the visualization of spherical patterns with symmetries of ...researcharticle the visualization of...
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Research ArticleThe Visualization of Spherical Patterns with Symmetries ofthe Wallpaper Group
Shihuan Liu ,1,2 Ming Leng,1 and Peichang Ouyang 1
1School of Mathematics & Physics, Jinggangshan University, Jiâan 343009, China2Sichuan Province Key Lab of Signal and Information Processing, Southwest Jiaotong University, Chengdu 611756, China
Correspondence should be addressed to Peichang Ouyang; g [email protected]
Received 17 October 2017; Accepted 1 January 2018; Published 12 February 2018
Academic Editor: Michele Scarpiniti
Copyright © 2018 Shihuan Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By constructing invariant mappings associated with wallpaper groups, this paper presents a simple and efficient method to generatecolorful wallpaper patterns. Although the constructed mappings have simple form and only two parameters, combined with thecolor scheme of orbit trap algorithm, such mappings can create a great variety of aesthetic wallpaper patterns. The resultingwallpaper patterns are further projected by central projection onto the sphere. This creates the interesting spherical patterns thatpossess infinite symmetries in a finite space.
1. Introduction
Wallpaper groups (or plane crystallographic groups) aremathematical classification of two-dimensional repetitivepatterns. The first systematic proof that there were only 17possible wallpaper patterns was carried out by Fedorov in1891 [1] and later derived independently by PoÌlya 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 [3â5]. It is surprising that the three-dimensional 230 crystallographic groups were enumeratedbefore the planar wallpaper groups.
The art of M. C. Escher features the rigorous mathe-matical structure and elegant artistic charm, which mightbe the one and only in the history of art. After his journeyto the Alhambra, La Mezquita, and Cordoba, he createdmany mathematically inspired arts and became a master increating wallpaper arts [6]. With the development of moderncomputers, there is considerable research on the automaticgeneration of wallpaper patterns. In [7], Field and Golubitskyfirst proposed the conception of equivariant mappings. Theyconstructed equivariant mapping to generated chaotic cyclic,dihedral, and wallpaper attractors. Carter et al. developed aneasier method that used equivariant truncated 2-dimensional
Fourier series to achieve it [8]. Chung and Chan [9] andLu et al. [10] later presented similar ideas to create colorfulwallpaper patterns. Recently, Douglas and John discovered avery simple approach to yield interesting wallpaper patternsof fractal characteristic [11].
The key idea behind [7â10] is equivariant mapping,which is not easy to achieve, since such mapping must becommutable with respect to symmetry group. In this paper,we present a simple invariant method to create wallpaperpatterns. It has independent mapping form and only twoparameters. Combined with the color scheme of orbit trapalgorithm, our approach can be conveniently utilized to yieldrich wallpaper patterns.
Escherâs Circle Limits IâIV are unusual and visuallyattractive because they realized infinity in a finite unit disc.Inspired by his arts, we use central projection to projectwallpaper patterns onto the finite sphere. This obtains theaesthetic patterns of infinite symmetry structure in thefinite sphere space. Such patterns look beautiful. Combinedwith simulation and printing technologies, these computer-generated patterns could be utilized in wallpaper, textiles,ceramics, carpet, stained glasswindows, and so on, producingboth economic and aesthetic benefits.
The remainder of this paper is organized as follows. InSection 2, we first introduce some basic conceptions and the
HindawiComplexityVolume 2018, Article ID 7315695, 8 pageshttps://doi.org/10.1155/2018/7315695
http://orcid.org/0000-0003-3003-1919http://orcid.org/0000-0003-0447-3190https://doi.org/10.1155/2018/7315695
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Table 1: The concrete invariant mappingð»ððŽ,ðµ(ð¥) forms associated with 17 wallpaper groups. In the fourth column, the subscripts ðŽ and ðµidentify the lattice kind (ð¿ ð represents square lattice, while ð¿ð represents diamond lattice) and wallpaper group type, respectively.Wallpapergroup
Pointgroup Extra symmetry set Invariant mapping
p1 ð¶1 None ð»ðð¿ð ,ð1 (ð¥) = âðâð¶1ðð¿ð [ð (ð¥)]p2 ð¶2 None ð»ðð¿ð ,ð2 (ð¥) = âðâð¶2ðð¿ð [ð (ð¥)]ðð ð·1 ð1 (ð, ð) = (ð, âð) ð»ðð¿ð ,ðð (ð¥) = âðâð·1ðð¿ð [ð (ð¥)] + âðâð·1ðð¿ð [(ð1ð) (ð¥)]ððð ð·2 ð1(ð, ð) = (ð, âð),ð2(ð, ð) = (âð, ð) ð»ðð¿ð ,ððð (ð¥) = âðâð·2ðð¿ð [ð (ð¥)] + 2âð=1{ âðâð·2ðð¿ð [(ððð) (ð¥)]}ðð ð·1 ð1 (ð, ð) = (ð + ð, âð) ð»ðð¿ð ,ðð (ð¥) = âðâð·1ðð¿ð [ð (ð¥)] + âðâð·1ðð¿ð [(ð1ð) (ð¥)]ððð ð·2 ð1(ð, ð) = (ð + ð, âð),ð2(ð, ð) = (âð, ð) ð»ðð¿ð ,ððð (ð¥) = âðâð·2ðð¿ð [ð (ð¥)] + 2âð=1{ âðâð·2ðð¿ð [(ððð) (ð¥)]}ð¶ð ð·1 ð1(ð, ð) = (ð, âð),ð2(ð, ð) = (ð + ð, ð â ð) ð»ðð¿ð ,ðð (ð¥) = âðâð·1ðð¿ð [ð (ð¥)] + 2âð=1{ âðâð·1ðð¿ð [(ððð) (ð¥)]}ð¶ðð ð·2 ð1(ð, ð) = (ð, âð),ð2(ð, ð) = (ð â ð, ð + ð)ð3(ð, ð) = (ð + ð, ð â ð),ð4(ð, ð) = (âð, ð) ð»ðð¿ð ,ððð (ð¥) = âðâð·2ðð¿ð [ð (ð¥)] +
4âð=1
{ âðâð·2
ðð¿ð [(ððð) (ð¥)]}p4 ð¶4 None ð»ðð¿ð ,ð4 (ð¥) = âðâð¶4ðð¿ð [ð (ð¥)]ð4ð ð·4 ð1 (ð, ð) = (ð + ð, âð) ð»ðð¿ð ,ð4ð (ð¥) = âðâð·4ðð¿ð [ð (ð¥)] + âðâð·4ðð¿ð [(ð1ð) (ð¥)]ð4ð ð·4 ð1 (ð, ð) = (ð, âð) ð»ðð¿ð ,ð4ð (ð¥) = âðâð·4ðð¿ð [ð (ð¥)] + âðâð·4ðð¿ðð [(ð1ð) (ð¥)]ððð ð·2 ð1(ð, ð) = (ð + ð, ð â ð),ð2(ð, ð) = (ð â ð, ð + ð) ð»ðð¿ð ,ððð (ð¥) = âðâð·2ðð¿ð [ð (ð¥)] + 2âð=1{ âðâð·2ðð¿ð [(ððð) (ð¥)]}p3 ð¶3 None ð»ðð¿ð ,ð3 (ð¥) = âðâð¶3ðð¿ð [ð (ð¥)]p3m1 ð·3 ð1 (ð, ð) = (âð, ð) ð»ðð¿ð ,ð3ð1 (ð¥) = âðâð·3ðð¿ð [ð (ð¥)] + âðâð·3ðð¿ð [(ð1ð) (ð¥)]p31m ð·3 ð1 (ð, ð) = (ð, âð) ð»ðð¿ð ,ð31ð (ð¥) = âðâð·3ðð¿ð [ð (ð¥)] + âðâð·3ðð¿ð [(ð1ð) (ð¥)]p6 ð¶6 None ð»ðð¿ð ,ð6 (ð¥) = âðâð¶6ðð¿ð [ð (ð¥)]ð6ð ð·6 ð1 (ð, ð) = (ð, âð) ð»ðð¿ð ,ð6ð (ð¥) = âðâð·6ðð¿ð [ð (ð¥)] + âðâð·6ðð¿ð [(ð1ð) (ð¥)]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 ð 2 is essentially a subgroup of ð 2. Or,equivalently, for any basis vectors of ð 2, the subgroup of alllinear combinations with integer coefficients of the vectors
forms a lattice [12, 13]. Since a lattice is a finitely generated freeabelian group, it is isomorphic to ð2 and fully spans the realvector spaceð 2 [14]. A latticemay be viewed as a regular tilingof a space by a primitive cell. Lattices have many significantapplications in pure mathematics, particularly in connectionto Lie algebras, number theory, and group theory [15].
In this section, wemainly introduce the lattices associatedwith wallpaper group. Firstly, we introduce some basicconceptions.
The symmetry group of an object is the set of all isometriesunder which the object is invariant with composition as thegroup operation. A point group (sometimes called rosettegroup) is a group of isometries that keep at least one pointfixed.
Point groups in ð 2 come in two infinite families: dihedralgroup ð·ð which is the symmetry group of a regular polygon
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and cyclic groupð¶ð that only comprises rotation transforma-tions of ð·ð. Let ð ð = ( cos(2ð/ð) â sin(2ð/ð)sin(2ð/ð) cos(2ð/ð) ) and ð = ( â1 00 1 ).Then their matrix group can be represented as ð¶ð = {ð ðð, ð =1, 2, 3, . . . , ð} andð·ð = ð¶ð ⪠{ðð ðð, ð = 1, 2, 3, . . . , ð}.
Awallpaper group is a type of topologically discrete groupin ð 2 which contains two linearly independent translations.A lattice in ð 2 is the symmetry group of discrete translationalsymmetry in two independent directions. A tiling with thislattice of translational symmetry cannot have more but mayhave less symmetry than the lattice itself. Let ð¿ be a lattice inð 2. A lattice ð¿â is called the dual lattice of ð¿ if, âð¢ â ð¿ andâV â ð¿â, the inner product ð¢ â V is an integer, where ð¢ and Vare vectors inð 2. Letð be amapping fromð 2 toð 2 and letðºbe a symmetry group in ð 2;ð is called an invariant mappingwith respect to ðº if, âð¥ â ð 2 and âð â ðº, ð(ð¥) = ð(ðð¥).
By the crystallographic restriction theorem, there areonly 5 lattice types in ð 2 [16]. Although wallpaper groupshave totally 17 types, their lattices can be simplified into twolattices: square and diamond lattices. For convenience, werequire that the inner product of the mutual dual lattice of awallpaper group be an integermultiple of 2ð.Throughout thepaper, for square lattice, we choose lattice ð¿ ð = {(1, 0), (0, 1)}with dual lattice ð¿âð = {2ð(1, 0), 2ð(0, 1)}; for diamond lattice,we choose lattice ð¿ð = {(1, 0), (1/2)(â1,â3)}with dual latticeð¿âð = {(2ð/â3)(â3, â1), 2ð(0, â2/â3)}.
In this paper, we use standard crystallographic notationsof wallpaper groups [16, 17]. Among 17 wallpaper groups, ð1,ð2, ðð, ððð, ðð, ððð, ðð, ððð, ð4, ððð, ð4ð, and ð4ðpossess square lattice, while ð3, ð3ð1, ð31ð, ð6, and ð6ðpossess 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 ðð (ð = 1, 2, 3, 4) are sine or cosinefunctions, ðº is a wallpaper group with lattice ð¿ = {ðŽ, ðµ}, ð¿â ={ðŽâ, ðµâ} is the dual lattice of ð¿, and ð and ð are real numbers.Then mapping
ðð¿ (ð¥) = ( ðð1 {âVâð¿ð2 (ð¥ â V) + âVâð¿ (ð¥ â V)}ðð3 {âVâð¿ð4 (ð¥ â V) + â
Vâð¿(ð¥ â V)} ),
âð¥ â ð 2,(1)
is invariant with respect to ð¿â, or ðð¿(ð¥) has translationinvariance of ð¿â; that is,
(ðð1 {âVâð¿ð2 ((ð¢ + ð¥) â V) + âVâð¿ ((ð¢ + ð¥) â V)}ðð3 {âVâð¿
ð4 ((ð¢ + ð¥) â V) + âVâð¿
((ð¢ + ð¥) â V)})
=(ðð1 {âVâð¿ð2 (ð¥ â V) + âVâð¿ (ð¥ â V)}ðð3 {âVâð¿
ð4 (ð¥ â V) + âVâð¿
(ð¥ â V)}) = ðð¿ (ð¥) ,(2)
where ð¢ = ððŽâ + ððµâ, ð, ð â ð.Proof. Since ð¿â is the dual lattice of ð¿, âV â ð¿, we have ð¢ â V =(ððŽâ + ððµâ) â V = ð(ðŽâ â V) + ð(ðµâ â V) = 2ðð for certainð â ð. Thus we get ðð{âVâð¿ ðð((ð¢+ð¥) â V)+âVâð¿((ð¢+ð¥) â V)} =ðð{âVâð¿ ðð(ð¥ â V)+âVâð¿(ð¥ â V)}, since ðð and ðð are functions ofperiod 2ð (ð, ð = 1, 2, 3, 4). Consequently, the mapping ðð¿(ð¥)constructed by ðð (ð = 1, 2, 3, 4) satisfies (2). This completesthe proof.
Essentially Lemma 1 says that ðð¿(ð¥) is a double periodmapping (of period 2ð) along the independent translationaldirections of ð¿â.Theorem 2. Let ðº be a finite group realized by 2 à 2 matricesacting on ð 2 by multiplication on the right and let ð be anarbitrary mapping from ð 2 to ð 2. Then mappingð»ð,ðº (ð¥) = â
ðâðº
ð [ð (ð¥)] , ð¥ â ð 2, (3)is an invariant mapping with respect to ðº.Proof. For ð â ðº, by closure of the group operation, we seethat ðð runs through ðº as ð does. Therefore we haveð»ð,ðº [ð (ð¥)] = â
ðâðº
ð [ð (ðð¥)] = âðââðº
ð [ðâ (ð¥)]= ð»ð,ðº (ð¥) , (4)
where ðâ = ðð â ðº. This means thatð»ð,ðº(ð¥) is an invariantmapping with respect to ðº.
Combining Lemma 1 and Theorem 2, we immediatelyderive the following theorem.
Theorem 3. Let ð in Theorem 2 have the form ðð¿ as inLemma 1. Suppose that ðº is a cyclic group ð¶ð or dihedralgroup ð·ð with lattice ð¿; ð¿â is the dual lattice associated withð¿. Assume that ð»ðð¿,ðº(ð¥) is a mapping from ð 2 to ð 2 of thefollowing form:ð»ðð¿,ðº (ð¥) = â
ðâðº
ðð¿ [ð (ð¥)] , ð¥ â ð 2. (5)Thenð»ðð¿,ðº(ð¥) is an invariant mapping with respect to both ðºand ð¿â.
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 ð»ðð¿,ðº(ð¥) in
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BEGINstart x = 0end x = 6 â 3.1415926start y = 0end y = 6 â 3.1415926 //Set pi = 3.1415926step x = (end x â start x)/X res //Xres is the resolution in X directionstep y = (end y â 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 â step xy = start y + j â step yFOR k = 1 TOMaxIter//MaxIter is the number of iterations, the default set is 100
/âGiven a invariant mappingð»ðð¿,ð€(ð¥) associated with a wallpaper group ð€ as iterationfunction and initial point (x, y), function Iteration (x, y) iterates MaxIter times. The iteratedsequences are stored in the array Sequenceâ/.
Sequence [ð] = Iteration (x, y)END FOR
/âInputting Sequence, the color scheme OrbitTrap outputs the color [r, g, b]â/[r, g, b] = OrbitTrap (Sequence)Set color [r, g, b] to point (x, y)
END FOREND FOR
END
Algorithm 1: CreatingWallpaperPattern()// algorithm for creating patterns with the wallpaper symmetry.
Theorem 3 can be used to create wallpaper patterns well.However, except for the symmetries of a point group, somewallpaper groups may possess other symmetries. For exam-ple, except for symmetries of dihedral group ð·3, wallpapergroup ð31ð still has a reflection along horizontal direction,say symmetry ð1(ð, ð) = (ð, âð) ((ð, ð) â ð 2); besidesð·2 symmetry, wallpaper group ððð contains perpendicularreflections; that is, ð1(ð, ð) = (ð, âð) and ð2(ð, ð) = (âð, ð).
For a wallpaper group ð€ with extra symmetry set Î ={ð1, ð2, ð3, . . . , ðð}, based on mapping (5), we add properterms so that the resulting mappingð»ðð¿,ð€(ð¥) is also invariantwith respect to ð€. This is summarized inTheorem 4.Theorem 4. Let ð inTheorem 2 have the form ðð¿ in Lemma 1.Suppose that ð€ is a wallpaper group with symmetry group ðºand extra symmetry set Î = {ð1, ð2, ð3, . . . , ðð}. Assume that ð¿is the lattice of ð€ and ð¿â is the dual lattice associated with ð¿.Letð»ðð¿,ð€(ð¥) be a mapping from ð 2 to ð 2 of the following form:ð»ðð¿,ð€ (ð¥) = â
ðâðº
ðð¿ [ð (ð¥)]+ ðâð=1
{{{ âðâðº,ððâÎðð¿ [(ððð) (ð¥)]}}} , ð¥ â ð 2.(6)
Thenð»ðð¿,ð€(ð¥) is an invariant mapping with respect to both ð€and ð¿â.
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 [18â23]. Color scheme is analgorithm that is used to determine the color of a point. Givena color scheme and domainð·, by iterating invariantmappingð»ðð¿,ð€(ð¥), ð¥ â ð·, one can determine the color of ð¥. Coloringpoints in ð· pointwise, one can obtain a colorful pattern inð· with symmetries of the wallpaper group ð€. Figures 1-2are four wallpaper patterns obtained in this manner. Thesepatterns were created by VC++ 6.0 on a PC (SVGA). InAlgorithm 1, we provide the pseudocode so that the interestedreader can create their own colorful wallpaper patterns.
The color scheme used in this paper is called orbit trapalgorithm [10].We refer the reader to [10, 20] for more detailsabout the algorithm (the algorithm is named as functionOrbitTrap() in Algorithm 1). It hasmany parameters to adjustcolor, which could enhance the visual appeal of patternseffectively. Compared with the complex equivariant mappingconstructed in [7â10], 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 ð»ðð¿ð ,ð4ð(ð¥) and ð»ðð¿ð ,ð3ð1(ð¥), respectively, inwhich the specific mappings ðð¿ð and ðð¿ð were
ðð¿ð =(2.12 cos{âVâð¿ð cos (ð¥ â V) + âVâð¿ð (ð¥ â V)}1.03 cos{âVâð¿ð
sin (ð¥ â V) + âVâð¿ð
(ð¥ â V)}), (7)
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(a) (b)
Figure 1: Colorful wallpaper patterns with the ð4ð (a) and ð6ð (b) symmetry.
(a) (b)
Figure 2: Colorful wallpaper patterns with the ððð (a) and ð3ð1 (b) symmetry.ðð¿ð =( 1.1 cos{âVâð¿ð sin (ð¥ â V) + âVâð¿ð (ð¥ â V)}â0.52 sin{â
Vâð¿ð
cos (ð¥ â V) + âVâð¿ð
(ð¥ â V)}). (8)It seems that the deference between (7) and (8) is not verysignificant. However, by Table 1, mappings ð»ðð¿ð ,ð4ð(ð¥) andð»ðð¿ð ,ð3ð1(ð¥) have 16 and 12 summation terms, respectively.The cumulative difference will be very obvious, which isenough to produce different style patterns.
5. Spherical Wallpaper Patterns byCentral Projection
In this section, we introduce central projection to yieldspherical patterns of the wallpaper symmetry.
Let ð2 = {(ð, ð, ð) â ð 3 | ð2+ð2+ð2 = 1} be the unit spherein ð 3, let ð = ð¹ be a projection plane, where ð¹ is a negativeconstant. Assume that ð(ð, ð, ð) â ð 3; then ð(ð, ð, ð) â ð 3andð(âð, âð, âð) â ð 3 are a pair of antipodal points. For anypoint ð(ð, ð, ð) â ð 3, there exist a unique line ð¿ through theorigin (0, 0, 0) and ð (and ð) which intersects the projectionplane ð = ð¹ at point (ðŒ, ðœ, ð¹). Denote the projection by ð. Byanalytic geometry, it is easy to check that
[[[ðŒðœð¹ ]]] = ð (ð, ð, ð) = ð (âð, âð, âð) = [[[
ððð ]]] ð¹ð . (9)Because the projection point is at the center of ð2, we call ð ascentral projection.
The choice of the planeð = ð¹ has a great influence on thespherical patterns. If plane ð = ð¹ is too close to coordinateplane ððð, the resulting spherical pattern only shows a few
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(a) (b)
Figure 3: Two spherical wallpaper patterns with the ð4ð symmetry, in which the projection plane was set as ð = â2ð (a) and ð = â4ð (b).
(a) (b)
Figure 4: Colorful spherical wallpaper patterns with the ð4ð (a) and ð6ð (b) symmetry.periods of the wallpaper pattern. However, if plane ð = ð¹is too far away from coordinate plane ððð, 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 ð = ð¹.
Given a wallpaper pattern, by central projection ð, we canmap it onto the sphere ð2 and obtain a corresponding spher-ical wallpaper pattern. We next explain how to implement itin more detail.
Suppose that (ð, ð, ð) â ð 3 and ð»ðð¿,ð€(ð¥) is an invari-ant mapping compatible with the symmetry of wallpapergroup ð€. First, by central projection ð, we obtain a corre-sponding point ((ðŽ/ð)ð, (ðŽ/ð)ð, ð¹) on the projection planeð = ð¹. Second, let ð»ðð¿,ð€(ð¥) be iteration function and letð¥((ðŽ/ð)ð, (ðŽ/ð)ð) be initial point; using the color schemeof orbit trap, we assign a color to point ((ðŽ/ð)ð, (ðŽ/ð)ð, ð¹).Finally, repeat the second step; by coloring unit sphere ð2pointwise, we obtain a spherical pattern of the wallpapergroup ð€ symmetry.
Figures 3â7 are ten patterns obtained by this manner.Except for Figure 3(b) in which the projection plane wasset as ð = â4ð, all the other projection planes were set asð = â2ð. We utilized the wallpaper patterns of Figure 1 toproduce spherical patterns shown in Figure 4. For beauty, allthe camera views are perpendicular to plane ððð and passthe origin, except for Figure 7(b), where the camera view aimsat the equator of ð2. To better understand the effect of centralprojection, Figure 7 demonstrates spherical patterns that areobserved from different perspectives.
Additional Points
Theartistic patterns created in this article have significant aes-thetic and economic value.We plan to produce somematerialobjects with the help of simulation and printing technologies.We produced Figures 1â7 in the VC++ 6.0 programmingenvironment with the aid of OpenGL, a powerful graphicssoftware package.
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(a) (b)
Figure 5: Colorful spherical wallpaper patterns with the ð6ð (a) and ððð (b) symmetry.
(a) (b)
Figure 6: Colorful spherical wallpaper patterns with the ð4 (a) and ð6ð (b) symmetry.
(a) (b)
Figure 7: Two spherical wallpaper patterns with the ð3ð symmetry. The camera view of (a) is perpendicular to plane ððð and passed theorigin, while (b) aims at equator.
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Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper.
Acknowledgments
The authors acknowledge Adobe and Microsoft for theirfriendly technical support. This work was supported by theNationalNatural Science Foundation of China (nos. 11461035,11761039, and 61363014), Young Scientist Training Program ofJiangxi Province (20153BCB23003), Science and TechnologyPlan Project of Jiangxi Provincial Education Department(no. GJJ160749), and Doctoral Startup Fund of JinggangshanUniversity (no. JZB1303).
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