Research ArticleThe Visualization of Spherical Patterns with Symmetries ofthe Wallpaper Group
Shihuan Liu ,1,2 Ming Leng,1 and Peichang Ouyang 1
1School of Mathematics & Physics, Jinggangshan University, Jiβan 343009, China2Sichuan 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 Β© 2018 Shihuan Liu et al. This 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 efficient method to generatecolorful wallpaper patterns. Although the constructed mappings have simple form and only two parameters, combined with thecolor scheme of orbit trap algorithm, such mappings can create a great variety of aesthetic wallpaper patterns. The resultingwallpaper patterns are further projected by central projection onto the sphere. This creates the interesting spherical patterns thatpossess infinite symmetries in a finite space.
1. Introduction
Wallpaper groups (or plane crystallographic groups) aremathematical classification of two-dimensional repetitivepatterns. The first systematic proof that there were only 17possible wallpaper patterns was carried out by Fedorov in1891 [1] and later derived independently by PoΜlya in 1924 [2].Wallpaper groups are characterized by translations in twoindependent directions, which give rise to a lattice. Patternswith wallpaper symmetry can be widely found in architectureand decorative art [3β5]. It is surprising that the three-dimensional 230 crystallographic groups were enumeratedbefore the planar wallpaper groups.
The art of M. C. Escher features the rigorous mathe-matical structure and elegant artistic charm, which mightbe the one and only in the history of art. After his journeyto the Alhambra, La Mezquita, and Cordoba, he createdmany mathematically inspired arts and became a master increating wallpaper arts [6]. With the development of moderncomputers, there is considerable research on the automaticgeneration of wallpaper patterns. In [7], Field and Golubitskyfirst proposed the conception of equivariant mappings. Theyconstructed equivariant mapping to generated chaotic cyclic,dihedral, and wallpaper attractors. Carter et al. developed aneasier method that used equivariant truncated 2-dimensional
Fourier series to achieve it [8]. Chung and Chan [9] andLu et al. [10] later presented similar ideas to create colorfulwallpaper patterns. Recently, Douglas and John discovered avery simple approach to yield interesting wallpaper patternsof fractal characteristic [11].
The key idea behind [7β10] is equivariant mapping,which is not easy to achieve, since such mapping must becommutable with respect to symmetry group. In this paper,we present a simple invariant method to create wallpaperpatterns. It has independent mapping form and only twoparameters. Combined with the color scheme of orbit trapalgorithm, our approach can be conveniently utilized to yieldrich wallpaper patterns.
Escherβs Circle Limits IβIV are unusual and visuallyattractive because they realized infinity in a finite unit disc.Inspired by his arts, we use central projection to projectwallpaper patterns onto the finite sphere. This obtains theaesthetic patterns of infinite symmetry structure in thefinite sphere space. Such patterns look beautiful. Combinedwith simulation and printing technologies, these computer-generated patterns could be utilized in wallpaper, textiles,ceramics, carpet, stained glasswindows, and so on, producingboth economic and aesthetic benefits.
The remainder of this paper is organized as follows. InSection 2, we first introduce some basic conceptions and the
HindawiComplexityVolume 2018, Article ID 7315695, 8 pageshttps://doi.org/10.1155/2018/7315695
http://orcid.org/0000-0003-3003-1919http://orcid.org/0000-0003-0447-3190https://doi.org/10.1155/2018/7315695
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Table 1: The concrete invariant mappingπ»ππ΄,π΅(π₯) forms associated with 17 wallpaper groups. In the fourth column, the subscripts π΄ and π΅identify the lattice kind (πΏ π represents square lattice, while πΏπ represents diamond lattice) and wallpaper group type, respectively.Wallpapergroup
Pointgroup Extra symmetry set Invariant mapping
p1 πΆ1 None π»ππΏπ ,π1 (π₯) = βπβπΆ1ππΏπ [π (π₯)]p2 πΆ2 None π»ππΏπ ,π2 (π₯) = βπβπΆ2ππΏπ [π (π₯)]ππ π·1 π1 (π, π) = (π, βπ) π»ππΏπ ,ππ (π₯) = βπβπ·1ππΏπ [π (π₯)] + βπβπ·1ππΏπ [(π1π) (π₯)]πππ π·2 π1(π, π) = (π, βπ),π2(π, π) = (βπ, π) π»ππΏπ ,πππ (π₯) = βπβπ·2ππΏπ [π (π₯)] + 2βπ=1{ βπβπ·2ππΏπ [(πππ) (π₯)]}ππ π·1 π1 (π, π) = (π + π, βπ) π»ππΏπ ,ππ (π₯) = βπβπ·1ππΏπ [π (π₯)] + βπβπ·1ππΏπ [(π1π) (π₯)]πππ π·2 π1(π, π) = (π + π, βπ),π2(π, π) = (βπ, π) π»ππΏπ ,πππ (π₯) = βπβπ·2ππΏπ [π (π₯)] + 2βπ=1{ βπβπ·2ππΏπ [(πππ) (π₯)]}πΆπ π·1 π1(π, π) = (π, βπ),π2(π, π) = (π + π, π β π) π»ππΏπ ,ππ (π₯) = βπβπ·1ππΏπ [π (π₯)] + 2βπ=1{ βπβπ·1ππΏπ [(πππ) (π₯)]}πΆππ π·2 π1(π, π) = (π, βπ),π2(π, π) = (π β π, π + π)π3(π, π) = (π + π, π β π),π4(π, π) = (βπ, π) π»ππΏπ ,πππ (π₯) = βπβπ·2ππΏπ [π (π₯)] +
4βπ=1
{ βπβπ·2
ππΏπ [(πππ) (π₯)]}p4 πΆ4 None π»ππΏπ ,π4 (π₯) = βπβπΆ4ππΏπ [π (π₯)]π4π π·4 π1 (π, π) = (π + π, βπ) π»ππΏπ ,π4π (π₯) = βπβπ·4ππΏπ [π (π₯)] + βπβπ·4ππΏπ [(π1π) (π₯)]π4π π·4 π1 (π, π) = (π, βπ) π»ππΏπ ,π4π (π₯) = βπβπ·4ππΏπ [π (π₯)] + βπβπ·4ππΏππ [(π1π) (π₯)]πππ π·2 π1(π, π) = (π + π, π β π),π2(π, π) = (π β π, π + π) π»ππΏπ ,πππ (π₯) = βπβπ·2ππΏπ [π (π₯)] + 2βπ=1{ βπβπ·2ππΏπ [(πππ) (π₯)]}p3 πΆ3 None π»ππΏπ ,π3 (π₯) = βπβπΆ3ππΏπ [π (π₯)]p3m1 π·3 π1 (π, π) = (βπ, π) π»ππΏπ ,π3π1 (π₯) = βπβπ·3ππΏπ [π (π₯)] + βπβπ·3ππΏπ [(π1π) (π₯)]p31m π·3 π1 (π, π) = (π, βπ) π»ππΏπ ,π31π (π₯) = βπβπ·3ππΏπ [π (π₯)] + βπβπ·3ππΏπ [(π1π) (π₯)]p6 πΆ6 None π»ππΏπ ,π6 (π₯) = βπβπΆ6ππΏπ [π (π₯)]π6π π·6 π1 (π, π) = (π, βπ) π»ππΏπ ,π6π (π₯) = βπβπ·6ππΏπ [π (π₯)] + βπβπ·6ππΏπ [(π1π) (π₯)]lattices with respect to wallpaper groups. To create patternswith symmetries of the wallpaper group, we will explicitlyconstruct invariant mappings associated with 17 wallpapergroups (the concrete mapping forms are summarized inTable 1) in Section 3. In Section 4, we describe how tocreate colorful wallpaper patterns. Finally, we show somespherical wallpaper patterns obtained by central projectionin Section 5.
2. The Lattice of Wallpaper Groups
In geometry and group theory, a lattice in 2-dimensionalEuclidean plane π 2 is essentially a subgroup of π 2. Or,equivalently, for any basis vectors of π 2, the subgroup of alllinear combinations with integer coefficients of the vectors
forms a lattice [12, 13]. Since a lattice is a finitely generated freeabelian group, it is isomorphic to π2 and fully spans the realvector spaceπ 2 [14]. A latticemay be viewed as a regular tilingof a space by a primitive cell. Lattices have many significantapplications in pure mathematics, particularly in connectionto Lie algebras, number theory, and group theory [15].
In this section, wemainly introduce the lattices associatedwith wallpaper group. Firstly, we introduce some basicconceptions.
The symmetry group of an object is the set of all isometriesunder which the object is invariant with composition as thegroup operation. A point group (sometimes called rosettegroup) is a group of isometries that keep at least one pointfixed.
Point groups in π 2 come in two infinite families: dihedralgroup π·π which is the symmetry group of a regular polygon
Complexity 3
and cyclic groupπΆπ that only comprises rotation transforma-tions of π·π. Let π π = ( cos(2π/π) β sin(2π/π)sin(2π/π) cos(2π/π) ) and π = ( β1 00 1 ).Then their matrix group can be represented as πΆπ = {π ππ, π =1, 2, 3, . . . , π} andπ·π = πΆπ βͺ {ππ ππ, π = 1, 2, 3, . . . , π}.
Awallpaper group is a type of topologically discrete groupin π 2 which contains two linearly independent translations.A lattice in π 2 is the symmetry group of discrete translationalsymmetry in two independent directions. A tiling with thislattice of translational symmetry cannot have more but mayhave less symmetry than the lattice itself. Let πΏ be a lattice inπ 2. A lattice πΏβ is called the dual lattice of πΏ if, βπ’ β πΏ andβV β πΏβ, the inner product π’ β V is an integer, where π’ and Vare vectors inπ 2. Letπ be amapping fromπ 2 toπ 2 and letπΊbe a symmetry group in π 2;π is called an invariant mappingwith respect to πΊ if, βπ₯ β π 2 and βπ β πΊ, π(π₯) = π(ππ₯).
By the crystallographic restriction theorem, there areonly 5 lattice types in π 2 [16]. Although wallpaper groupshave totally 17 types, their lattices can be simplified into twolattices: square and diamond lattices. For convenience, werequire that the inner product of the mutual dual lattice of awallpaper group be an integermultiple of 2π.Throughout thepaper, for square lattice, we choose lattice πΏ π = {(1, 0), (0, 1)}with dual lattice πΏβπ = {2π(1, 0), 2π(0, 1)}; for diamond lattice,we choose lattice πΏπ = {(1, 0), (1/2)(β1,β3)}with dual latticeπΏβπ = {(2π/β3)(β3, β1), 2π(0, β2/β3)}.
In this paper, we use standard crystallographic notationsof wallpaper groups [16, 17]. Among 17 wallpaper groups, π1,π2, ππ, πππ, ππ, πππ, ππ, πππ, π4, πππ, π4π, and π4πpossess square lattice, while π3, π3π1, π31π, π6, and π6πpossess diamond lattice.
3. Invariant Mapping with respect toWallpaper Groups
In this section, we explicitly construct invariant mappingsassociated with wallpaper groups. To this end, we first provethe following lemma.
Lemma 1. Suppose that ππ (π = 1, 2, 3, 4) are sine or cosinefunctions, πΊ is a wallpaper group with lattice πΏ = {π΄, π΅}, πΏβ ={π΄β, π΅β} is the dual lattice of πΏ, and π and π are real numbers.Then mapping
ππΏ (π₯) = ( ππ1 {βVβπΏπ2 (π₯ β V) + βVβπΏ (π₯ β V)}ππ3 {βVβπΏπ4 (π₯ β V) + β
VβπΏ(π₯ β V)} ),
βπ₯ β π 2,(1)
is invariant with respect to πΏβ, or ππΏ(π₯) has translationinvariance of πΏβ; that is,
(ππ1 {βVβπΏπ2 ((π’ + π₯) β V) + βVβπΏ ((π’ + π₯) β V)}ππ3 {βVβπΏ
π4 ((π’ + π₯) β V) + βVβπΏ
((π’ + π₯) β V)})
=(ππ1 {βVβπΏπ2 (π₯ β V) + βVβπΏ (π₯ β V)}ππ3 {βVβπΏ
π4 (π₯ β V) + βVβπΏ
(π₯ β V)}) = ππΏ (π₯) ,(2)
where π’ = ππ΄β + ππ΅β, π, π β π.Proof. Since πΏβ is the dual lattice of πΏ, βV β πΏ, we have π’ β V =(ππ΄β + ππ΅β) β V = π(π΄β β V) + π(π΅β β V) = 2ππ for certainπ β π. Thus we get ππ{βVβπΏ ππ((π’+π₯) β V)+βVβπΏ((π’+π₯) β V)} =ππ{βVβπΏ ππ(π₯ β V)+βVβπΏ(π₯ β V)}, since ππ and ππ are functions ofperiod 2π (π, π = 1, 2, 3, 4). Consequently, the mapping ππΏ(π₯)constructed by ππ (π = 1, 2, 3, 4) satisfies (2). This completesthe proof.
Essentially Lemma 1 says that ππΏ(π₯) is a double periodmapping (of period 2π) along the independent translationaldirections of πΏβ.Theorem 2. Let πΊ be a finite group realized by 2 Γ 2 matricesacting on π 2 by multiplication on the right and let π be anarbitrary mapping from π 2 to π 2. Then mappingπ»π,πΊ (π₯) = β
πβπΊ
π [π (π₯)] , π₯ β π 2, (3)is an invariant mapping with respect to πΊ.Proof. For π β πΊ, by closure of the group operation, we seethat ππ runs through πΊ as π does. Therefore we haveπ»π,πΊ [π (π₯)] = β
πβπΊ
π [π (ππ₯)] = βπββπΊ
π [πβ (π₯)]= π»π,πΊ (π₯) , (4)
where πβ = ππ β πΊ. This means thatπ»π,πΊ(π₯) is an invariantmapping with respect to πΊ.
Combining Lemma 1 and Theorem 2, we immediatelyderive the following theorem.
Theorem 3. Let π in Theorem 2 have the form ππΏ as inLemma 1. Suppose that πΊ is a cyclic group πΆπ or dihedralgroup π·π with lattice πΏ; πΏβ is the dual lattice associated withπΏ. Assume that π»ππΏ,πΊ(π₯) is a mapping from π 2 to π 2 of thefollowing form:π»ππΏ,πΊ (π₯) = β
πβπΊ
ππΏ [π (π₯)] , π₯ β π 2. (5)Thenπ»ππΏ,πΊ(π₯) is an invariant mapping with respect to both πΊand πΏβ.
Wallpaper groups possess globally translation symmetryalong two independent directions as well as locally pointgroup symmetry. For the wallpaper groups that only havesymmetries of a certain point group, mapping π»ππΏ,πΊ(π₯) in
4 Complexity
BEGINstart x = 0end x = 6 β 3.1415926start y = 0end y = 6 β 3.1415926 //Set pi = 3.1415926step x = (end x β start x)/X res //Xres is the resolution in X directionstep y = (end y β start y)/Y res //Yres is the resolution in Y directionFOR i = 0 TO X res DO
FOR j = 0 TO Y res DOx = start x + i β step xy = start y + j β step yFOR k = 1 TOMaxIter//MaxIter is the number of iterations, the default set is 100
/βGiven a invariant mappingπ»ππΏ,π€(π₯) associated with a wallpaper group π€ as iterationfunction and initial point (x, y), function Iteration (x, y) iterates MaxIter times. The iteratedsequences are stored in the array Sequenceβ/.
Sequence [π] = Iteration (x, y)END FOR
/βInputting Sequence, the color scheme OrbitTrap outputs the color [r, g, b]β/[r, g, b] = OrbitTrap (Sequence)Set color [r, g, b] to point (x, y)
END FOREND FOR
END
Algorithm 1: CreatingWallpaperPattern()// algorithm for creating patterns with the wallpaper symmetry.
Theorem 3 can be used to create wallpaper patterns well.However, except for the symmetries of a point group, somewallpaper groups may possess other symmetries. For exam-ple, except for symmetries of dihedral group π·3, wallpapergroup π31π still has a reflection along horizontal direction,say symmetry π1(π, π) = (π, βπ) ((π, π) β π 2); besidesπ·2 symmetry, wallpaper group πππ contains perpendicularreflections; that is, π1(π, π) = (π, βπ) and π2(π, π) = (βπ, π).
For a wallpaper group π€ with extra symmetry set Ξ ={π1, π2, π3, . . . , ππ}, based on mapping (5), we add properterms so that the resulting mappingπ»ππΏ,π€(π₯) is also invariantwith respect to π€. This is summarized inTheorem 4.Theorem 4. Let π inTheorem 2 have the form ππΏ in Lemma 1.Suppose that π€ is a wallpaper group with symmetry group πΊand extra symmetry set Ξ = {π1, π2, π3, . . . , ππ}. Assume that πΏis the lattice of π€ and πΏβ is the dual lattice associated with πΏ.Letπ»ππΏ,π€(π₯) be a mapping from π 2 to π 2 of the following form:π»ππΏ,π€ (π₯) = β
πβπΊ
ππΏ [π (π₯)]+ πβπ=1
{{{ βπβπΊ,ππβΞππΏ [(πππ) (π₯)]}}} , π₯ β π 2.(6)
Thenπ»ππΏ,π€(π₯) is an invariant mapping with respect to both π€and πΏβ.
We refer the reader to [7, 8, 17] for more detailed descrip-tion about the extra symmetry set of wallpaper groups. ByTheorems 3-4, we list the invariant mappings associated with17 wallpaper groups in Table 1.
4. Colorful Wallpaper Patterns fromInvariant Mappings
Invariant mapping method is a common approach used increating symmetric patterns [18β23]. Color scheme is analgorithm that is used to determine the color of a point. Givena color scheme and domainπ·, by iterating invariantmappingπ»ππΏ,π€(π₯), π₯ β π·, one can determine the color of π₯. Coloringpoints in π· pointwise, one can obtain a colorful pattern inπ· with symmetries of the wallpaper group π€. Figures 1-2are four wallpaper patterns obtained in this manner. Thesepatterns were created by VC++ 6.0 on a PC (SVGA). InAlgorithm 1, we provide the pseudocode so that the interestedreader can create their own colorful wallpaper patterns.
The color scheme used in this paper is called orbit trapalgorithm [10].We refer the reader to [10, 20] for more detailsabout the algorithm (the algorithm is named as functionOrbitTrap() in Algorithm 1). It hasmany parameters to adjustcolor, which could enhance the visual appeal of patternseffectively. Compared with the complex equivariant mappingconstructed in [7β10], our invariant mappings possess notonly simple form but also sensitive dynamical system prop-erty, which can be used to produce infinite wallpaper patternseasily. For example, Figures 1(a) and 2(b) were createdby mappings π»ππΏπ ,π4π(π₯) and π»ππΏπ ,π3π1(π₯), respectively, inwhich the specific mappings ππΏπ and ππΏπ were
ππΏπ =(2.12 cos{βVβπΏπ cos (π₯ β V) + βVβπΏπ (π₯ β V)}1.03 cos{βVβπΏπ
sin (π₯ β V) + βVβπΏπ
(π₯ β V)}), (7)
Complexity 5
(a) (b)
Figure 1: Colorful wallpaper patterns with the π4π (a) and π6π (b) symmetry.
(a) (b)
Figure 2: Colorful wallpaper patterns with the πππ (a) and π3π1 (b) symmetry.ππΏπ =( 1.1 cos{βVβπΏπ sin (π₯ β V) + βVβπΏπ (π₯ β V)}β0.52 sin{β
VβπΏπ
cos (π₯ β V) + βVβπΏπ
(π₯ β V)}). (8)It seems that the deference between (7) and (8) is not verysignificant. However, by Table 1, mappings π»ππΏπ ,π4π(π₯) andπ»ππΏπ ,π3π1(π₯) have 16 and 12 summation terms, respectively.The cumulative difference will be very obvious, which isenough to produce different style patterns.
5. Spherical Wallpaper Patterns byCentral Projection
In this section, we introduce central projection to yieldspherical patterns of the wallpaper symmetry.
Let π2 = {(π, π, π) β π 3 | π2+π2+π2 = 1} be the unit spherein π 3, let π = πΉ be a projection plane, where πΉ is a negativeconstant. Assume that π(π, π, π) β π 3; then π(π, π, π) β π 3andπ(βπ, βπ, βπ) β π 3 are a pair of antipodal points. For anypoint π(π, π, π) β π 3, there exist a unique line πΏ through theorigin (0, 0, 0) and π (and π) which intersects the projectionplane π = πΉ at point (πΌ, π½, πΉ). Denote the projection by π. Byanalytic geometry, it is easy to check that
[[[πΌπ½πΉ ]]] = π (π, π, π) = π (βπ, βπ, βπ) = [[[
πππ ]]] πΉπ . (9)Because the projection point is at the center of π2, we call π ascentral projection.
The choice of the planeπ = πΉ has a great influence on thespherical patterns. If plane π = πΉ is too close to coordinateplane πππ, the resulting spherical pattern only shows a few
6 Complexity
(a) (b)
Figure 3: Two spherical wallpaper patterns with the π4π symmetry, in which the projection plane was set as π = β2π (a) and π = β4π (b).
(a) (b)
Figure 4: Colorful spherical wallpaper patterns with the π4π (a) and π6π (b) symmetry.periods of the wallpaper pattern. However, if plane π = πΉis too far away from coordinate plane πππ, the wallpaperpattern on the sphere may appear small so that we cannotidentify symmetries of the wallpaper pattern well. Figure 3illustrates the contrast effect of the setting of plane π = πΉ.
Given a wallpaper pattern, by central projection π, we canmap it onto the sphere π2 and obtain a corresponding spher-ical wallpaper pattern. We next explain how to implement itin more detail.
Suppose that (π, π, π) β π 3 and π»ππΏ,π€(π₯) is an invari-ant mapping compatible with the symmetry of wallpapergroup π€. First, by central projection π, we obtain a corre-sponding point ((π΄/π)π, (π΄/π)π, πΉ) on the projection planeπ = πΉ. Second, let π»ππΏ,π€(π₯) be iteration function and letπ₯((π΄/π)π, (π΄/π)π) be initial point; using the color schemeof orbit trap, we assign a color to point ((π΄/π)π, (π΄/π)π, πΉ).Finally, repeat the second step; by coloring unit sphere π2pointwise, we obtain a spherical pattern of the wallpapergroup π€ symmetry.
Figures 3β7 are ten patterns obtained by this manner.Except for Figure 3(b) in which the projection plane wasset as π = β4π, all the other projection planes were set asπ = β2π. We utilized the wallpaper patterns of Figure 1 toproduce spherical patterns shown in Figure 4. For beauty, allthe camera views are perpendicular to plane πππ and passthe origin, except for Figure 7(b), where the camera view aimsat the equator of π2. To better understand the effect of centralprojection, Figure 7 demonstrates spherical patterns that areobserved from different perspectives.
Additional Points
Theartistic patterns created in this article have significant aes-thetic and economic value.We plan to produce somematerialobjects with the help of simulation and printing technologies.We produced Figures 1β7 in the VC++ 6.0 programmingenvironment with the aid of OpenGL, a powerful graphicssoftware package.
Complexity 7
(a) (b)
Figure 5: Colorful spherical wallpaper patterns with the π6π (a) and πππ (b) symmetry.
(a) (b)
Figure 6: Colorful spherical wallpaper patterns with the π4 (a) and π6π (b) symmetry.
(a) (b)
Figure 7: Two spherical wallpaper patterns with the π3π symmetry. The camera view of (a) is perpendicular to plane πππ and passed theorigin, while (b) aims at equator.
8 Complexity
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper.
Acknowledgments
The authors acknowledge Adobe and Microsoft for theirfriendly technical support. This work was supported by theNationalNatural Science Foundation of China (nos. 11461035,11761039, and 61363014), Young Scientist Training Program ofJiangxi Province (20153BCB23003), Science and TechnologyPlan Project of Jiangxi Provincial Education Department(no. GJJ160749), and Doctoral Startup Fund of JinggangshanUniversity (no. JZB1303).
References
[1] E. Fedorov, βSymmetry in the plane,β Proceedings of the ImperialSt. Petersburg Mineralogical Society, vol. 2, pp. 245β291, 1891(Russian).
[2] G. PoΜlya, βXII. UΜber die analogie der kristallsymmetrie in derebene,β Zeitschrift fuΜr KristallographieβCrystalline Materials,vol. 60, no. 1-6, 1924.
[3] J. Owen, The Grammar of Ornament, Van Nostrand Reinhold,1910.
[4] P. S. Stevens,Handbook of Regular Patterns, MIT Press, London,UK, 1981.
[5] B. GruΜnbaum and G. C. Shephard, Tilings and Patterns, Cam-bridge University Press, Cambridge, UK, 1987.
[6] M. C. Escher, K. Ford, and J. W. Vermeulen, Escher on Escher:Exploring the Infinity, N. Harry, Ed., Abrams, New York, NY,USA, 1989.
[7] M. Field and M. Golubitsky, Symmetry in Chaos, OxfordUniversity Press, Oxford, UK, 1992.
[8] N. C. Carter, R. L. Eagles, S. . Grimes, A. C. Hahn, and C.A. Reiter, βChaotic attractors with discrete planar symmetries,βChaos, Solitons & Fractals, vol. 9, no. 12, pp. 2031β2054, 1998.
[9] K. W. Chung and H. S. Y. Chan, βSymmetrical patterns fromdynamics,β Computer Graphics Forum, vol. 12, no. 1, pp. 33β40,1993.
[10] J. Lu, Z. Ye, Y. Zou, and R. Ye, βOrbit trap renderingmethods forgenerating artistic images with crystallographic symmetries,βComputers and Graphics, vol. 29, no. 5, pp. 794β801, 2005.
[11] D. Douglas and S. John, βThe art of random fractals,β in Pro-ceedings of Bridges 2014: Mathematics, Music, Art, Architecture,Culture, pp. 79β86, 2014.
[12] J. L. Alperin, Groups and Symmetry. Mathematics Today TwelveInformal Essays, Springer, New York, NY, USA, 1978.
[13] H. S. M. Coxeter and W. O. J. Moser, Generators and Relationsfor Discrete Groups, Springer, New York, NY, USA, 1965.
[14] I. R. Shafarevich and A. O. Remizov, βLinear algebra andgeometry,β in Gordon and Breach Science PUB, Springer, NewYork, NY, USA, 1981.
[15] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices andgroups, Springer, New York, NY, USA, 1993.
[16] T. Hahn, International Tables for Crystallography, Published forthe International Union of Crystallography, Kluwer AcademicPublishers, 1987.
[17] V. E. Armstrong, Groups and Symmetry, Springer, New York,NY, USA, 1987.
[18] K. W. Chung and H. M. Ma, βAutomatic generation of aestheticpatterns on fractal tilings by means of dynamical systems,βChaos, Solitons & Fractals, vol. 24, no. 4, pp. 1145β1158, 2005.
[19] P. Ouyang and X. Wang, βBeautiful mathβaesthetic patternsbased on logarithmic spirals,β IEEE Computer Graphics andApplications, vol. 33, no. 6, pp. 21β23, 2013.
[20] P. Ouyang, D. Cheng, Y. Cao, and X. Zhan, βThe visualizationof hyperbolic patterns from invariant mapping method,β Com-puters and Graphics, vol. 36, no. 2, pp. 92β100, 2012.
[21] P. Ouyang and R.W. Fathauer, βBeautiful math, part 2: aestheticpatterns based on fractal tilings,β IEEE Computer Graphics andApplications, vol. 34, no. 1, pp. 68β76, 2014.
[22] P. Ouyang and K. Chung, βBeautiful math, part 3: hyperbolicaesthetic patterns based on conformal mappings,β IEEE Com-puter Graphics and Applications, vol. 34, no. 2, pp. 72β79, 2014.
[23] P. Ouyang, L. Wang, T. Yu, and X. Huang, βAesthetic patternswith symmetries of the regular polyhedron,β Symmetry, vol. 9,no. 2, article no. 21, 2017.
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