ) which intersects the projection �=� (�,�,�) � It seems that the deference between (7) and (8) is not very plane at point . Denote the projection by .By analytic geometry, it is easy to check that signifcant. However, by Table 1, mappings �� ,�4�(�) and �� � (�) �� ,�3�1 have 16 and 12 summation terms, respectively. � � � [ ] [ ] � Te cumulative diference will be very obvious, which is [ � ] =�(�, �, �) =�(−�, −�, −�) = [ � ] . (9) enough to produce diferent style patterns. � [ � ] [ � ]

2 5. Spherical Wallpaper Patterns by Because the projection point is at the center of � ,wecall� as Central Projection central projection. Te choice of the plane �=�hasagreatinfuenceonthe In this section, we introduce central projection to yield spherical patterns. If plane �=�is too close to coordinate spherical patterns of the wallpaper symmetry. plane ���, the resulting spherical pattern only shows a few 6 Complexity

(a) (b)

Figure 3: Two spherical wallpaper patterns with the �4� symmetry, in which the projection plane was set as �=−2�(a) and �=−4�(b).

(a) (b)

Figure 4: Colorful spherical wallpaper patterns with the �4� (a) and �6� (b) symmetry. periodsofthewallpaperpattern.However,ifplane�=� Figures 3–7 are ten patterns obtained by this manner. is too far away from coordinate plane ���, the wallpaper Except for Figure 3(b) in which the projection plane was patternonthespheremayappearsmallsothatwecannot set as �=−4�, all the other projection planes were set as identify symmetries of the wallpaper pattern well. Figure 3 �=−2�. We utilized the wallpaper patterns of Figure 1 to illustrates the contrast efect of the setting of plane �=�. produce spherical patterns shown in Figure 4. For beauty, all Given a wallpaper pattern, by central projection �,wecan the camera views are perpendicular to plane ��� and pass 2 map it onto the sphere � and obtain a corresponding spher- the origin, except for Figure 7(b), where the camera view aims 2 ical wallpaper pattern. We next explain how to implement it at the equator of � . To better understand the efect of central in more detail. projection, Figure 7 demonstrates spherical patterns that are (�, �, �) ∈ �3 � (�) Suppose that and ��,� is an invari- observed from diferent perspectives. ant mapping compatible with the symmetry of wallpaper group �. First, by central projection �,weobtainacorre- Additional Points sponding point ((�/�)�, (�/�)�, �) on the projection plane �=� � (�) .Second,let ��,� be iteration function and let Te artistic patterns created in this article have signifcant aes- �((�/�)�, (�/�)�) be initial point; using the color scheme theticandeconomicvalue.Weplantoproducesomematerial of orbit trap, we assign a color to point ((�/�)�, (�/�)�, �). objects with the help of simulation and printing technologies. 2 Finally, repeat the second step; by coloring unit sphere � We produced Figures 1–7 in the VC++ 6.0 programming pointwise, we obtain a spherical pattern of the wallpaper environment with the aid of OpenGL, a powerful graphics group � symmetry. sofware package. Complexity 7

(a) (b)

Figure 5: Colorful spherical wallpaper patterns with the �6� (a) and ��� (b) symmetry.

(a) (b)

Figure 6: Colorful spherical wallpaper patterns with the �4 (a) and �6� (b) symmetry.

(a) (b)

Figure 7: Two spherical wallpaper patterns with the �3� symmetry. Te camera view of (a) is perpendicular to plane ��� and passed the origin, while (b) aims at equator. 8 Complexity

Conflicts of Interest [17] V. E. Armstrong, Groups and Symmetry, Springer, New York, NY, USA, 1987. Te authors declare that there are no conficts of interest [18] K. W. Chung and H. M. Ma, “Automatic generation of aesthetic regarding the publication of this paper. patterns on fractal tilings by means of dynamical systems,” Chaos, Solitons & Fractals, vol. 24, no. 4, pp. 1145–1158, 2005. Acknowledgments [19] P. Ouyang and X. Wang, “Beautiful math—aesthetic patterns based on logarithmic spirals,” IEEE Computer Graphics and Te authors acknowledge Adobe and Microsof for their Applications,vol.33,no.6,pp.21–23,2013. friendly technical support. Tis work was supported by the [20]P.Ouyang,D.Cheng,Y.Cao,andX.Zhan,“Tevisualization National Natural Science Foundation of China (nos. 11461035, of hyperbolic patterns from invariant mapping method,” Com- 11761039, and 61363014), Young Scientist Training Program of puters and Graphics,vol.36,no.2,pp.92–100,2012. JiangxiProvince(20153BCB23003),ScienceandTechnology [21] P.Ouyang and R. W.Fathauer, “Beautiful math, part 2: aesthetic Plan Project of Jiangxi Provincial Education Department patterns based on fractal tilings,” IEEE Computer Graphics and (no. GJJ160749), and Doctoral Startup Fund of Jinggangshan Applications,vol.34,no.1,pp.68–76,2014. University (no. JZB1303). [22] P. Ouyang and K. Chung, “Beautiful math, part 3: hyperbolic aesthetic patterns based on conformal mappings,” IEEE Com- puter Graphics and Applications,vol.34,no.2,pp.72–79,2014. References [23] P. Ouyang, L. Wang, T. Yu, and X. Huang, “Aesthetic patterns with symmetries of the regular polyhedron,” Symmetry,vol.9, [1] E. Fedorov, “Symmetry in the plane,” Proceedings of the Imperial no. 2, article no. 21, 2017. St. Petersburg Mineralogical Society,vol.2,pp.245–291,1891 (Russian). [2] G. Polya,´ “XII. Uber¨ die analogie der kristallsymmetrie in der ebene,” Zeitschrif fur¨ Kristallographie—Crystalline Materials, vol. 60, no. 1-6, 1924. [3] J. Owen, Te Grammar of Ornament, Van Nostrand Reinhold, 1910. [4] P.S.Stevens,Handbook of Regular Patterns, MIT Press, London, UK, 1981. [5] B. Grunbaum¨ and G. C. Shephard, Tilings and Patterns,Cam- bridge University Press, Cambridge, UK, 1987. [6]M.C.Escher,K.Ford,andJ.W.Vermeulen,Escher on Escher: Exploring the Infnity,N.Harry,Ed.,Abrams,NewYork,NY, USA, 1989. [7] M. Field and M. Golubitsky, Symmetry in Chaos,Oxford University Press, Oxford, UK, 1992. [8]N.C.Carter,R.L.Eagles,S..Grimes,A.C.Hahn,andC. A. Reiter, “Chaotic attractors with discrete planar symmetries,” Chaos, Solitons & Fractals, vol. 9, no. 12, pp. 2031–2054, 1998. [9]K.W.ChungandH.S.Y.Chan,“Symmetricalpatternsfrom dynamics,” Computer Graphics Forum,vol.12,no.1,pp.33–40, 1993. [10] J.Lu,Z.Ye,Y.Zou,andR.Ye,“Orbittraprenderingmethodsfor generating artistic images with crystallographic symmetries,” Computers and Graphics,vol.29,no.5,pp.794–801,2005. [11] D. Douglas and S. John, “Te art of random fractals,” in Pro- ceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture,pp.79–86,2014. [12] J. L. Alperin, Groups and Symmetry. Mathematics Today Twelve Informal Essays, Springer, New York, NY, USA, 1978. [13]H.S.M.CoxeterandW.O.J.Moser,Generators and Relations for Discrete Groups, Springer, New York, NY, USA, 1965. [14] I. R. Shafarevich and A. O. Remizov, “Linear algebra and geometry,” in Gordon and Breach Science PUB, Springer, New York, NY, USA, 1981. [15] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and groups, Springer, New York, NY, USA, 1993. [16] T. Hahn, International Tables for Crystallography, Published for the International Union of Crystallography, Kluwer Academic Publishers, 1987. Advances in Advances in Journal of The Scientifc Journal of Operations Research Decision Sciences Applied Mathematics World Journal Probability and Statistics Hindawi Hindawi Hindawi Hindawi Publishing Corporation Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018

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