# The Visualization of Spherical Patterns with Symmetries of the Wallpaper Group

Authors: Shihuan Liu; Ming Leng; Peichang Ouyang
Journal: Complexity
 (2018)

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## Abstract

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. The resulting wallpaper patterns are further projected by central projection onto the sphere. This creates the interesting spherical patterns that possess infinite symmetries in a finite space.


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## Body

## 1. Introduction

Wallpaper groups (or plane crystallographic groups) are mathematical classification of two-dimensional repetitive patterns. The first systematic proof that there were only 17 possible wallpaper patterns was carried out by Fedorov in 1891 [1] and later derived independently by Pólya 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.The art of M. C. Escher features the rigorous mathematical structure and elegant artistic charm, which might be the one and only in the history of art. After 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. They 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].The 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. This 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.The remainder of this paper is organized as follows. In Section2, we first introduce some basic conceptions and the lattices with respect to wallpaper groups. To create patterns with symmetries of the wallpaper group, we will explicitly construct invariant mappings associated with 17 wallpaper groups (the concrete mapping forms are summarized in Table 1) in Section 3. In Section 4, we describe how to create colorful wallpaper patterns. Finally, we show some spherical wallpaper patterns obtained by central projection in Section 5.Table 1
The concrete invariant mappingHfA,Bx forms associated with 17 wallpaper groups. In the fourth column, the subscripts A and B identify the lattice kind (Ls represents square lattice, while Ld represents diamond lattice) and wallpaper group type, respectively.
Wallpaper group
Point group
Extra symmetry set
Invariant mapping
p1
C
1
None
H
f
L
s
,
p
1
x
=
∑
g
∈
C
1
f
L
s
g
x
p2
C
2
None
H
f
L
s
,
p
2
x
=
∑
g
∈
C
2
f
L
s
g
x
p
m
D
1
σ
1
a
,
b
=
a
,
-
b
H
f
L
s
,
p
m
x
=
∑
g
∈
D
1
f
L
s
g
x
+
∑
g
∈
D
1
f
L
s
σ
1
g
x
p
m
m
D
2
σ
1
a
,
b
=
a
,
-
b, σ2a,b=-a,b
H
f
L
s
,
p
m
m
x
=
∑
g
∈
D
2
f
L
s
g
x
+
∑
i
=
1
2
∑
g
∈
D
2
f
L
s
σ
i
g
x
p
g
D
1
σ
1
a
,
b
=
π
+
a
,
-
b
H
f
L
s
,
p
g
x
=
∑
g
∈
D
1
f
L
s
g
x
+
∑
g
∈
D
1
f
L
s
σ
1
g
x
p
m
g
D
2
σ
1
a
,
b
=
π
+
a
,
-
b, σ2a,b=-a,b
H
f
L
s
,
p
m
g
x
=
∑
g
∈
D
2
f
L
s
g
x
+
∑
i
=
1
2
∑
g
∈
D
2
f
L
s
σ
i
g
x
C
m
D
1
σ
1
a
,
b
=
a
,
-
b, σ2a,b=π+a,π-b
H
f
L
s
,
c
m
x
=
∑
g
∈
D
1
f
L
s
g
x
+
∑
i
=
1
2
∑
g
∈
D
1
f
L
s
σ
i
g
x
C
m
m
D
2
σ
1
a
,
b
=
a
,
-
b, σ2a,b=π-a,π+b σ3a,b=π+a,π-b, σ4a,b=-a,b
H
f
L
s
,
c
m
m
x
=
∑
g
∈
D
2
f
L
s
g
x
+
∑
i
=
1
4
∑
g
∈
D
2
f
L
s
σ
i
g
x
p4
C
4
None
H
f
L
s
,
p
4
x
=
∑
g
∈
C
4
f
L
s
g
x
p4g
D
4
σ
1
a
,
b
=
π
+
a
,
-
b
H
f
L
s
,
p
4
g
x
=
∑
g
∈
D
4
f
L
s
g
x
+
∑
g
∈
D
4
f
L
s
σ
1
g
x
p4m
D
4
σ
1
a
,
b
=
a
,
-
b
H
f
L
s
,
p
4
m
x
=
∑
g
∈
D
4
f
L
s
g
x
+
∑
g
∈
D
4
f
L
d
s
σ
1
g
x
p
g
g
D
2
σ
1
a
,
b
=
π
+
a
,
π
-
b, σ2a,b=π-a,π+b
H
f
L
s
,
p
g
g
x
=
∑
g
∈
D
2
f
L
s
g
x
+
∑
i
=
1
2
∑
g
∈
D
2
f
L
s
σ
i
g
x
p3
C
3
None
H
f
L
d
,
p
3
x
=
∑
g
∈
C
3
f
L
d
g
x
p3m1
D
3
σ
1
a
,
b
=
-
a
,
b
H
f
L
d
,
p
3
m
1
x
=
∑
g
∈
D
3
f
L
d
g
x
+
∑
g
∈
D
3
f
L
d
σ
1
g
x
p31m
D
3
σ
1
a
,
b
=
a
,
-
b
H
f
L
d
,
p
31
m
x
=
∑
g
∈
D
3
f
L
d
g
x
+
∑
g
∈
D
3
f
L
d
σ
1
g
x
p6
C
6
None
H
f
L
d
,
p
6
x
=
∑
g
∈
C
6
f
L
d
g
x
p6m
D
6
σ
1
a
,
b
=
a
,
-
b
H
f
L
d
,
p
6
m
x
=
∑
g
∈
D
6
f
L
d
g
x
+
∑
g
∈
D
6
f
L
d
σ
1
g
x


## 2. The Lattice of Wallpaper Groups

In geometry and group theory, a lattice in 2-dimensional Euclidean planeR2 is essentially a subgroup of R2. Or, equivalently, for any basis vectors of R2, the subgroup of all linear combinations with integer coefficients of the vectors forms a lattice [12, 13]. Since a lattice is a finitely generated free abelian group, it is isomorphic to Z2 and fully spans the real vector space R2 [14]. A lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory, and group theory [15].In this section, we mainly introduce the lattices associated with wallpaper group. Firstly, we introduce some basic conceptions.Thesymmetry group of an object is the set of all isometries under which the object is invariant with composition as the group operation. Apoint group (sometimes called rosette group) is a group of isometries that keep at least one point fixed.Point groups inR2 come in two infinite families: dihedral group Dn which is the symmetry group of a regular polygon and cyclic group Cn that only comprises rotation transformations of Dn. Let Rn=cos⁡2π/n-sin⁡2π/nsin⁡2π/ncos⁡2π/n and T=-1001. Then their matrix group can be represented as Cn=Rni,i=1,2,3,…,n and Dn=Cn∪TRni,i=1,2,3,…,n.Awallpaper group is a type of topologically discrete group in R2 which contains two linearly independent translations. Alattice in R2 is the symmetry group of discrete translational symmetry in two independent directions. A tiling with this lattice of translational symmetry cannot have more but may have less symmetry than the lattice itself. Let L be a lattice in R2. A lattice L∗ is called thedual lattice of L if, ∀u∈L and ∀v∈L∗, the inner product u·v is an integer, where u and v are vectors in R2. Let M be a mapping from R2 to R2 and let G be a symmetry group in R2; M is called aninvariant mapping with respect to G if, ∀x∈R2 and ∀g∈G,Mx=Mgx.By the crystallographic restriction theorem, there are only 5 lattice types inR2 [16]. Although wallpaper groups have totally 17 types, their lattices can be simplified into two lattices: square and diamond lattices. For convenience, we require that the inner product of the mutual dual lattice of a wallpaper group be an integer multiple of 2π. Throughout the paper, for square lattice, we choose lattice Ls=1,0,0,1 with dual lattice Ls∗=2π1,0,2π0,1; for diamond lattice, we choose lattice Ld=1,0,1/2-1,3 with dual lattice Ld∗=2π/33,-1,2π0,-2/3.In this paper, we use standard crystallographic notations of wallpaper groups [16, 17]. Among 17 wallpaper groups, p1, p2, pm, pmm, pg, pmg, cm, cmm, p4, pgg, p4g, and p4m possess square lattice, while p3,p3m1,p31m,p6,andp6m possess diamond lattice.


## 3. Invariant Mapping with respect to Wallpaper Groups

In this section, we explicitly construct invariant mappings associated with wallpaper groups. To this end, we first prove the following lemma.Lemma 1.
Suppose thatfii=1,2,3,4 are sine or cosine functions, G is a wallpaper group with lattice L=A,B, L∗=A∗,B∗ is the dual lattice of L, and a and b are real numbers. Then mapping(1)fLx=af1∑v∈Lf2x·v+∑v∈Lx·vbf3∑v∈Lf4x·v+∑v∈Lx·v,∀x∈R2,is invariant with respect to L∗, or fLx has translation invariance of L∗; that is, (2)af1∑v∈Lf2u+x·v+∑v∈Lu+x·vbf3∑v∈Lf4u+x·v+∑v∈Lu+x·v=af1∑v∈Lf2x·v+∑v∈Lx·vbf3∑v∈Lf4x·v+∑v∈Lx·v=fLx,where u=mA∗+nB∗,m,n∈Z.Proof.
SinceL∗ is the dual lattice of L,∀v∈L, we have u·v=mA∗+nB∗·v=m(A∗·v)+n(B∗·v)=2kπ for certain k∈Z. Thus we get fi∑v∈Lfju+x·v+∑v∈Lu+x·v=fi{∑v∈Lfjx·v+∑v∈Lx·v}, since fi and fj are functions of period 2πi,j=1,2,3,4. Consequently, the mapping fLx constructed by fii=1,2,3,4 satisfies (2). This completes the proof.Essentially Lemma1 says that fLx is a double period mapping (of period 2π) along the independent translational directions of L∗.Theorem 2.
LetG be a finite group realized by 2×2 matrices acting on R2 by multiplication on the right and let f be an arbitrary mapping from R2 to R2. Then mapping(3)Hf,Gx=∑g∈Gfgx,x∈R2,is an invariant mapping with respect to G.Proof.
Forσ∈G, by closure of the group operation, we see that gσ runs through G as g does. Therefore we have(4)Hf,Gσx=∑g∈Gfσgx=∑g∗∈Gfg∗x=Hf,Gx,where g∗=σg∈G. This means that Hf,Gx is an invariant mapping with respect to G.Combining Lemma1 and Theorem 2, we immediately derive the following theorem.Theorem 3.
Letf in Theorem 2 have the form fL as in Lemma 1. Suppose that G is a cyclic group Cn or dihedral group Dn with lattice L; L∗ is the dual lattice associated with L. Assume that HfL,Gx is a mapping from R2 to R2 of the following form:(5)HfL,Gx=∑g∈GfLgx,x∈R2.Then HfL,Gx is an invariant mapping with respect to both G and L∗.Wallpaper groups possess globally translation symmetry along two independent directions as well as locally point group symmetry. For the wallpaper groups that only have symmetries of a certain point group, mappingHfL,Gx in Theorem 3 can be used to create wallpaper patterns well. However, except for the symmetries of a point group, some wallpaper groups may possess other symmetries. For example, except for symmetries of dihedral group D3, wallpaper group p31m still has a reflection along horizontal direction, say symmetry σ1a,b=a,-ba,b∈R2; besides D2 symmetry, wallpaper group pmm contains perpendicular reflections; that is, σ1a,b=a,-b and σ2a,b=-a,b.For a wallpaper groupw with extra symmetry set Δ=σ1,σ2,σ3,…,σn, based on mapping (5), we add proper terms so that the resulting mapping HfL,wx is also invariant with respect to w. This is summarized in Theorem 4.Theorem 4.
Letf in Theorem 2 have the form fL in Lemma 1. Suppose that w is a wallpaper group with symmetry group G and extra symmetry set Δ=σ1,σ2,σ3,…,σn. Assume that L is the lattice of w and L∗ is the dual lattice associated with L. Let HfL,wx be a mapping from R2 to R2 of the following form:(6)HfL,wx=∑g∈GfLgx+∑i=1n∑g∈G,σi∈ΔfLσigx,x∈R2.Then HfL,wx is an invariant mapping with respect to both w and L∗.We refer the reader to [7, 8, 17] for more detailed description about the extra symmetry set of wallpaper groups. By Theorems 3-4, we list the invariant mappings associated with 17 wallpaper groups in Table 1.


## 4. Colorful Wallpaper Patterns from Invariant Mappings

Invariant mapping method is a common approach used in creating symmetric patterns [18–23]. Color scheme is an algorithm that is used to determine the color of a point. Given a color scheme and domain D, by iterating invariant mapping HfL,wx, x∈D, one can determine the color of x. Coloring points in D pointwise, one can obtain a colorful pattern in D with symmetries of the wallpaper group w. Figures 1-2 are four wallpaper patterns obtained in this manner. These patterns were created by VC++ 6.0 on a PC (SVGA). In Algorithm 1, we provide the pseudocode so that the interested reader can create their own colorful wallpaper patterns.Algorithm 1:CreatingWallpaperPattern()// algorithm for creating patterns with the wallpaper symmetry.
BEGIN
start x = 0
end x = 6∗3.1415926
start y = 0
end y = 6∗3.1415926//Set pi = 3.1415926
step x = (end x – start x)/Xres//Xres is the resolution in X direction
step y = (end y – start y)/Yres//Yres is the resolution in Y direction
FOR i = 0 TO X res DO
FOR j = 0 TO Y res DO
x = start x +i∗ step x
y = start y + j∗ step y
FOR k = 1 TO MaxIter//MaxIter is the number of iterations, the default set is 100
/∗Given a invariant mapping HfL,wx associated with a wallpaper group w as iteration
function and initial point (x, y), function Iteration (x, y) iterates MaxIter times. The iterated
sequences are stored in the array Sequence∗/.
Sequence k = Iteration (x, y)
END FOR
/∗Inputting Sequence, the color scheme OrbitTrap outputs the color [r,g,b]∗/
[
r
,
g
,
b
] = OrbitTrap (Sequence)
Set color [r,g,b] to point (x, y)
END FOR
END FOR
ENDFigure 1
Colorful wallpaper patterns with thep4m (a) and p6m (b) symmetry.
(a)
(b)Figure 2
Colorful wallpaper patterns with thepmm (a) and p3m1 (b) symmetry.
(a)
(b)The color scheme used in this paper is called orbit trap algorithm [10]. We refer the reader to [10, 20] for more details about the algorithm (the algorithm is named as function OrbitTrap() in Algorithm 1). It has many parameters to adjust color, which could enhance the visual appeal of patterns effectively. Compared with the complex equivariant mapping constructed in [7–10], our invariant mappings possess not only simple form but also sensitive dynamical system property, which can be used to produce infinite wallpaper patterns easily. For example, Figures 1(a) and 2(b) were created by mappings HfLs,p4mx and HfLd,p3m1x, respectively, in which the specific mappings fLs and fLd were(7)fLs=2.12cos⁡⁡∑v∈Lscos⁡x·v+∑v∈Lsx·v1.03cos⁡∑v∈Lssin⁡x·v+∑v∈Lsx·v,(8)fLd=1.1cos⁡∑v∈Ldsin⁡x·v+∑v∈Ldx·v-0.52sin⁡∑v∈Ldcos⁡x·v+∑v∈Ldx·v.It seems that the deference between (7) and (8) is not very significant. However, by Table 1, mappings HfLs,p4mx and HfLd,p3m1x have 16 and 12 summation terms, respectively. The cumulative difference will be very obvious, which is enough to produce different style patterns.


## 5. Spherical Wallpaper Patterns by Central Projection

In this section, we introduce central projection to yield spherical patterns of the wallpaper symmetry.LetS2=a,b,c∈R3∣a2+b2+c2=1 be the unit sphere in R3, let Z=F be a projection plane, where F is a negative constant. Assume that P(a,b,c)∈R3; then P(a,b,c)∈R3 and P′(-a,-b,-c)∈R3 are a pair of antipodal points. For any point P(a,b,c)∈R3, there exist a unique line L through the origin (0,0,0) and P (and P′) which intersects the projection plane Z=F at point (α,β,F). Denote the projection by τ. By analytic geometry, it is easy to check that(9)αβF=τa,b,c=τ-a,-b,-c=abcFc.Because the projection point is at the center of S2, we call τ as central projection.The choice of the planeZ=F has a great influence on the spherical patterns. If plane Z=F is too close to coordinate plane XOY, the resulting spherical pattern only shows a few periods of the wallpaper pattern. However, if plane Z=F is too far away from coordinate plane XOY, the wallpaper pattern on the sphere may appear small so that we cannot identify symmetries of the wallpaper pattern well. Figure 3 illustrates the contrast effect of the setting of plane Z=F.Figure 3
Two spherical wallpaper patterns with thep4m symmetry, in which the projection plane was set as Z=-2π (a) and Z=-4π (b).
(a)
(b)Given a wallpaper pattern, by central projectionτ, we can map it onto the sphere S2 and obtain a corresponding spherical wallpaper pattern. We next explain how to implement it in more detail.Suppose thata,b,c∈R3 and HfL,wx is an invariant mapping compatible with the symmetry of wallpaper group w. First, by central projection τ, we obtain a corresponding point A/ca,A/cb,F on the projection plane Z=F. Second, let HfL,wx be iteration function and let xA/ca,A/cb be initial point; using the color scheme of orbit trap, we assign a color to point (A/c)a,(A/c)b,F. Finally, repeat the second step; by coloring unit sphere S2 pointwise, we obtain a spherical pattern of the wallpaper group w symmetry.Figures3–7 are ten patterns obtained by this manner. Except for Figure 3(b) in which the projection plane was set as Z=-4π, all the other projection planes were set as Z=-2π. We utilized the wallpaper patterns of Figure 1 to produce spherical patterns shown in Figure 4. For beauty, all the camera views are perpendicular to plane XOY and pass the origin, except for Figure 7(b), where the camera view aims at the equator of S2. To better understand the effect of central projection, Figure 7 demonstrates spherical patterns that are observed from different perspectives.Figure 4
Colorful spherical wallpaper patterns with thep4m (a) and p6m (b) symmetry.
(a)
(b)Figure 5
Colorful spherical wallpaper patterns with thep6m (a) and pmm (b) symmetry.
(a)
(b)Figure 6
Colorful spherical wallpaper patterns with thep4 (a) and p6m (b) symmetry.
(a)
(b)Figure 7
Two spherical wallpaper patterns with thep3m symmetry. The camera view of (a) is perpendicular to plane XOY and passed the origin, while (b) aims at equator.
(a)
(b)



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