The Visualization of Spherical Patterns with Symmetries of...
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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 2 Sichuan 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. is 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 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. e resulting wallpaper patterns are further projected by central projection onto the sphere. is creates the interesting spherical patterns that possess infinite symmetries in a finite space. 1. Introduction Wallpaper groups (or plane crystallographic groups) are mathematical classification of two-dimensional repetitive patterns. e first systematic proof that there were only 17 possible wallpaper patterns was carried out by Fedorov in 1891 [1] and later derived independently by P´ olya in 1924 [2]. Wallpaper groups are characterized by translations in two independent directions, which give rise to a lattice. Patterns with wallpaper symmetry can be widely found in architecture and decorative art [3–5]. It is surprising that the three- dimensional 230 crystallographic groups were enumerated before the planar wallpaper groups. e art of M. C. Escher features the rigorous mathe- matical structure and elegant artistic charm, which might be the one and only in the history of art. Aſter his journey to the Alhambra, La Mezquita, and Cordoba, he created many mathematically inspired arts and became a master in creating wallpaper arts [6]. With the development of modern computers, there is considerable research on the automatic generation of wallpaper patterns. In [7], Field and Golubitsky first proposed the conception of equivariant mappings. ey constructed equivariant mapping to generated chaotic cyclic, dihedral, and wallpaper attractors. Carter et al. developed an easier method that used equivariant truncated 2-dimensional Fourier series to achieve it [8]. Chung and Chan [9] and Lu et al. [10] later presented similar ideas to create colorful wallpaper patterns. Recently, Douglas and John discovered a very simple approach to yield interesting wallpaper patterns of fractal characteristic [11]. e key idea behind [7–10] is equivariant mapping, which is not easy to achieve, since such mapping must be commutable with respect to symmetry group. In this paper, we present a simple invariant method to create wallpaper patterns. It has independent mapping form and only two parameters. Combined with the color scheme of orbit trap algorithm, our approach can be conveniently utilized to yield rich wallpaper patterns. Escher’s Circle Limits I–IV are unusual and visually attractive because they realized infinity in a finite unit disc. Inspired by his arts, we use central projection to project wallpaper patterns onto the finite sphere. is obtains the aesthetic patterns of infinite symmetry structure in the finite sphere space. Such patterns look beautiful. Combined with simulation and printing technologies, these computer- generated patterns could be utilized in wallpaper, textiles, ceramics, carpet, stained glass windows, and so on, producing both economic and aesthetic benefits. e remainder of this paper is organized as follows. In Section 2, we first introduce some basic conceptions and the Hindawi Complexity Volume 2018, Article ID 7315695, 8 pages https://doi.org/10.1155/2018/7315695
Transcript of The Visualization of Spherical Patterns with Symmetries of...
Research ArticleThe Visualization of Spherical Patterns with Symmetries ofthe Wallpaper Group
Shihuan Liu 12 Ming Leng1 and Peichang Ouyang 1
1School of Mathematics amp Physics Jinggangshan University Jirsquoan 343009 China2Sichuan Province Key Lab of Signal and Information Processing Southwest Jiaotong University Chengdu 611756 China
Correspondence should be addressed to Peichang Ouyang g fcayang163com
Received 17 October 2017 Accepted 1 January 2018 Published 12 February 2018
Academic Editor Michele Scarpiniti
Copyright copy 2018 Shihuan Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich 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 Polya 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 [3ndash5] 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 cyclicdihedral 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 [7ndash10] is equivariant mappingwhich is not easy to achieve since such mapping must becommutable with respect to symmetry group In this paperwe 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
Escherrsquos Circle Limits IndashIV are unusual and visuallyattractive because they realized infinity in a finite unit discInspired 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 textilesceramics 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 pageshttpsdoiorg10115520187315695
2 Complexity
Table 1 The concrete invariant mapping119867119891119860119861(119909) forms associated with 17 wallpaper groups In the fourth column the subscripts 119860 and 119861identify the lattice kind (119871 119904 represents square lattice while 119871119889 represents diamond lattice) and wallpaper group type respectively
Wallpapergroup
Pointgroup Extra symmetry set Invariant mapping
p1 1198621 None 119867119891119871119904 1199011 (119909) = sum119892isin1198621
119891119871119904 [119892 (119909)]p2 1198622 None 119867119891119871119904 1199012 (119909) = sum
119892isin1198622
119891119871119904 [119892 (119909)]119901119898 1198631 1205901 (119886 119887) = (119886 minus119887) 119867119891119871119904 119901119898 (119909) = sum119892isin1198631
119891119871119904 [119892 (119909)] + sum119892isin1198631
119891119871119904 [(1205901119892) (119909)]119901119898119898 1198632 1205901(119886 119887) = (119886 minus119887)1205902(119886 119887) = (minus119886 119887) 119867119891119871119904 119901119898119898 (119909) = sum119892isin1198632
119891119871119904 [119892 (119909)] + 2sum119894=1
sum119892isin1198632
119891119871119904 [(120590119894119892) (119909)]119901119892 1198631 1205901 (119886 119887) = (120587 + 119886 minus119887) 119867119891119871119904 119901119892 (119909) = sum119892isin1198631
119891119871119904 [119892 (119909)] + sum119892isin1198631
119891119871119904 [(1205901119892) (119909)]119901119898119892 1198632 1205901(119886 119887) = (120587 + 119886 minus119887)1205902(119886 119887) = (minus119886 119887) 119867119891119871119904 119901119898119892 (119909) = sum119892isin1198632
119891119871119904 [119892 (119909)] + 2sum119894=1
sum119892isin1198632
119891119871119904 [(120590119894119892) (119909)]119862119898 1198631 1205901(119886 119887) = (119886 minus119887)1205902(119886 119887) = (120587 + 119886 120587 minus 119887) 119867119891119871119904 119888119898 (119909) = sum119892isin1198631
119891119871119904 [119892 (119909)] + 2sum119894=1
sum119892isin1198631
119891119871119904 [(120590119894119892) (119909)]119862119898119898 1198632 1205901(119886 119887) = (119886 minus119887)1205902(119886 119887) = (120587 minus 119886 120587 + 119887)1205903(119886 119887) = (120587 + 119886 120587 minus 119887)1205904(119886 119887) = (minus119886 119887) 119867119891119871119904 119888119898119898 (119909) = sum
119892isin1198632
119891119871119904 [119892 (119909)] + 4sum119894=1
sum119892isin1198632
119891119871119904 [(120590119894119892) (119909)]p4 1198624 None 119867119891119871119904 1199014 (119909) = sum
119892isin1198624
119891119871119904 [119892 (119909)]1199014119892 1198634 1205901 (119886 119887) = (120587 + 119886 minus119887) 119867119891119871119904 1199014119892 (119909) = sum119892isin1198634
119891119871119904 [119892 (119909)] + sum119892isin1198634
119891119871119904 [(1205901119892) (119909)]1199014119898 1198634 1205901 (119886 119887) = (119886 minus119887) 119867119891119871119904 1199014119898 (119909) = sum119892isin1198634
119891119871119904 [119892 (119909)] + sum119892isin1198634
119891119871119889119904 [(1205901119892) (119909)]119901119892119892 1198632 1205901(119886 119887) = (120587 + 119886 120587 minus 119887)1205902(119886 119887) = (120587 minus 119886 120587 + 119887) 119867119891119871119904 119901119892119892 (119909) = sum119892isin1198632
119891119871119904 [119892 (119909)] + 2sum119894=1
sum119892isin1198632
119891119871119904 [(120590119894119892) (119909)]p3 1198623 None 119867119891119871119889 1199013 (119909) = sum
119892isin1198623
119891119871119889 [119892 (119909)]p3m1 1198633 1205901 (119886 119887) = (minus119886 119887) 119867119891119871119889 11990131198981 (119909) = sum
119892isin1198633
119891119871119889 [119892 (119909)] + sum119892isin1198633
119891119871119889 [(1205901119892) (119909)]p31m 1198633 1205901 (119886 119887) = (119886 minus119887) 119867119891119871119889 11990131119898 (119909) = sum
119892isin1198633
119891119871119889 [119892 (119909)] + sum119892isin1198633
119891119871119889 [(1205901119892) (119909)]p6 1198626 None 119867119891119871119889 1199016 (119909) = sum
119892isin1198626
119891119871119889 [119892 (119909)]1199016119898 1198636 1205901 (119886 119887) = (119886 minus119887) 119867119891119871119889 1199016119898 (119909) = sum119892isin1198636
119891119871119889 [119892 (119909)] + sum119892isin1198636
119891119871119889 [(1205901119892) (119909)]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 1198772 is essentially a subgroup of 1198772 Orequivalently for any basis vectors of 1198772 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 1198852 and fully spans the realvector space1198772 [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 1198772 come in two infinite families dihedralgroup 119863119899 which is the symmetry group of a regular polygon
Complexity 3
and cyclic group119862119899 that only comprises rotation transforma-tions of 119863119899 Let 119877119899 = ( cos(2120587119899) minus sin(2120587119899)
sin(2120587119899) cos(2120587119899) ) and 119879 = ( minus1 00 1 )Then their matrix group can be represented as 119862119899 = 119877119894119899 119894 =1 2 3 119899 and119863119899 = 119862119899 cup 119879119877119894119899 119894 = 1 2 3 119899
Awallpaper group is a type of topologically discrete groupin 1198772 which contains two linearly independent translationsA lattice in 1198772 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 119871 be a lattice in1198772 A lattice 119871lowast is called the dual lattice of 119871 if forall119906 isin 119871 andforallV isin 119871lowast the inner product 119906 sdot V is an integer where 119906 and Vare vectors in1198772 Let119872 be amapping from1198772 to1198772 and let119866be a symmetry group in 1198772119872 is called an invariant mappingwith respect to 119866 if forall119909 isin 1198772 and forall119892 isin 119866 119872(119909) = 119872(119892119909)
By the crystallographic restriction theorem there areonly 5 lattice types in 1198772 [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 2120587Throughout thepaper for square lattice we choose lattice 119871 119904 = (1 0) (0 1)with dual lattice 119871lowast119904 = 2120587(1 0) 2120587(0 1) for diamond latticewe choose lattice 119871119889 = (1 0) (12)(minus1radic3)with dual lattice119871lowast119889 = (2120587radic3)(radic3 minus1) 2120587(0 minus2radic3)
In this paper we use standard crystallographic notationsof wallpaper groups [16 17] Among 17 wallpaper groups 11990111199012 119901119898 119901119898119898 119901119892 119901119898119892 119888119898 119888119898119898 1199014 119901119892119892 1199014119892 and 1199014119898possess square lattice while 1199013 11990131198981 11990131119898 1199016 and 1199016119898possess 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 119891119894 (119894 = 1 2 3 4) are sine or cosinefunctions 119866 is a wallpaper group with lattice 119871 = 119860 119861 119871lowast =119860lowast 119861lowast is the dual lattice of 119871 and 119886 and 119887 are real numbersThen mapping
119891119871 (119909) = ( 1198861198911 sumVisin1198711198912 (119909 sdot V) + sum
Visin119871(119909 sdot V)1198871198913 sum
Visin1198711198914 (119909 sdot V) + sum
Visin119871(119909 sdot V) )
forall119909 isin 1198772(1)
is invariant with respect to 119871lowast or 119891119871(119909) has translationinvariance of 119871lowast that is
(1198861198911 sumVisin119871
1198912 ((119906 + 119909) sdot V) + sumVisin119871
((119906 + 119909) sdot V)1198871198913 sumVisin119871
1198914 ((119906 + 119909) sdot V) + sumVisin119871
((119906 + 119909) sdot V))
=(1198861198911 sumVisin119871
1198912 (119909 sdot V) + sumVisin119871
(119909 sdot V)1198871198913 sumVisin119871
1198914 (119909 sdot V) + sumVisin119871
(119909 sdot V)) = 119891119871 (119909) (2)
where 119906 = 119898119860lowast + 119899119861lowast 119898 119899 isin 119885Proof Since 119871lowast is the dual lattice of 119871 forallV isin 119871 we have 119906 sdotV =(119898119860lowast + 119899119861lowast) sdot V = 119898(119860lowast sdot V) + 119899(119861lowast sdot V) = 2119896120587 for certain119896 isin 119885 Thus we get 119891119894sumVisin119871 119891119895((119906+119909) sdotV)+sumVisin119871((119906+119909) sdotV) =119891119894sumVisin119871 119891119895(119909 sdotV)+sumVisin119871(119909 sdotV) since 119891119894 and 119891119895 are functions ofperiod 2120587 (119894 119895 = 1 2 3 4) Consequently the mapping 119891119871(119909)constructed by 119891119894 (119894 = 1 2 3 4) satisfies (2) This completesthe proof
Essentially Lemma 1 says that 119891119871(119909) is a double periodmapping (of period 2120587) along the independent translationaldirections of 119871lowastTheorem 2 Let 119866 be a finite group realized by 2 times 2 matricesacting on 1198772 by multiplication on the right and let 119891 be anarbitrary mapping from 1198772 to 1198772 Then mapping119867119891119866 (119909) = sum
119892isin119866
119891 [119892 (119909)] 119909 isin 1198772 (3)
is an invariant mapping with respect to 119866Proof For 120590 isin 119866 by closure of the group operation we seethat 119892120590 runs through 119866 as 119892 does Therefore we have119867119891119866 [120590 (119909)] = sum
119892isin119866
119891 [120590 (119892119909)] = sum119892lowastisin119866
119891 [119892lowast (119909)]= 119867119891119866 (119909) (4)
where 119892lowast = 120590119892 isin 119866 This means that119867119891119866(119909) is an invariantmapping with respect to 119866
Combining Lemma 1 and Theorem 2 we immediatelyderive the following theorem
Theorem 3 Let 119891 in Theorem 2 have the form 119891119871 as inLemma 1 Suppose that 119866 is a cyclic group 119862119899 or dihedralgroup 119863119899 with lattice 119871 119871lowast is the dual lattice associated with119871 Assume that 119867119891119871119866(119909) is a mapping from 1198772 to 1198772 of thefollowing form119867119891119871119866 (119909) = sum
119892isin119866
119891119871 [119892 (119909)] 119909 isin 1198772 (5)
Then119867119891119871119866(119909) is an invariant mapping with respect to both 119866and 119871lowast
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 119867119891119871119866(119909) in
4 Complexity
BEGINstart x = 0end x = 6 lowast 31415926start y = 0end y = 6 lowast 31415926 Set pi = 31415926step x = (end x ndash start x)X res Xres is the resolution in X directionstep y = (end y ndash 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 lowast step xy = start y + j lowast step yFOR k = 1 TOMaxIterMaxIter is the number of iterations the default set is 100
lowastGiven a invariant mapping119867119891119871119908(119909) associated with a wallpaper group 119908 as iterationfunction and initial point (x y) function Iteration (x y) iterates MaxIter times The iteratedsequences are stored in the array Sequencelowast
Sequence [119896] = Iteration (x y)END FOR
lowastInputting Sequence the color scheme OrbitTrap outputs the color [r g b]lowast[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 wellHowever except for the symmetries of a point group somewallpaper groups may possess other symmetries For exam-ple except for symmetries of dihedral group 1198633 wallpapergroup 11990131119898 still has a reflection along horizontal directionsay symmetry 1205901(119886 119887) = (119886 minus119887) ((119886 119887) isin 1198772) besides1198632 symmetry wallpaper group 119901119898119898 contains perpendicularreflections that is 1205901(119886 119887) = (119886 minus119887) and 1205902(119886 119887) = (minus119886 119887)
For a wallpaper group 119908 with extra symmetry set Δ =1205901 1205902 1205903 120590119899 based on mapping (5) we add properterms so that the resulting mapping119867119891119871119908(119909) is also invariantwith respect to 119908 This is summarized inTheorem 4
Theorem 4 Let 119891 inTheorem 2 have the form 119891119871 in Lemma 1Suppose that 119908 is a wallpaper group with symmetry group 119866and extra symmetry set Δ = 1205901 1205902 1205903 120590119899 Assume that 119871is the lattice of 119908 and 119871lowast is the dual lattice associated with 119871Let119867119891119871119908(119909) be a mapping from 1198772 to 1198772 of the following form119867119891119871119908 (119909) = sum
119892isin119866
119891119871 [119892 (119909)]+ 119899sum119894=1
sum119892isin119866120590119894isinΔ
119891119871 [(120590119894119892) (119909)] 119909 isin 1198772 (6)
Then119867119891119871119908(119909) is an invariant mapping with respect to both 119908and 119871lowast
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 [18ndash23] Color scheme is analgorithm that is used to determine the color of a point Givena color scheme and domain119863 by iterating invariantmapping119867119891119871119908(119909) 119909 isin 119863 one can determine the color of 119909 Coloringpoints in 119863 pointwise one can obtain a colorful pattern in119863 with symmetries of the wallpaper group 119908 Figures 1-2are four wallpaper patterns obtained in this manner Thesepatterns were created by VC++ 60 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 [7ndash10] 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 119867119891119871119904 1199014119898(119909) and 119867119891119871119889 11990131198981(119909) respectively inwhich the specific mappings 119891119871119904 and 119891119871119889 were
119891119871119904 =(212 cossumVisin119871119904
cos (119909 sdot V) + sumVisin119871119904
(119909 sdot V)103 cossum
Visin119871119904
sin (119909 sdot V) + sumVisin119871119904
(119909 sdot V)) (7)
Complexity 5
(a) (b)
Figure 1 Colorful wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 2 Colorful wallpaper patterns with the 119901119898119898 (a) and 11990131198981 (b) symmetry
119891119871119889 =( 11 cossumVisin119871119889
sin (119909 sdot V) + sumVisin119871119889
(119909 sdot V)minus052 sinsum
Visin119871119889
cos (119909 sdot V) + sumVisin119871119889
(119909 sdot V)) (8)
It seems that the deference between (7) and (8) is not verysignificant However by Table 1 mappings 119867119891119871119904 1199014119898(119909) and119867119891119871119889 11990131198981(119909) have 16 and 12 summation terms respectivelyThe 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 1198782 = (119886 119887 119888) isin 1198773 | 1198862+1198872+1198882 = 1 be the unit spherein 1198773 let 119885 = 119865 be a projection plane where 119865 is a negativeconstant Assume that 119875(119886 119887 119888) isin 1198773 then 119875(119886 119887 119888) isin 1198773and1198751015840(minus119886 minus119887 minus119888) isin 1198773 are a pair of antipodal points For anypoint 119875(119886 119887 119888) isin 1198773 there exist a unique line 119871 through theorigin (0 0 0) and 119875 (and 1198751015840) which intersects the projectionplane 119885 = 119865 at point (120572 120573 119865) Denote the projection by 120591 Byanalytic geometry it is easy to check that
[[[120572120573119865 ]]] = 120591 (119886 119887 119888) = 120591 (minus119886 minus119887 minus119888) = [[[
119886119887119888 ]]] 119865119888 (9)
Because the projection point is at the center of 1198782 we call 120591 ascentral projection
The choice of the plane119885 = 119865 has a great influence on thespherical patterns If plane 119885 = 119865 is too close to coordinateplane 119883119874119884 the resulting spherical pattern only shows a few
6 Complexity
(a) (b)
Figure 3 Two spherical wallpaper patterns with the 1199014119898 symmetry in which the projection plane was set as 119885 = minus2120587 (a) and 119885 = minus4120587 (b)
(a) (b)
Figure 4 Colorful spherical wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
periods of the wallpaper pattern However if plane 119885 = 119865is too far away from coordinate plane 119883119874119884 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 119885 = 119865
Given a wallpaper pattern by central projection 120591 we canmap it onto the sphere 1198782 and obtain a corresponding spher-ical wallpaper pattern We next explain how to implement itin more detail
Suppose that (119886 119887 119888) isin 1198773 and 119867119891119871119908(119909) is an invari-ant mapping compatible with the symmetry of wallpapergroup 119908 First by central projection 120591 we obtain a corre-sponding point ((119860119888)119886 (119860119888)119887 119865) on the projection plane119885 = 119865 Second let 119867119891119871119908(119909) be iteration function and let119909((119860119888)119886 (119860119888)119887) be initial point using the color schemeof orbit trap we assign a color to point ((119860119888)119886 (119860119888)119887 119865)Finally repeat the second step by coloring unit sphere 1198782pointwise we obtain a spherical pattern of the wallpapergroup 119908 symmetry
Figures 3ndash7 are ten patterns obtained by this mannerExcept for Figure 3(b) in which the projection plane wasset as 119885 = minus4120587 all the other projection planes were set as119885 = minus2120587 We utilized the wallpaper patterns of Figure 1 toproduce spherical patterns shown in Figure 4 For beauty allthe camera views are perpendicular to plane 119883119874119884 and passthe origin except for Figure 7(b) where the camera view aimsat the equator of 1198782 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 valueWe plan to produce somematerialobjects with the help of simulation and printing technologiesWe produced Figures 1ndash7 in the VC++ 60 programmingenvironment with the aid of OpenGL a powerful graphicssoftware package
Complexity 7
(a) (b)
Figure 5 Colorful spherical wallpaper patterns with the 1199016119898 (a) and 119901119898119898 (b) symmetry
(a) (b)
Figure 6 Colorful spherical wallpaper patterns with the 1199014 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 7 Two spherical wallpaper patterns with the 1199013119898 symmetry The camera view of (a) is perpendicular to plane 119883119874119884 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 1146103511761039 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 ldquoSymmetry in the planerdquo Proceedings of the ImperialSt Petersburg Mineralogical Society vol 2 pp 245ndash291 1891(Russian)
[2] G Polya ldquoXII Uber die analogie der kristallsymmetrie in derebenerdquo Zeitschrift fur KristallographiemdashCrystalline Materialsvol 60 no 1-6 1924
[3] J Owen The Grammar of Ornament Van Nostrand Reinhold1910
[4] P S StevensHandbook of Regular Patterns MIT Press LondonUK 1981
[5] B Grunbaum 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 EscherExploring the Infinity N Harry Ed Abrams New York NYUSA 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 CA Reiter ldquoChaotic attractors with discrete planar symmetriesrdquoChaos Solitons amp Fractals vol 9 no 12 pp 2031ndash2054 1998
[9] K W Chung and H S Y Chan ldquoSymmetrical patterns fromdynamicsrdquo Computer Graphics Forum vol 12 no 1 pp 33ndash401993
[10] J Lu Z Ye Y Zou and R Ye ldquoOrbit trap renderingmethods forgenerating artistic images with crystallographic symmetriesrdquoComputers and Graphics vol 29 no 5 pp 794ndash801 2005
[11] D Douglas and S John ldquoThe art of random fractalsrdquo in Pro-ceedings of Bridges 2014 Mathematics Music Art ArchitectureCulture pp 79ndash86 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 ldquoLinear algebra andgeometryrdquo 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 YorkNY USA 1987
[18] K W Chung and H M Ma ldquoAutomatic generation of aestheticpatterns on fractal tilings by means of dynamical systemsrdquoChaos Solitons amp Fractals vol 24 no 4 pp 1145ndash1158 2005
[19] P Ouyang and X Wang ldquoBeautiful mathmdashaesthetic patternsbased on logarithmic spiralsrdquo IEEE Computer Graphics andApplications vol 33 no 6 pp 21ndash23 2013
[20] P Ouyang D Cheng Y Cao and X Zhan ldquoThe visualizationof hyperbolic patterns from invariant mapping methodrdquo Com-puters and Graphics vol 36 no 2 pp 92ndash100 2012
[21] P Ouyang and RW Fathauer ldquoBeautiful math part 2 aestheticpatterns based on fractal tilingsrdquo IEEE Computer Graphics andApplications vol 34 no 1 pp 68ndash76 2014
[22] P Ouyang and K Chung ldquoBeautiful math part 3 hyperbolicaesthetic patterns based on conformal mappingsrdquo IEEE Com-puter Graphics and Applications vol 34 no 2 pp 72ndash79 2014
[23] P Ouyang L Wang T Yu and X Huang ldquoAesthetic patternswith symmetries of the regular polyhedronrdquo Symmetry vol 9no 2 article no 21 2017
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2 Complexity
Table 1 The concrete invariant mapping119867119891119860119861(119909) forms associated with 17 wallpaper groups In the fourth column the subscripts 119860 and 119861identify the lattice kind (119871 119904 represents square lattice while 119871119889 represents diamond lattice) and wallpaper group type respectively
Wallpapergroup
Pointgroup Extra symmetry set Invariant mapping
p1 1198621 None 119867119891119871119904 1199011 (119909) = sum119892isin1198621
119891119871119904 [119892 (119909)]p2 1198622 None 119867119891119871119904 1199012 (119909) = sum
119892isin1198622
119891119871119904 [119892 (119909)]119901119898 1198631 1205901 (119886 119887) = (119886 minus119887) 119867119891119871119904 119901119898 (119909) = sum119892isin1198631
119891119871119904 [119892 (119909)] + sum119892isin1198631
119891119871119904 [(1205901119892) (119909)]119901119898119898 1198632 1205901(119886 119887) = (119886 minus119887)1205902(119886 119887) = (minus119886 119887) 119867119891119871119904 119901119898119898 (119909) = sum119892isin1198632
119891119871119904 [119892 (119909)] + 2sum119894=1
sum119892isin1198632
119891119871119904 [(120590119894119892) (119909)]119901119892 1198631 1205901 (119886 119887) = (120587 + 119886 minus119887) 119867119891119871119904 119901119892 (119909) = sum119892isin1198631
119891119871119904 [119892 (119909)] + sum119892isin1198631
119891119871119904 [(1205901119892) (119909)]119901119898119892 1198632 1205901(119886 119887) = (120587 + 119886 minus119887)1205902(119886 119887) = (minus119886 119887) 119867119891119871119904 119901119898119892 (119909) = sum119892isin1198632
119891119871119904 [119892 (119909)] + 2sum119894=1
sum119892isin1198632
119891119871119904 [(120590119894119892) (119909)]119862119898 1198631 1205901(119886 119887) = (119886 minus119887)1205902(119886 119887) = (120587 + 119886 120587 minus 119887) 119867119891119871119904 119888119898 (119909) = sum119892isin1198631
119891119871119904 [119892 (119909)] + 2sum119894=1
sum119892isin1198631
119891119871119904 [(120590119894119892) (119909)]119862119898119898 1198632 1205901(119886 119887) = (119886 minus119887)1205902(119886 119887) = (120587 minus 119886 120587 + 119887)1205903(119886 119887) = (120587 + 119886 120587 minus 119887)1205904(119886 119887) = (minus119886 119887) 119867119891119871119904 119888119898119898 (119909) = sum
119892isin1198632
119891119871119904 [119892 (119909)] + 4sum119894=1
sum119892isin1198632
119891119871119904 [(120590119894119892) (119909)]p4 1198624 None 119867119891119871119904 1199014 (119909) = sum
119892isin1198624
119891119871119904 [119892 (119909)]1199014119892 1198634 1205901 (119886 119887) = (120587 + 119886 minus119887) 119867119891119871119904 1199014119892 (119909) = sum119892isin1198634
119891119871119904 [119892 (119909)] + sum119892isin1198634
119891119871119904 [(1205901119892) (119909)]1199014119898 1198634 1205901 (119886 119887) = (119886 minus119887) 119867119891119871119904 1199014119898 (119909) = sum119892isin1198634
119891119871119904 [119892 (119909)] + sum119892isin1198634
119891119871119889119904 [(1205901119892) (119909)]119901119892119892 1198632 1205901(119886 119887) = (120587 + 119886 120587 minus 119887)1205902(119886 119887) = (120587 minus 119886 120587 + 119887) 119867119891119871119904 119901119892119892 (119909) = sum119892isin1198632
119891119871119904 [119892 (119909)] + 2sum119894=1
sum119892isin1198632
119891119871119904 [(120590119894119892) (119909)]p3 1198623 None 119867119891119871119889 1199013 (119909) = sum
119892isin1198623
119891119871119889 [119892 (119909)]p3m1 1198633 1205901 (119886 119887) = (minus119886 119887) 119867119891119871119889 11990131198981 (119909) = sum
119892isin1198633
119891119871119889 [119892 (119909)] + sum119892isin1198633
119891119871119889 [(1205901119892) (119909)]p31m 1198633 1205901 (119886 119887) = (119886 minus119887) 119867119891119871119889 11990131119898 (119909) = sum
119892isin1198633
119891119871119889 [119892 (119909)] + sum119892isin1198633
119891119871119889 [(1205901119892) (119909)]p6 1198626 None 119867119891119871119889 1199016 (119909) = sum
119892isin1198626
119891119871119889 [119892 (119909)]1199016119898 1198636 1205901 (119886 119887) = (119886 minus119887) 119867119891119871119889 1199016119898 (119909) = sum119892isin1198636
119891119871119889 [119892 (119909)] + sum119892isin1198636
119891119871119889 [(1205901119892) (119909)]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 1198772 is essentially a subgroup of 1198772 Orequivalently for any basis vectors of 1198772 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 1198852 and fully spans the realvector space1198772 [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 1198772 come in two infinite families dihedralgroup 119863119899 which is the symmetry group of a regular polygon
Complexity 3
and cyclic group119862119899 that only comprises rotation transforma-tions of 119863119899 Let 119877119899 = ( cos(2120587119899) minus sin(2120587119899)
sin(2120587119899) cos(2120587119899) ) and 119879 = ( minus1 00 1 )Then their matrix group can be represented as 119862119899 = 119877119894119899 119894 =1 2 3 119899 and119863119899 = 119862119899 cup 119879119877119894119899 119894 = 1 2 3 119899
Awallpaper group is a type of topologically discrete groupin 1198772 which contains two linearly independent translationsA lattice in 1198772 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 119871 be a lattice in1198772 A lattice 119871lowast is called the dual lattice of 119871 if forall119906 isin 119871 andforallV isin 119871lowast the inner product 119906 sdot V is an integer where 119906 and Vare vectors in1198772 Let119872 be amapping from1198772 to1198772 and let119866be a symmetry group in 1198772119872 is called an invariant mappingwith respect to 119866 if forall119909 isin 1198772 and forall119892 isin 119866 119872(119909) = 119872(119892119909)
By the crystallographic restriction theorem there areonly 5 lattice types in 1198772 [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 2120587Throughout thepaper for square lattice we choose lattice 119871 119904 = (1 0) (0 1)with dual lattice 119871lowast119904 = 2120587(1 0) 2120587(0 1) for diamond latticewe choose lattice 119871119889 = (1 0) (12)(minus1radic3)with dual lattice119871lowast119889 = (2120587radic3)(radic3 minus1) 2120587(0 minus2radic3)
In this paper we use standard crystallographic notationsof wallpaper groups [16 17] Among 17 wallpaper groups 11990111199012 119901119898 119901119898119898 119901119892 119901119898119892 119888119898 119888119898119898 1199014 119901119892119892 1199014119892 and 1199014119898possess square lattice while 1199013 11990131198981 11990131119898 1199016 and 1199016119898possess 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 119891119894 (119894 = 1 2 3 4) are sine or cosinefunctions 119866 is a wallpaper group with lattice 119871 = 119860 119861 119871lowast =119860lowast 119861lowast is the dual lattice of 119871 and 119886 and 119887 are real numbersThen mapping
119891119871 (119909) = ( 1198861198911 sumVisin1198711198912 (119909 sdot V) + sum
Visin119871(119909 sdot V)1198871198913 sum
Visin1198711198914 (119909 sdot V) + sum
Visin119871(119909 sdot V) )
forall119909 isin 1198772(1)
is invariant with respect to 119871lowast or 119891119871(119909) has translationinvariance of 119871lowast that is
(1198861198911 sumVisin119871
1198912 ((119906 + 119909) sdot V) + sumVisin119871
((119906 + 119909) sdot V)1198871198913 sumVisin119871
1198914 ((119906 + 119909) sdot V) + sumVisin119871
((119906 + 119909) sdot V))
=(1198861198911 sumVisin119871
1198912 (119909 sdot V) + sumVisin119871
(119909 sdot V)1198871198913 sumVisin119871
1198914 (119909 sdot V) + sumVisin119871
(119909 sdot V)) = 119891119871 (119909) (2)
where 119906 = 119898119860lowast + 119899119861lowast 119898 119899 isin 119885Proof Since 119871lowast is the dual lattice of 119871 forallV isin 119871 we have 119906 sdotV =(119898119860lowast + 119899119861lowast) sdot V = 119898(119860lowast sdot V) + 119899(119861lowast sdot V) = 2119896120587 for certain119896 isin 119885 Thus we get 119891119894sumVisin119871 119891119895((119906+119909) sdotV)+sumVisin119871((119906+119909) sdotV) =119891119894sumVisin119871 119891119895(119909 sdotV)+sumVisin119871(119909 sdotV) since 119891119894 and 119891119895 are functions ofperiod 2120587 (119894 119895 = 1 2 3 4) Consequently the mapping 119891119871(119909)constructed by 119891119894 (119894 = 1 2 3 4) satisfies (2) This completesthe proof
Essentially Lemma 1 says that 119891119871(119909) is a double periodmapping (of period 2120587) along the independent translationaldirections of 119871lowastTheorem 2 Let 119866 be a finite group realized by 2 times 2 matricesacting on 1198772 by multiplication on the right and let 119891 be anarbitrary mapping from 1198772 to 1198772 Then mapping119867119891119866 (119909) = sum
119892isin119866
119891 [119892 (119909)] 119909 isin 1198772 (3)
is an invariant mapping with respect to 119866Proof For 120590 isin 119866 by closure of the group operation we seethat 119892120590 runs through 119866 as 119892 does Therefore we have119867119891119866 [120590 (119909)] = sum
119892isin119866
119891 [120590 (119892119909)] = sum119892lowastisin119866
119891 [119892lowast (119909)]= 119867119891119866 (119909) (4)
where 119892lowast = 120590119892 isin 119866 This means that119867119891119866(119909) is an invariantmapping with respect to 119866
Combining Lemma 1 and Theorem 2 we immediatelyderive the following theorem
Theorem 3 Let 119891 in Theorem 2 have the form 119891119871 as inLemma 1 Suppose that 119866 is a cyclic group 119862119899 or dihedralgroup 119863119899 with lattice 119871 119871lowast is the dual lattice associated with119871 Assume that 119867119891119871119866(119909) is a mapping from 1198772 to 1198772 of thefollowing form119867119891119871119866 (119909) = sum
119892isin119866
119891119871 [119892 (119909)] 119909 isin 1198772 (5)
Then119867119891119871119866(119909) is an invariant mapping with respect to both 119866and 119871lowast
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 119867119891119871119866(119909) in
4 Complexity
BEGINstart x = 0end x = 6 lowast 31415926start y = 0end y = 6 lowast 31415926 Set pi = 31415926step x = (end x ndash start x)X res Xres is the resolution in X directionstep y = (end y ndash 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 lowast step xy = start y + j lowast step yFOR k = 1 TOMaxIterMaxIter is the number of iterations the default set is 100
lowastGiven a invariant mapping119867119891119871119908(119909) associated with a wallpaper group 119908 as iterationfunction and initial point (x y) function Iteration (x y) iterates MaxIter times The iteratedsequences are stored in the array Sequencelowast
Sequence [119896] = Iteration (x y)END FOR
lowastInputting Sequence the color scheme OrbitTrap outputs the color [r g b]lowast[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 wellHowever except for the symmetries of a point group somewallpaper groups may possess other symmetries For exam-ple except for symmetries of dihedral group 1198633 wallpapergroup 11990131119898 still has a reflection along horizontal directionsay symmetry 1205901(119886 119887) = (119886 minus119887) ((119886 119887) isin 1198772) besides1198632 symmetry wallpaper group 119901119898119898 contains perpendicularreflections that is 1205901(119886 119887) = (119886 minus119887) and 1205902(119886 119887) = (minus119886 119887)
For a wallpaper group 119908 with extra symmetry set Δ =1205901 1205902 1205903 120590119899 based on mapping (5) we add properterms so that the resulting mapping119867119891119871119908(119909) is also invariantwith respect to 119908 This is summarized inTheorem 4
Theorem 4 Let 119891 inTheorem 2 have the form 119891119871 in Lemma 1Suppose that 119908 is a wallpaper group with symmetry group 119866and extra symmetry set Δ = 1205901 1205902 1205903 120590119899 Assume that 119871is the lattice of 119908 and 119871lowast is the dual lattice associated with 119871Let119867119891119871119908(119909) be a mapping from 1198772 to 1198772 of the following form119867119891119871119908 (119909) = sum
119892isin119866
119891119871 [119892 (119909)]+ 119899sum119894=1
sum119892isin119866120590119894isinΔ
119891119871 [(120590119894119892) (119909)] 119909 isin 1198772 (6)
Then119867119891119871119908(119909) is an invariant mapping with respect to both 119908and 119871lowast
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 [18ndash23] Color scheme is analgorithm that is used to determine the color of a point Givena color scheme and domain119863 by iterating invariantmapping119867119891119871119908(119909) 119909 isin 119863 one can determine the color of 119909 Coloringpoints in 119863 pointwise one can obtain a colorful pattern in119863 with symmetries of the wallpaper group 119908 Figures 1-2are four wallpaper patterns obtained in this manner Thesepatterns were created by VC++ 60 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 [7ndash10] 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 119867119891119871119904 1199014119898(119909) and 119867119891119871119889 11990131198981(119909) respectively inwhich the specific mappings 119891119871119904 and 119891119871119889 were
119891119871119904 =(212 cossumVisin119871119904
cos (119909 sdot V) + sumVisin119871119904
(119909 sdot V)103 cossum
Visin119871119904
sin (119909 sdot V) + sumVisin119871119904
(119909 sdot V)) (7)
Complexity 5
(a) (b)
Figure 1 Colorful wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 2 Colorful wallpaper patterns with the 119901119898119898 (a) and 11990131198981 (b) symmetry
119891119871119889 =( 11 cossumVisin119871119889
sin (119909 sdot V) + sumVisin119871119889
(119909 sdot V)minus052 sinsum
Visin119871119889
cos (119909 sdot V) + sumVisin119871119889
(119909 sdot V)) (8)
It seems that the deference between (7) and (8) is not verysignificant However by Table 1 mappings 119867119891119871119904 1199014119898(119909) and119867119891119871119889 11990131198981(119909) have 16 and 12 summation terms respectivelyThe 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 1198782 = (119886 119887 119888) isin 1198773 | 1198862+1198872+1198882 = 1 be the unit spherein 1198773 let 119885 = 119865 be a projection plane where 119865 is a negativeconstant Assume that 119875(119886 119887 119888) isin 1198773 then 119875(119886 119887 119888) isin 1198773and1198751015840(minus119886 minus119887 minus119888) isin 1198773 are a pair of antipodal points For anypoint 119875(119886 119887 119888) isin 1198773 there exist a unique line 119871 through theorigin (0 0 0) and 119875 (and 1198751015840) which intersects the projectionplane 119885 = 119865 at point (120572 120573 119865) Denote the projection by 120591 Byanalytic geometry it is easy to check that
[[[120572120573119865 ]]] = 120591 (119886 119887 119888) = 120591 (minus119886 minus119887 minus119888) = [[[
119886119887119888 ]]] 119865119888 (9)
Because the projection point is at the center of 1198782 we call 120591 ascentral projection
The choice of the plane119885 = 119865 has a great influence on thespherical patterns If plane 119885 = 119865 is too close to coordinateplane 119883119874119884 the resulting spherical pattern only shows a few
6 Complexity
(a) (b)
Figure 3 Two spherical wallpaper patterns with the 1199014119898 symmetry in which the projection plane was set as 119885 = minus2120587 (a) and 119885 = minus4120587 (b)
(a) (b)
Figure 4 Colorful spherical wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
periods of the wallpaper pattern However if plane 119885 = 119865is too far away from coordinate plane 119883119874119884 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 119885 = 119865
Given a wallpaper pattern by central projection 120591 we canmap it onto the sphere 1198782 and obtain a corresponding spher-ical wallpaper pattern We next explain how to implement itin more detail
Suppose that (119886 119887 119888) isin 1198773 and 119867119891119871119908(119909) is an invari-ant mapping compatible with the symmetry of wallpapergroup 119908 First by central projection 120591 we obtain a corre-sponding point ((119860119888)119886 (119860119888)119887 119865) on the projection plane119885 = 119865 Second let 119867119891119871119908(119909) be iteration function and let119909((119860119888)119886 (119860119888)119887) be initial point using the color schemeof orbit trap we assign a color to point ((119860119888)119886 (119860119888)119887 119865)Finally repeat the second step by coloring unit sphere 1198782pointwise we obtain a spherical pattern of the wallpapergroup 119908 symmetry
Figures 3ndash7 are ten patterns obtained by this mannerExcept for Figure 3(b) in which the projection plane wasset as 119885 = minus4120587 all the other projection planes were set as119885 = minus2120587 We utilized the wallpaper patterns of Figure 1 toproduce spherical patterns shown in Figure 4 For beauty allthe camera views are perpendicular to plane 119883119874119884 and passthe origin except for Figure 7(b) where the camera view aimsat the equator of 1198782 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 valueWe plan to produce somematerialobjects with the help of simulation and printing technologiesWe produced Figures 1ndash7 in the VC++ 60 programmingenvironment with the aid of OpenGL a powerful graphicssoftware package
Complexity 7
(a) (b)
Figure 5 Colorful spherical wallpaper patterns with the 1199016119898 (a) and 119901119898119898 (b) symmetry
(a) (b)
Figure 6 Colorful spherical wallpaper patterns with the 1199014 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 7 Two spherical wallpaper patterns with the 1199013119898 symmetry The camera view of (a) is perpendicular to plane 119883119874119884 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 1146103511761039 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 ldquoSymmetry in the planerdquo Proceedings of the ImperialSt Petersburg Mineralogical Society vol 2 pp 245ndash291 1891(Russian)
[2] G Polya ldquoXII Uber die analogie der kristallsymmetrie in derebenerdquo Zeitschrift fur KristallographiemdashCrystalline Materialsvol 60 no 1-6 1924
[3] J Owen The Grammar of Ornament Van Nostrand Reinhold1910
[4] P S StevensHandbook of Regular Patterns MIT Press LondonUK 1981
[5] B Grunbaum 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 EscherExploring the Infinity N Harry Ed Abrams New York NYUSA 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 CA Reiter ldquoChaotic attractors with discrete planar symmetriesrdquoChaos Solitons amp Fractals vol 9 no 12 pp 2031ndash2054 1998
[9] K W Chung and H S Y Chan ldquoSymmetrical patterns fromdynamicsrdquo Computer Graphics Forum vol 12 no 1 pp 33ndash401993
[10] J Lu Z Ye Y Zou and R Ye ldquoOrbit trap renderingmethods forgenerating artistic images with crystallographic symmetriesrdquoComputers and Graphics vol 29 no 5 pp 794ndash801 2005
[11] D Douglas and S John ldquoThe art of random fractalsrdquo in Pro-ceedings of Bridges 2014 Mathematics Music Art ArchitectureCulture pp 79ndash86 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 ldquoLinear algebra andgeometryrdquo 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 YorkNY USA 1987
[18] K W Chung and H M Ma ldquoAutomatic generation of aestheticpatterns on fractal tilings by means of dynamical systemsrdquoChaos Solitons amp Fractals vol 24 no 4 pp 1145ndash1158 2005
[19] P Ouyang and X Wang ldquoBeautiful mathmdashaesthetic patternsbased on logarithmic spiralsrdquo IEEE Computer Graphics andApplications vol 33 no 6 pp 21ndash23 2013
[20] P Ouyang D Cheng Y Cao and X Zhan ldquoThe visualizationof hyperbolic patterns from invariant mapping methodrdquo Com-puters and Graphics vol 36 no 2 pp 92ndash100 2012
[21] P Ouyang and RW Fathauer ldquoBeautiful math part 2 aestheticpatterns based on fractal tilingsrdquo IEEE Computer Graphics andApplications vol 34 no 1 pp 68ndash76 2014
[22] P Ouyang and K Chung ldquoBeautiful math part 3 hyperbolicaesthetic patterns based on conformal mappingsrdquo IEEE Com-puter Graphics and Applications vol 34 no 2 pp 72ndash79 2014
[23] P Ouyang L Wang T Yu and X Huang ldquoAesthetic patternswith symmetries of the regular polyhedronrdquo Symmetry vol 9no 2 article no 21 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 3
and cyclic group119862119899 that only comprises rotation transforma-tions of 119863119899 Let 119877119899 = ( cos(2120587119899) minus sin(2120587119899)
sin(2120587119899) cos(2120587119899) ) and 119879 = ( minus1 00 1 )Then their matrix group can be represented as 119862119899 = 119877119894119899 119894 =1 2 3 119899 and119863119899 = 119862119899 cup 119879119877119894119899 119894 = 1 2 3 119899
Awallpaper group is a type of topologically discrete groupin 1198772 which contains two linearly independent translationsA lattice in 1198772 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 119871 be a lattice in1198772 A lattice 119871lowast is called the dual lattice of 119871 if forall119906 isin 119871 andforallV isin 119871lowast the inner product 119906 sdot V is an integer where 119906 and Vare vectors in1198772 Let119872 be amapping from1198772 to1198772 and let119866be a symmetry group in 1198772119872 is called an invariant mappingwith respect to 119866 if forall119909 isin 1198772 and forall119892 isin 119866 119872(119909) = 119872(119892119909)
By the crystallographic restriction theorem there areonly 5 lattice types in 1198772 [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 2120587Throughout thepaper for square lattice we choose lattice 119871 119904 = (1 0) (0 1)with dual lattice 119871lowast119904 = 2120587(1 0) 2120587(0 1) for diamond latticewe choose lattice 119871119889 = (1 0) (12)(minus1radic3)with dual lattice119871lowast119889 = (2120587radic3)(radic3 minus1) 2120587(0 minus2radic3)
In this paper we use standard crystallographic notationsof wallpaper groups [16 17] Among 17 wallpaper groups 11990111199012 119901119898 119901119898119898 119901119892 119901119898119892 119888119898 119888119898119898 1199014 119901119892119892 1199014119892 and 1199014119898possess square lattice while 1199013 11990131198981 11990131119898 1199016 and 1199016119898possess 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 119891119894 (119894 = 1 2 3 4) are sine or cosinefunctions 119866 is a wallpaper group with lattice 119871 = 119860 119861 119871lowast =119860lowast 119861lowast is the dual lattice of 119871 and 119886 and 119887 are real numbersThen mapping
119891119871 (119909) = ( 1198861198911 sumVisin1198711198912 (119909 sdot V) + sum
Visin119871(119909 sdot V)1198871198913 sum
Visin1198711198914 (119909 sdot V) + sum
Visin119871(119909 sdot V) )
forall119909 isin 1198772(1)
is invariant with respect to 119871lowast or 119891119871(119909) has translationinvariance of 119871lowast that is
(1198861198911 sumVisin119871
1198912 ((119906 + 119909) sdot V) + sumVisin119871
((119906 + 119909) sdot V)1198871198913 sumVisin119871
1198914 ((119906 + 119909) sdot V) + sumVisin119871
((119906 + 119909) sdot V))
=(1198861198911 sumVisin119871
1198912 (119909 sdot V) + sumVisin119871
(119909 sdot V)1198871198913 sumVisin119871
1198914 (119909 sdot V) + sumVisin119871
(119909 sdot V)) = 119891119871 (119909) (2)
where 119906 = 119898119860lowast + 119899119861lowast 119898 119899 isin 119885Proof Since 119871lowast is the dual lattice of 119871 forallV isin 119871 we have 119906 sdotV =(119898119860lowast + 119899119861lowast) sdot V = 119898(119860lowast sdot V) + 119899(119861lowast sdot V) = 2119896120587 for certain119896 isin 119885 Thus we get 119891119894sumVisin119871 119891119895((119906+119909) sdotV)+sumVisin119871((119906+119909) sdotV) =119891119894sumVisin119871 119891119895(119909 sdotV)+sumVisin119871(119909 sdotV) since 119891119894 and 119891119895 are functions ofperiod 2120587 (119894 119895 = 1 2 3 4) Consequently the mapping 119891119871(119909)constructed by 119891119894 (119894 = 1 2 3 4) satisfies (2) This completesthe proof
Essentially Lemma 1 says that 119891119871(119909) is a double periodmapping (of period 2120587) along the independent translationaldirections of 119871lowastTheorem 2 Let 119866 be a finite group realized by 2 times 2 matricesacting on 1198772 by multiplication on the right and let 119891 be anarbitrary mapping from 1198772 to 1198772 Then mapping119867119891119866 (119909) = sum
119892isin119866
119891 [119892 (119909)] 119909 isin 1198772 (3)
is an invariant mapping with respect to 119866Proof For 120590 isin 119866 by closure of the group operation we seethat 119892120590 runs through 119866 as 119892 does Therefore we have119867119891119866 [120590 (119909)] = sum
119892isin119866
119891 [120590 (119892119909)] = sum119892lowastisin119866
119891 [119892lowast (119909)]= 119867119891119866 (119909) (4)
where 119892lowast = 120590119892 isin 119866 This means that119867119891119866(119909) is an invariantmapping with respect to 119866
Combining Lemma 1 and Theorem 2 we immediatelyderive the following theorem
Theorem 3 Let 119891 in Theorem 2 have the form 119891119871 as inLemma 1 Suppose that 119866 is a cyclic group 119862119899 or dihedralgroup 119863119899 with lattice 119871 119871lowast is the dual lattice associated with119871 Assume that 119867119891119871119866(119909) is a mapping from 1198772 to 1198772 of thefollowing form119867119891119871119866 (119909) = sum
119892isin119866
119891119871 [119892 (119909)] 119909 isin 1198772 (5)
Then119867119891119871119866(119909) is an invariant mapping with respect to both 119866and 119871lowast
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 119867119891119871119866(119909) in
4 Complexity
BEGINstart x = 0end x = 6 lowast 31415926start y = 0end y = 6 lowast 31415926 Set pi = 31415926step x = (end x ndash start x)X res Xres is the resolution in X directionstep y = (end y ndash 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 lowast step xy = start y + j lowast step yFOR k = 1 TOMaxIterMaxIter is the number of iterations the default set is 100
lowastGiven a invariant mapping119867119891119871119908(119909) associated with a wallpaper group 119908 as iterationfunction and initial point (x y) function Iteration (x y) iterates MaxIter times The iteratedsequences are stored in the array Sequencelowast
Sequence [119896] = Iteration (x y)END FOR
lowastInputting Sequence the color scheme OrbitTrap outputs the color [r g b]lowast[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 wellHowever except for the symmetries of a point group somewallpaper groups may possess other symmetries For exam-ple except for symmetries of dihedral group 1198633 wallpapergroup 11990131119898 still has a reflection along horizontal directionsay symmetry 1205901(119886 119887) = (119886 minus119887) ((119886 119887) isin 1198772) besides1198632 symmetry wallpaper group 119901119898119898 contains perpendicularreflections that is 1205901(119886 119887) = (119886 minus119887) and 1205902(119886 119887) = (minus119886 119887)
For a wallpaper group 119908 with extra symmetry set Δ =1205901 1205902 1205903 120590119899 based on mapping (5) we add properterms so that the resulting mapping119867119891119871119908(119909) is also invariantwith respect to 119908 This is summarized inTheorem 4
Theorem 4 Let 119891 inTheorem 2 have the form 119891119871 in Lemma 1Suppose that 119908 is a wallpaper group with symmetry group 119866and extra symmetry set Δ = 1205901 1205902 1205903 120590119899 Assume that 119871is the lattice of 119908 and 119871lowast is the dual lattice associated with 119871Let119867119891119871119908(119909) be a mapping from 1198772 to 1198772 of the following form119867119891119871119908 (119909) = sum
119892isin119866
119891119871 [119892 (119909)]+ 119899sum119894=1
sum119892isin119866120590119894isinΔ
119891119871 [(120590119894119892) (119909)] 119909 isin 1198772 (6)
Then119867119891119871119908(119909) is an invariant mapping with respect to both 119908and 119871lowast
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 [18ndash23] Color scheme is analgorithm that is used to determine the color of a point Givena color scheme and domain119863 by iterating invariantmapping119867119891119871119908(119909) 119909 isin 119863 one can determine the color of 119909 Coloringpoints in 119863 pointwise one can obtain a colorful pattern in119863 with symmetries of the wallpaper group 119908 Figures 1-2are four wallpaper patterns obtained in this manner Thesepatterns were created by VC++ 60 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 [7ndash10] 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 119867119891119871119904 1199014119898(119909) and 119867119891119871119889 11990131198981(119909) respectively inwhich the specific mappings 119891119871119904 and 119891119871119889 were
119891119871119904 =(212 cossumVisin119871119904
cos (119909 sdot V) + sumVisin119871119904
(119909 sdot V)103 cossum
Visin119871119904
sin (119909 sdot V) + sumVisin119871119904
(119909 sdot V)) (7)
Complexity 5
(a) (b)
Figure 1 Colorful wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 2 Colorful wallpaper patterns with the 119901119898119898 (a) and 11990131198981 (b) symmetry
119891119871119889 =( 11 cossumVisin119871119889
sin (119909 sdot V) + sumVisin119871119889
(119909 sdot V)minus052 sinsum
Visin119871119889
cos (119909 sdot V) + sumVisin119871119889
(119909 sdot V)) (8)
It seems that the deference between (7) and (8) is not verysignificant However by Table 1 mappings 119867119891119871119904 1199014119898(119909) and119867119891119871119889 11990131198981(119909) have 16 and 12 summation terms respectivelyThe 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 1198782 = (119886 119887 119888) isin 1198773 | 1198862+1198872+1198882 = 1 be the unit spherein 1198773 let 119885 = 119865 be a projection plane where 119865 is a negativeconstant Assume that 119875(119886 119887 119888) isin 1198773 then 119875(119886 119887 119888) isin 1198773and1198751015840(minus119886 minus119887 minus119888) isin 1198773 are a pair of antipodal points For anypoint 119875(119886 119887 119888) isin 1198773 there exist a unique line 119871 through theorigin (0 0 0) and 119875 (and 1198751015840) which intersects the projectionplane 119885 = 119865 at point (120572 120573 119865) Denote the projection by 120591 Byanalytic geometry it is easy to check that
[[[120572120573119865 ]]] = 120591 (119886 119887 119888) = 120591 (minus119886 minus119887 minus119888) = [[[
119886119887119888 ]]] 119865119888 (9)
Because the projection point is at the center of 1198782 we call 120591 ascentral projection
The choice of the plane119885 = 119865 has a great influence on thespherical patterns If plane 119885 = 119865 is too close to coordinateplane 119883119874119884 the resulting spherical pattern only shows a few
6 Complexity
(a) (b)
Figure 3 Two spherical wallpaper patterns with the 1199014119898 symmetry in which the projection plane was set as 119885 = minus2120587 (a) and 119885 = minus4120587 (b)
(a) (b)
Figure 4 Colorful spherical wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
periods of the wallpaper pattern However if plane 119885 = 119865is too far away from coordinate plane 119883119874119884 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 119885 = 119865
Given a wallpaper pattern by central projection 120591 we canmap it onto the sphere 1198782 and obtain a corresponding spher-ical wallpaper pattern We next explain how to implement itin more detail
Suppose that (119886 119887 119888) isin 1198773 and 119867119891119871119908(119909) is an invari-ant mapping compatible with the symmetry of wallpapergroup 119908 First by central projection 120591 we obtain a corre-sponding point ((119860119888)119886 (119860119888)119887 119865) on the projection plane119885 = 119865 Second let 119867119891119871119908(119909) be iteration function and let119909((119860119888)119886 (119860119888)119887) be initial point using the color schemeof orbit trap we assign a color to point ((119860119888)119886 (119860119888)119887 119865)Finally repeat the second step by coloring unit sphere 1198782pointwise we obtain a spherical pattern of the wallpapergroup 119908 symmetry
Figures 3ndash7 are ten patterns obtained by this mannerExcept for Figure 3(b) in which the projection plane wasset as 119885 = minus4120587 all the other projection planes were set as119885 = minus2120587 We utilized the wallpaper patterns of Figure 1 toproduce spherical patterns shown in Figure 4 For beauty allthe camera views are perpendicular to plane 119883119874119884 and passthe origin except for Figure 7(b) where the camera view aimsat the equator of 1198782 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 valueWe plan to produce somematerialobjects with the help of simulation and printing technologiesWe produced Figures 1ndash7 in the VC++ 60 programmingenvironment with the aid of OpenGL a powerful graphicssoftware package
Complexity 7
(a) (b)
Figure 5 Colorful spherical wallpaper patterns with the 1199016119898 (a) and 119901119898119898 (b) symmetry
(a) (b)
Figure 6 Colorful spherical wallpaper patterns with the 1199014 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 7 Two spherical wallpaper patterns with the 1199013119898 symmetry The camera view of (a) is perpendicular to plane 119883119874119884 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 1146103511761039 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 ldquoSymmetry in the planerdquo Proceedings of the ImperialSt Petersburg Mineralogical Society vol 2 pp 245ndash291 1891(Russian)
[2] G Polya ldquoXII Uber die analogie der kristallsymmetrie in derebenerdquo Zeitschrift fur KristallographiemdashCrystalline Materialsvol 60 no 1-6 1924
[3] J Owen The Grammar of Ornament Van Nostrand Reinhold1910
[4] P S StevensHandbook of Regular Patterns MIT Press LondonUK 1981
[5] B Grunbaum 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 EscherExploring the Infinity N Harry Ed Abrams New York NYUSA 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 CA Reiter ldquoChaotic attractors with discrete planar symmetriesrdquoChaos Solitons amp Fractals vol 9 no 12 pp 2031ndash2054 1998
[9] K W Chung and H S Y Chan ldquoSymmetrical patterns fromdynamicsrdquo Computer Graphics Forum vol 12 no 1 pp 33ndash401993
[10] J Lu Z Ye Y Zou and R Ye ldquoOrbit trap renderingmethods forgenerating artistic images with crystallographic symmetriesrdquoComputers and Graphics vol 29 no 5 pp 794ndash801 2005
[11] D Douglas and S John ldquoThe art of random fractalsrdquo in Pro-ceedings of Bridges 2014 Mathematics Music Art ArchitectureCulture pp 79ndash86 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 ldquoLinear algebra andgeometryrdquo 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 YorkNY USA 1987
[18] K W Chung and H M Ma ldquoAutomatic generation of aestheticpatterns on fractal tilings by means of dynamical systemsrdquoChaos Solitons amp Fractals vol 24 no 4 pp 1145ndash1158 2005
[19] P Ouyang and X Wang ldquoBeautiful mathmdashaesthetic patternsbased on logarithmic spiralsrdquo IEEE Computer Graphics andApplications vol 33 no 6 pp 21ndash23 2013
[20] P Ouyang D Cheng Y Cao and X Zhan ldquoThe visualizationof hyperbolic patterns from invariant mapping methodrdquo Com-puters and Graphics vol 36 no 2 pp 92ndash100 2012
[21] P Ouyang and RW Fathauer ldquoBeautiful math part 2 aestheticpatterns based on fractal tilingsrdquo IEEE Computer Graphics andApplications vol 34 no 1 pp 68ndash76 2014
[22] P Ouyang and K Chung ldquoBeautiful math part 3 hyperbolicaesthetic patterns based on conformal mappingsrdquo IEEE Com-puter Graphics and Applications vol 34 no 2 pp 72ndash79 2014
[23] P Ouyang L Wang T Yu and X Huang ldquoAesthetic patternswith symmetries of the regular polyhedronrdquo Symmetry vol 9no 2 article no 21 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
4 Complexity
BEGINstart x = 0end x = 6 lowast 31415926start y = 0end y = 6 lowast 31415926 Set pi = 31415926step x = (end x ndash start x)X res Xres is the resolution in X directionstep y = (end y ndash 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 lowast step xy = start y + j lowast step yFOR k = 1 TOMaxIterMaxIter is the number of iterations the default set is 100
lowastGiven a invariant mapping119867119891119871119908(119909) associated with a wallpaper group 119908 as iterationfunction and initial point (x y) function Iteration (x y) iterates MaxIter times The iteratedsequences are stored in the array Sequencelowast
Sequence [119896] = Iteration (x y)END FOR
lowastInputting Sequence the color scheme OrbitTrap outputs the color [r g b]lowast[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 wellHowever except for the symmetries of a point group somewallpaper groups may possess other symmetries For exam-ple except for symmetries of dihedral group 1198633 wallpapergroup 11990131119898 still has a reflection along horizontal directionsay symmetry 1205901(119886 119887) = (119886 minus119887) ((119886 119887) isin 1198772) besides1198632 symmetry wallpaper group 119901119898119898 contains perpendicularreflections that is 1205901(119886 119887) = (119886 minus119887) and 1205902(119886 119887) = (minus119886 119887)
For a wallpaper group 119908 with extra symmetry set Δ =1205901 1205902 1205903 120590119899 based on mapping (5) we add properterms so that the resulting mapping119867119891119871119908(119909) is also invariantwith respect to 119908 This is summarized inTheorem 4
Theorem 4 Let 119891 inTheorem 2 have the form 119891119871 in Lemma 1Suppose that 119908 is a wallpaper group with symmetry group 119866and extra symmetry set Δ = 1205901 1205902 1205903 120590119899 Assume that 119871is the lattice of 119908 and 119871lowast is the dual lattice associated with 119871Let119867119891119871119908(119909) be a mapping from 1198772 to 1198772 of the following form119867119891119871119908 (119909) = sum
119892isin119866
119891119871 [119892 (119909)]+ 119899sum119894=1
sum119892isin119866120590119894isinΔ
119891119871 [(120590119894119892) (119909)] 119909 isin 1198772 (6)
Then119867119891119871119908(119909) is an invariant mapping with respect to both 119908and 119871lowast
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 [18ndash23] Color scheme is analgorithm that is used to determine the color of a point Givena color scheme and domain119863 by iterating invariantmapping119867119891119871119908(119909) 119909 isin 119863 one can determine the color of 119909 Coloringpoints in 119863 pointwise one can obtain a colorful pattern in119863 with symmetries of the wallpaper group 119908 Figures 1-2are four wallpaper patterns obtained in this manner Thesepatterns were created by VC++ 60 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 [7ndash10] 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 119867119891119871119904 1199014119898(119909) and 119867119891119871119889 11990131198981(119909) respectively inwhich the specific mappings 119891119871119904 and 119891119871119889 were
119891119871119904 =(212 cossumVisin119871119904
cos (119909 sdot V) + sumVisin119871119904
(119909 sdot V)103 cossum
Visin119871119904
sin (119909 sdot V) + sumVisin119871119904
(119909 sdot V)) (7)
Complexity 5
(a) (b)
Figure 1 Colorful wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 2 Colorful wallpaper patterns with the 119901119898119898 (a) and 11990131198981 (b) symmetry
119891119871119889 =( 11 cossumVisin119871119889
sin (119909 sdot V) + sumVisin119871119889
(119909 sdot V)minus052 sinsum
Visin119871119889
cos (119909 sdot V) + sumVisin119871119889
(119909 sdot V)) (8)
It seems that the deference between (7) and (8) is not verysignificant However by Table 1 mappings 119867119891119871119904 1199014119898(119909) and119867119891119871119889 11990131198981(119909) have 16 and 12 summation terms respectivelyThe 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 1198782 = (119886 119887 119888) isin 1198773 | 1198862+1198872+1198882 = 1 be the unit spherein 1198773 let 119885 = 119865 be a projection plane where 119865 is a negativeconstant Assume that 119875(119886 119887 119888) isin 1198773 then 119875(119886 119887 119888) isin 1198773and1198751015840(minus119886 minus119887 minus119888) isin 1198773 are a pair of antipodal points For anypoint 119875(119886 119887 119888) isin 1198773 there exist a unique line 119871 through theorigin (0 0 0) and 119875 (and 1198751015840) which intersects the projectionplane 119885 = 119865 at point (120572 120573 119865) Denote the projection by 120591 Byanalytic geometry it is easy to check that
[[[120572120573119865 ]]] = 120591 (119886 119887 119888) = 120591 (minus119886 minus119887 minus119888) = [[[
119886119887119888 ]]] 119865119888 (9)
Because the projection point is at the center of 1198782 we call 120591 ascentral projection
The choice of the plane119885 = 119865 has a great influence on thespherical patterns If plane 119885 = 119865 is too close to coordinateplane 119883119874119884 the resulting spherical pattern only shows a few
6 Complexity
(a) (b)
Figure 3 Two spherical wallpaper patterns with the 1199014119898 symmetry in which the projection plane was set as 119885 = minus2120587 (a) and 119885 = minus4120587 (b)
(a) (b)
Figure 4 Colorful spherical wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
periods of the wallpaper pattern However if plane 119885 = 119865is too far away from coordinate plane 119883119874119884 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 119885 = 119865
Given a wallpaper pattern by central projection 120591 we canmap it onto the sphere 1198782 and obtain a corresponding spher-ical wallpaper pattern We next explain how to implement itin more detail
Suppose that (119886 119887 119888) isin 1198773 and 119867119891119871119908(119909) is an invari-ant mapping compatible with the symmetry of wallpapergroup 119908 First by central projection 120591 we obtain a corre-sponding point ((119860119888)119886 (119860119888)119887 119865) on the projection plane119885 = 119865 Second let 119867119891119871119908(119909) be iteration function and let119909((119860119888)119886 (119860119888)119887) be initial point using the color schemeof orbit trap we assign a color to point ((119860119888)119886 (119860119888)119887 119865)Finally repeat the second step by coloring unit sphere 1198782pointwise we obtain a spherical pattern of the wallpapergroup 119908 symmetry
Figures 3ndash7 are ten patterns obtained by this mannerExcept for Figure 3(b) in which the projection plane wasset as 119885 = minus4120587 all the other projection planes were set as119885 = minus2120587 We utilized the wallpaper patterns of Figure 1 toproduce spherical patterns shown in Figure 4 For beauty allthe camera views are perpendicular to plane 119883119874119884 and passthe origin except for Figure 7(b) where the camera view aimsat the equator of 1198782 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 valueWe plan to produce somematerialobjects with the help of simulation and printing technologiesWe produced Figures 1ndash7 in the VC++ 60 programmingenvironment with the aid of OpenGL a powerful graphicssoftware package
Complexity 7
(a) (b)
Figure 5 Colorful spherical wallpaper patterns with the 1199016119898 (a) and 119901119898119898 (b) symmetry
(a) (b)
Figure 6 Colorful spherical wallpaper patterns with the 1199014 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 7 Two spherical wallpaper patterns with the 1199013119898 symmetry The camera view of (a) is perpendicular to plane 119883119874119884 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 1146103511761039 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 ldquoSymmetry in the planerdquo Proceedings of the ImperialSt Petersburg Mineralogical Society vol 2 pp 245ndash291 1891(Russian)
[2] G Polya ldquoXII Uber die analogie der kristallsymmetrie in derebenerdquo Zeitschrift fur KristallographiemdashCrystalline Materialsvol 60 no 1-6 1924
[3] J Owen The Grammar of Ornament Van Nostrand Reinhold1910
[4] P S StevensHandbook of Regular Patterns MIT Press LondonUK 1981
[5] B Grunbaum 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 EscherExploring the Infinity N Harry Ed Abrams New York NYUSA 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 CA Reiter ldquoChaotic attractors with discrete planar symmetriesrdquoChaos Solitons amp Fractals vol 9 no 12 pp 2031ndash2054 1998
[9] K W Chung and H S Y Chan ldquoSymmetrical patterns fromdynamicsrdquo Computer Graphics Forum vol 12 no 1 pp 33ndash401993
[10] J Lu Z Ye Y Zou and R Ye ldquoOrbit trap renderingmethods forgenerating artistic images with crystallographic symmetriesrdquoComputers and Graphics vol 29 no 5 pp 794ndash801 2005
[11] D Douglas and S John ldquoThe art of random fractalsrdquo in Pro-ceedings of Bridges 2014 Mathematics Music Art ArchitectureCulture pp 79ndash86 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 ldquoLinear algebra andgeometryrdquo 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 YorkNY USA 1987
[18] K W Chung and H M Ma ldquoAutomatic generation of aestheticpatterns on fractal tilings by means of dynamical systemsrdquoChaos Solitons amp Fractals vol 24 no 4 pp 1145ndash1158 2005
[19] P Ouyang and X Wang ldquoBeautiful mathmdashaesthetic patternsbased on logarithmic spiralsrdquo IEEE Computer Graphics andApplications vol 33 no 6 pp 21ndash23 2013
[20] P Ouyang D Cheng Y Cao and X Zhan ldquoThe visualizationof hyperbolic patterns from invariant mapping methodrdquo Com-puters and Graphics vol 36 no 2 pp 92ndash100 2012
[21] P Ouyang and RW Fathauer ldquoBeautiful math part 2 aestheticpatterns based on fractal tilingsrdquo IEEE Computer Graphics andApplications vol 34 no 1 pp 68ndash76 2014
[22] P Ouyang and K Chung ldquoBeautiful math part 3 hyperbolicaesthetic patterns based on conformal mappingsrdquo IEEE Com-puter Graphics and Applications vol 34 no 2 pp 72ndash79 2014
[23] P Ouyang L Wang T Yu and X Huang ldquoAesthetic patternswith symmetries of the regular polyhedronrdquo Symmetry vol 9no 2 article no 21 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 5
(a) (b)
Figure 1 Colorful wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 2 Colorful wallpaper patterns with the 119901119898119898 (a) and 11990131198981 (b) symmetry
119891119871119889 =( 11 cossumVisin119871119889
sin (119909 sdot V) + sumVisin119871119889
(119909 sdot V)minus052 sinsum
Visin119871119889
cos (119909 sdot V) + sumVisin119871119889
(119909 sdot V)) (8)
It seems that the deference between (7) and (8) is not verysignificant However by Table 1 mappings 119867119891119871119904 1199014119898(119909) and119867119891119871119889 11990131198981(119909) have 16 and 12 summation terms respectivelyThe 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 1198782 = (119886 119887 119888) isin 1198773 | 1198862+1198872+1198882 = 1 be the unit spherein 1198773 let 119885 = 119865 be a projection plane where 119865 is a negativeconstant Assume that 119875(119886 119887 119888) isin 1198773 then 119875(119886 119887 119888) isin 1198773and1198751015840(minus119886 minus119887 minus119888) isin 1198773 are a pair of antipodal points For anypoint 119875(119886 119887 119888) isin 1198773 there exist a unique line 119871 through theorigin (0 0 0) and 119875 (and 1198751015840) which intersects the projectionplane 119885 = 119865 at point (120572 120573 119865) Denote the projection by 120591 Byanalytic geometry it is easy to check that
[[[120572120573119865 ]]] = 120591 (119886 119887 119888) = 120591 (minus119886 minus119887 minus119888) = [[[
119886119887119888 ]]] 119865119888 (9)
Because the projection point is at the center of 1198782 we call 120591 ascentral projection
The choice of the plane119885 = 119865 has a great influence on thespherical patterns If plane 119885 = 119865 is too close to coordinateplane 119883119874119884 the resulting spherical pattern only shows a few
6 Complexity
(a) (b)
Figure 3 Two spherical wallpaper patterns with the 1199014119898 symmetry in which the projection plane was set as 119885 = minus2120587 (a) and 119885 = minus4120587 (b)
(a) (b)
Figure 4 Colorful spherical wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
periods of the wallpaper pattern However if plane 119885 = 119865is too far away from coordinate plane 119883119874119884 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 119885 = 119865
Given a wallpaper pattern by central projection 120591 we canmap it onto the sphere 1198782 and obtain a corresponding spher-ical wallpaper pattern We next explain how to implement itin more detail
Suppose that (119886 119887 119888) isin 1198773 and 119867119891119871119908(119909) is an invari-ant mapping compatible with the symmetry of wallpapergroup 119908 First by central projection 120591 we obtain a corre-sponding point ((119860119888)119886 (119860119888)119887 119865) on the projection plane119885 = 119865 Second let 119867119891119871119908(119909) be iteration function and let119909((119860119888)119886 (119860119888)119887) be initial point using the color schemeof orbit trap we assign a color to point ((119860119888)119886 (119860119888)119887 119865)Finally repeat the second step by coloring unit sphere 1198782pointwise we obtain a spherical pattern of the wallpapergroup 119908 symmetry
Figures 3ndash7 are ten patterns obtained by this mannerExcept for Figure 3(b) in which the projection plane wasset as 119885 = minus4120587 all the other projection planes were set as119885 = minus2120587 We utilized the wallpaper patterns of Figure 1 toproduce spherical patterns shown in Figure 4 For beauty allthe camera views are perpendicular to plane 119883119874119884 and passthe origin except for Figure 7(b) where the camera view aimsat the equator of 1198782 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 valueWe plan to produce somematerialobjects with the help of simulation and printing technologiesWe produced Figures 1ndash7 in the VC++ 60 programmingenvironment with the aid of OpenGL a powerful graphicssoftware package
Complexity 7
(a) (b)
Figure 5 Colorful spherical wallpaper patterns with the 1199016119898 (a) and 119901119898119898 (b) symmetry
(a) (b)
Figure 6 Colorful spherical wallpaper patterns with the 1199014 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 7 Two spherical wallpaper patterns with the 1199013119898 symmetry The camera view of (a) is perpendicular to plane 119883119874119884 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 1146103511761039 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 ldquoSymmetry in the planerdquo Proceedings of the ImperialSt Petersburg Mineralogical Society vol 2 pp 245ndash291 1891(Russian)
[2] G Polya ldquoXII Uber die analogie der kristallsymmetrie in derebenerdquo Zeitschrift fur KristallographiemdashCrystalline Materialsvol 60 no 1-6 1924
[3] J Owen The Grammar of Ornament Van Nostrand Reinhold1910
[4] P S StevensHandbook of Regular Patterns MIT Press LondonUK 1981
[5] B Grunbaum 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 EscherExploring the Infinity N Harry Ed Abrams New York NYUSA 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 CA Reiter ldquoChaotic attractors with discrete planar symmetriesrdquoChaos Solitons amp Fractals vol 9 no 12 pp 2031ndash2054 1998
[9] K W Chung and H S Y Chan ldquoSymmetrical patterns fromdynamicsrdquo Computer Graphics Forum vol 12 no 1 pp 33ndash401993
[10] J Lu Z Ye Y Zou and R Ye ldquoOrbit trap renderingmethods forgenerating artistic images with crystallographic symmetriesrdquoComputers and Graphics vol 29 no 5 pp 794ndash801 2005
[11] D Douglas and S John ldquoThe art of random fractalsrdquo in Pro-ceedings of Bridges 2014 Mathematics Music Art ArchitectureCulture pp 79ndash86 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 ldquoLinear algebra andgeometryrdquo 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 YorkNY USA 1987
[18] K W Chung and H M Ma ldquoAutomatic generation of aestheticpatterns on fractal tilings by means of dynamical systemsrdquoChaos Solitons amp Fractals vol 24 no 4 pp 1145ndash1158 2005
[19] P Ouyang and X Wang ldquoBeautiful mathmdashaesthetic patternsbased on logarithmic spiralsrdquo IEEE Computer Graphics andApplications vol 33 no 6 pp 21ndash23 2013
[20] P Ouyang D Cheng Y Cao and X Zhan ldquoThe visualizationof hyperbolic patterns from invariant mapping methodrdquo Com-puters and Graphics vol 36 no 2 pp 92ndash100 2012
[21] P Ouyang and RW Fathauer ldquoBeautiful math part 2 aestheticpatterns based on fractal tilingsrdquo IEEE Computer Graphics andApplications vol 34 no 1 pp 68ndash76 2014
[22] P Ouyang and K Chung ldquoBeautiful math part 3 hyperbolicaesthetic patterns based on conformal mappingsrdquo IEEE Com-puter Graphics and Applications vol 34 no 2 pp 72ndash79 2014
[23] P Ouyang L Wang T Yu and X Huang ldquoAesthetic patternswith symmetries of the regular polyhedronrdquo Symmetry vol 9no 2 article no 21 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Complexity
(a) (b)
Figure 3 Two spherical wallpaper patterns with the 1199014119898 symmetry in which the projection plane was set as 119885 = minus2120587 (a) and 119885 = minus4120587 (b)
(a) (b)
Figure 4 Colorful spherical wallpaper patterns with the 1199014119898 (a) and 1199016119898 (b) symmetry
periods of the wallpaper pattern However if plane 119885 = 119865is too far away from coordinate plane 119883119874119884 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 119885 = 119865
Given a wallpaper pattern by central projection 120591 we canmap it onto the sphere 1198782 and obtain a corresponding spher-ical wallpaper pattern We next explain how to implement itin more detail
Suppose that (119886 119887 119888) isin 1198773 and 119867119891119871119908(119909) is an invari-ant mapping compatible with the symmetry of wallpapergroup 119908 First by central projection 120591 we obtain a corre-sponding point ((119860119888)119886 (119860119888)119887 119865) on the projection plane119885 = 119865 Second let 119867119891119871119908(119909) be iteration function and let119909((119860119888)119886 (119860119888)119887) be initial point using the color schemeof orbit trap we assign a color to point ((119860119888)119886 (119860119888)119887 119865)Finally repeat the second step by coloring unit sphere 1198782pointwise we obtain a spherical pattern of the wallpapergroup 119908 symmetry
Figures 3ndash7 are ten patterns obtained by this mannerExcept for Figure 3(b) in which the projection plane wasset as 119885 = minus4120587 all the other projection planes were set as119885 = minus2120587 We utilized the wallpaper patterns of Figure 1 toproduce spherical patterns shown in Figure 4 For beauty allthe camera views are perpendicular to plane 119883119874119884 and passthe origin except for Figure 7(b) where the camera view aimsat the equator of 1198782 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 valueWe plan to produce somematerialobjects with the help of simulation and printing technologiesWe produced Figures 1ndash7 in the VC++ 60 programmingenvironment with the aid of OpenGL a powerful graphicssoftware package
Complexity 7
(a) (b)
Figure 5 Colorful spherical wallpaper patterns with the 1199016119898 (a) and 119901119898119898 (b) symmetry
(a) (b)
Figure 6 Colorful spherical wallpaper patterns with the 1199014 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 7 Two spherical wallpaper patterns with the 1199013119898 symmetry The camera view of (a) is perpendicular to plane 119883119874119884 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 1146103511761039 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 ldquoSymmetry in the planerdquo Proceedings of the ImperialSt Petersburg Mineralogical Society vol 2 pp 245ndash291 1891(Russian)
[2] G Polya ldquoXII Uber die analogie der kristallsymmetrie in derebenerdquo Zeitschrift fur KristallographiemdashCrystalline Materialsvol 60 no 1-6 1924
[3] J Owen The Grammar of Ornament Van Nostrand Reinhold1910
[4] P S StevensHandbook of Regular Patterns MIT Press LondonUK 1981
[5] B Grunbaum 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 EscherExploring the Infinity N Harry Ed Abrams New York NYUSA 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 CA Reiter ldquoChaotic attractors with discrete planar symmetriesrdquoChaos Solitons amp Fractals vol 9 no 12 pp 2031ndash2054 1998
[9] K W Chung and H S Y Chan ldquoSymmetrical patterns fromdynamicsrdquo Computer Graphics Forum vol 12 no 1 pp 33ndash401993
[10] J Lu Z Ye Y Zou and R Ye ldquoOrbit trap renderingmethods forgenerating artistic images with crystallographic symmetriesrdquoComputers and Graphics vol 29 no 5 pp 794ndash801 2005
[11] D Douglas and S John ldquoThe art of random fractalsrdquo in Pro-ceedings of Bridges 2014 Mathematics Music Art ArchitectureCulture pp 79ndash86 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 ldquoLinear algebra andgeometryrdquo 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 YorkNY USA 1987
[18] K W Chung and H M Ma ldquoAutomatic generation of aestheticpatterns on fractal tilings by means of dynamical systemsrdquoChaos Solitons amp Fractals vol 24 no 4 pp 1145ndash1158 2005
[19] P Ouyang and X Wang ldquoBeautiful mathmdashaesthetic patternsbased on logarithmic spiralsrdquo IEEE Computer Graphics andApplications vol 33 no 6 pp 21ndash23 2013
[20] P Ouyang D Cheng Y Cao and X Zhan ldquoThe visualizationof hyperbolic patterns from invariant mapping methodrdquo Com-puters and Graphics vol 36 no 2 pp 92ndash100 2012
[21] P Ouyang and RW Fathauer ldquoBeautiful math part 2 aestheticpatterns based on fractal tilingsrdquo IEEE Computer Graphics andApplications vol 34 no 1 pp 68ndash76 2014
[22] P Ouyang and K Chung ldquoBeautiful math part 3 hyperbolicaesthetic patterns based on conformal mappingsrdquo IEEE Com-puter Graphics and Applications vol 34 no 2 pp 72ndash79 2014
[23] P Ouyang L Wang T Yu and X Huang ldquoAesthetic patternswith symmetries of the regular polyhedronrdquo Symmetry vol 9no 2 article no 21 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 7
(a) (b)
Figure 5 Colorful spherical wallpaper patterns with the 1199016119898 (a) and 119901119898119898 (b) symmetry
(a) (b)
Figure 6 Colorful spherical wallpaper patterns with the 1199014 (a) and 1199016119898 (b) symmetry
(a) (b)
Figure 7 Two spherical wallpaper patterns with the 1199013119898 symmetry The camera view of (a) is perpendicular to plane 119883119874119884 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 1146103511761039 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 ldquoSymmetry in the planerdquo Proceedings of the ImperialSt Petersburg Mineralogical Society vol 2 pp 245ndash291 1891(Russian)
[2] G Polya ldquoXII Uber die analogie der kristallsymmetrie in derebenerdquo Zeitschrift fur KristallographiemdashCrystalline Materialsvol 60 no 1-6 1924
[3] J Owen The Grammar of Ornament Van Nostrand Reinhold1910
[4] P S StevensHandbook of Regular Patterns MIT Press LondonUK 1981
[5] B Grunbaum 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 EscherExploring the Infinity N Harry Ed Abrams New York NYUSA 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 CA Reiter ldquoChaotic attractors with discrete planar symmetriesrdquoChaos Solitons amp Fractals vol 9 no 12 pp 2031ndash2054 1998
[9] K W Chung and H S Y Chan ldquoSymmetrical patterns fromdynamicsrdquo Computer Graphics Forum vol 12 no 1 pp 33ndash401993
[10] J Lu Z Ye Y Zou and R Ye ldquoOrbit trap renderingmethods forgenerating artistic images with crystallographic symmetriesrdquoComputers and Graphics vol 29 no 5 pp 794ndash801 2005
[11] D Douglas and S John ldquoThe art of random fractalsrdquo in Pro-ceedings of Bridges 2014 Mathematics Music Art ArchitectureCulture pp 79ndash86 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 ldquoLinear algebra andgeometryrdquo 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 YorkNY USA 1987
[18] K W Chung and H M Ma ldquoAutomatic generation of aestheticpatterns on fractal tilings by means of dynamical systemsrdquoChaos Solitons amp Fractals vol 24 no 4 pp 1145ndash1158 2005
[19] P Ouyang and X Wang ldquoBeautiful mathmdashaesthetic patternsbased on logarithmic spiralsrdquo IEEE Computer Graphics andApplications vol 33 no 6 pp 21ndash23 2013
[20] P Ouyang D Cheng Y Cao and X Zhan ldquoThe visualizationof hyperbolic patterns from invariant mapping methodrdquo Com-puters and Graphics vol 36 no 2 pp 92ndash100 2012
[21] P Ouyang and RW Fathauer ldquoBeautiful math part 2 aestheticpatterns based on fractal tilingsrdquo IEEE Computer Graphics andApplications vol 34 no 1 pp 68ndash76 2014
[22] P Ouyang and K Chung ldquoBeautiful math part 3 hyperbolicaesthetic patterns based on conformal mappingsrdquo IEEE Com-puter Graphics and Applications vol 34 no 2 pp 72ndash79 2014
[23] P Ouyang L Wang T Yu and X Huang ldquoAesthetic patternswith symmetries of the regular polyhedronrdquo Symmetry vol 9no 2 article no 21 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
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 1146103511761039 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 ldquoSymmetry in the planerdquo Proceedings of the ImperialSt Petersburg Mineralogical Society vol 2 pp 245ndash291 1891(Russian)
[2] G Polya ldquoXII Uber die analogie der kristallsymmetrie in derebenerdquo Zeitschrift fur KristallographiemdashCrystalline Materialsvol 60 no 1-6 1924
[3] J Owen The Grammar of Ornament Van Nostrand Reinhold1910
[4] P S StevensHandbook of Regular Patterns MIT Press LondonUK 1981
[5] B Grunbaum 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 EscherExploring the Infinity N Harry Ed Abrams New York NYUSA 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 CA Reiter ldquoChaotic attractors with discrete planar symmetriesrdquoChaos Solitons amp Fractals vol 9 no 12 pp 2031ndash2054 1998
[9] K W Chung and H S Y Chan ldquoSymmetrical patterns fromdynamicsrdquo Computer Graphics Forum vol 12 no 1 pp 33ndash401993
[10] J Lu Z Ye Y Zou and R Ye ldquoOrbit trap renderingmethods forgenerating artistic images with crystallographic symmetriesrdquoComputers and Graphics vol 29 no 5 pp 794ndash801 2005
[11] D Douglas and S John ldquoThe art of random fractalsrdquo in Pro-ceedings of Bridges 2014 Mathematics Music Art ArchitectureCulture pp 79ndash86 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 ldquoLinear algebra andgeometryrdquo 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 YorkNY USA 1987
[18] K W Chung and H M Ma ldquoAutomatic generation of aestheticpatterns on fractal tilings by means of dynamical systemsrdquoChaos Solitons amp Fractals vol 24 no 4 pp 1145ndash1158 2005
[19] P Ouyang and X Wang ldquoBeautiful mathmdashaesthetic patternsbased on logarithmic spiralsrdquo IEEE Computer Graphics andApplications vol 33 no 6 pp 21ndash23 2013
[20] P Ouyang D Cheng Y Cao and X Zhan ldquoThe visualizationof hyperbolic patterns from invariant mapping methodrdquo Com-puters and Graphics vol 36 no 2 pp 92ndash100 2012
[21] P Ouyang and RW Fathauer ldquoBeautiful math part 2 aestheticpatterns based on fractal tilingsrdquo IEEE Computer Graphics andApplications vol 34 no 1 pp 68ndash76 2014
[22] P Ouyang and K Chung ldquoBeautiful math part 3 hyperbolicaesthetic patterns based on conformal mappingsrdquo IEEE Com-puter Graphics and Applications vol 34 no 2 pp 72ndash79 2014
[23] P Ouyang L Wang T Yu and X Huang ldquoAesthetic patternswith symmetries of the regular polyhedronrdquo Symmetry vol 9no 2 article no 21 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
![Symmetries in 2HDM and beyond [2mm] Lecture 1: Describing ... · Lecture 2: symmetries in 2HDM Lecture 3: abelian symmetries in bSM models Lecture 4: non-abelian symmetries in NHDM](https://static.fdocuments.net/doc/165x107/6056c24cff523627a22196b1/symmetries-in-2hdm-and-beyond-2mm-lecture-1-describing-lecture-2-symmetries.jpg)
