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Tessellations with symmetries of the wallpaper groups and the modular group in the hyperbolic 3-space from dynamics
Computers & Graphics 25 (2001) 333}341
Chaos and Graphics
Tessellations with symmetries of the wallpaper groups and the modular group in the hyperbolic 3-space from dynamics K.W. Chung *, H.S.Y. Chan , B.N. Wang Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Department of Systems Ecology, Academia Sinica, Beijing, People's Republic of China
Abstract Automatic generation of colored patterns in the hyperbolic 3-space is considered from a dynamical system's point of view. The symmetries of various two-dimensional cross-sections of a tessellation are related to the 17 wallpaper groups and the modular group. The computer images generated in this paper reveal the exotic structures of the tessellations in the hyperbolic 3-space. A color scheme is described, which re#ects the rate of convergence of various orbits and, at the same time, enhances the artistic appeal of a generated image. This method can be used to create a great variety of aesthetic symmetrical patterns. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Dynamical systems; Hyperbolic 3-space; Wallpaper groups; Modular group
1. Introduction The studies of hyperbolic spaces appeared "rst in the work of Lobachevski in the "rst half of the 19th century [1]. The geometries underlying the hyperbolic spaces are of fundamental importance since they do not satisfy the axiom of parallels. The axiom of parallels states that, given a line and a point not on the line, there is exactly one parallel to the line through the point. This axiom holds in a Euclidean space. In the classical works, the hyperbolic 3-space was mainly studied by means of various concrete models such as the upper half-space model, the unit ball model, etc. [2]. In the upper half-space model, a point in the hyperbolic 3-space H is represented by a point u"(x, y, z)2 in the three-dimensional Euclidean space with z'0, i.e. H"u3Ru"(x, y, z)2 with z'0.
(1)
A hyperbolic line is represented by an upper half-circle or half-line which is orthogonal to the x}y plane. In the unit
* Corresponding author. Tel.: #852-2788-8671; fax: #8522788-8561. E-mail address: [email protected] (K.W. Chung).
ball model, a point in the hyperbolic 3-space is represented by a point inside an open unit sphere S. A hyperbolic line is represented by the segment in S of a circle or a straight line intersecting the boundary of S orthogonally. In the last 20 years, the subject has risen to particular importance due to the work of the mathematician Thurston on three-dimensional manifolds [3]. Tilings in the hyperbolic 3-space look rather complicated, but are well structured. For instance, beautiful images of tiling by dodecahedrals can be found at the web site of the Geometry Center at the University of Minnesota (http://www.geom.umn.edu/). Visual presentations of three-dimensional hyperbolic patterns reveal the exotic structure of this non-Euclidean geometry. In [4], the tessellations associated with the Picard group were considered. is the group of fractional linear transformation in H de"ned as w"z"(az#b)(cz#d )\, where w, z3H; 3 ; a, b, c, d3Z#Zi"Z and ad!bc"1. The multiplication on the right-hand side of the above equation obeys that of the quaternions. The parameters a, b, c and d are complex numbers of which both the real and imaginary parts are integers. Thus, the Picard group contains the well-known modular group as
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a subgroup. Various cross-sections of a tessellation associated with exhibit the symmetry of the modular group and also the P2 wallpaper symmetry. In [4], we considered the group action of on H and found that the symmetry of a horizontal crosssection of a tessellation was restricted to the P2 wallpaper symmetry only. In this paper, we consider the possible rotational group generators so that various horizontal cross-sections of the tessellations will exhibit the 17 wallpaper symmetries. A detailed description of the wallpaper groups is given in [5]. The approach considered in this paper is di!erent from that of [4] and the result is more general. In Section 2, we will describe the group generators for the creation of three-dimensional hyperbolic tessellations. The construction of equivariant mappings is considered in Section 3. A pseudo-code is given in Appendix A for the interested readers to create their own patterns.
The hyperbolic 3-space has various concrete models which all have certain advantages. We will consider in particular the upper half-space model. The upper halfspace gives a convenient model of the hyperbolic 3-space since its properties resemble closely the well-known upper half-plane which is a model for the plane hyperbolic geometry. Its relationship with the unit ball model will be outlined in Section 3. The tessellations considered in this paper will have the following symmetry properties: (i) an upper half-plane containing the x-axis exhibits the symmetry of the modular group; and (ii) a horizontal plane exhibits one of the 17 wallpaper symmetries. To exhibit the symmetry of the modular group in an upper half-plane containing the x-axis, the symmetric group under consideration should contain the two generators and given by
y " z
x#1 y z
1 2 x (3 y " 2 z 0
x
1 and y " x#y#z z
(3 ! 2 1 2
0
x
0
y
(3)
z
0
1
which is a counterclockwise rotation of /3 about the z-axis. Figs. 1 and 2 show the tessellations of at di!erent cross-sections. Consider next the fundamental region ; of H under . A fundamental region is a connected set whose transformed copies under cover the entire space without overlapping except at the boundaries. Let R; denote the boundaries of ;. For a P6 wallpaper pattern with periods 1 and (3 in the x- and y-directions, respectively, a fundamental region is given by 1 x) 2
2. Group generators
x
contains the generator
x and y) . (3
The following theorem describes a fundamental region ; of H under . Theorem 1. The region
1 x and u*1 ;" u"(x, y, z)23H x) , y) 2 (3
(4) is a fundamental region under . Proof. Assume that u is a point outside ;, i.e., u3H;. First of all we show that there exists 3 and v3; such that v" u. Let ; , ; and ; be three regions in
!x y
. (2)
z
When the actions of and are restricted to the x}z plane, these elements are the generators of the modular group as de"ned in [6]. To exhibit wallpaper symmetry in a horizontal plane, should contain other generators such as a rotation about the z-axis, a re#ection or a glide re#ection about a vertical plane. In the subsequent investigation, we will consider the P6 symmetry as an illustration. That is, a horizontal plane will exhibit a sixfold rotational symmetry about the origin. Thus, besides the two generators and de"ned in (2), also
Fig. 1. Upper half-space: x)0.92, y)1.2, z'0. The horizontal cross-sections are at z"1, 0.5 and 0.25.
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3. Colored tessellations from dynamics In this section, we investigate the construction of dynamical systems whose orbits exhibit the symmetry of and discuss the automatic generation of the associated colored tessellations. 3.1. Equivariancy of a dynamical system We restrict our consideration to iterative mappings F in H, i.e.
Fig. 2. Upper half-space: x#y#z)1, z'0. The vertical cross-sections are at x"0.1 and 0.4.
u "F(u ) where u 3H for i"0, 1, 2,2 . (5) G> G G An orbit of F is de"ned as the iterated sequence of points u , u , u ,2. As is shown in [6], the necessary and su$ cient condition for the orbits of F to exhibit the symmetry of a group is that F commutes with (or is equivariant), i.e. F " F for all 3.
H de"ned as, respectively,
1 ; " u"(x, y, z)23H x) 2
x and y) , (3
if F 50, (F , F , !F )2 if F (0. R In fact, the head operator ` a commutes with , and de"ned in (2) and (3), respectively. It follows that FK (u) is also a -equivariant function. As a result, it can be assumed that a -equivariant function always maps H to itself. FK (u)"
and
Furthermore, it is su$cient to check only whether the generators of satisfy (6). In general, F(u) may not lie in H. If F(u)"(F , F , F )2 where F (i"1, 2, 3) are scalar functions of x, y and z, we de"ne
; "u"(x, y, z)23H x)
(6)
x ; " u"(x, y, z)23H y) . (3 We note that ;L; "; ; . To determine , u is "rst translated in the x-direction to u 3; by as u"Lu"(x!W x# X, y, z)2, where !n"W x#X is the largest integer smaller than or equal to x#. If u ,; , then u is rotated to u 3; by . If u ,; , then u is translated in the x-direction again by to a point inside ; . Each time when a hori zontal translation is applied, the resulting point gets closer to the z-axis. Eventually, the image u will fall into ; . If u *1, then u 3;; otherwise u is transformed by . u will be transformed again by translations and rotations into ; . Each time when is applied, the resulting point will get higher and, eventually, the "nal image v will fall into ;. In this process, only the group elements , and are used in bringing any point u3H into ;. Therefore, for any point u outside ;, there exists 3 such that v" u3;. Furthermore, v is unique if it is inside ; and may have two values if it is on R;. In fact, under the generators , and , a point in ;R; is transformed to another point outside ;. Therefore, two points in ;R; cannot be related by any elements of . This completes the proof. 䊐
F(u)
3.2. Construction of -equivariant functions Let J be the part of the upper hemisphere in R;, i.e.
1 x J" u"(x, y, z)23H x) , y) and u"1 2 (3
(7) and K be a set of four points in J de"ned as
K" (0, 0, 1)2,
1 (3 2 , , 0, 2 2
1 1 2 2 , , ,( 2 2(3 3
1 1 2 2 ,! ,( . 2 3 2(3
The following theorem describes a basic property of K for a -equivariant mapping. Theorem 2. Let F be a -equivariant mapping on H. Then, the points in K are all xxed points for F.
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Proof. Consider the point u"(1/2, 1/2(3, (2/3)23K. Since u"u, we have, from the equivariant property, that F(u)"F(u)"F(u). Let F(u)"(F , F , F )2. From the above equation,
1 (3 ! F ! F #1 2 2
F (3 1 F " F ! F 2 2 F F
1 NF " 2
1 and F " . 2(3
To simplify the construction, we assume that F(u)"1. Eq. (10) becomes, f (u)"!f (u), (11a) f (u)"f (u), (11b) f (u)"f (u), (11c) If f (u) is expressed as a linear combination of g(Gu) and g(Gu) (i"0, 1,2, 5) where g(u) is an arbitrary scalar function of x and y, we obtain from (9) and (11a) that f (u)"a[g(u)!g(u)!g(u)#g(u)] #b[g(u)!g(u)!g(u)#g(u)] #(b!a)[g(u)!g(u)!g(u)#g(u)],
Furthermore, as u"u, we obtain F(u)"1. Therefore, u"(1/2, 1/2(3, (2/3)2 is a "xed point for F. Similarly, the other points in K can be shown to be "xed points for F from the following facts: (i) if u"(0, 0, 1)2, then u"u"u; (ii) if u"(1/2, 0, (3/2)2, then u"u"u; and (iii) if u"(1/2, !1/2(3, (2/3)2, then u"\u"u. 䊐 If a horizontal plane is tiled by regular hexagons, it follows from Theorem 2 that, for each hexagon, the center, its vertices and the midpoints of the edges are all "xed points for a -equivariant mapping. We "rst consider explicitly the construction of F on J. For u"(x, y, (1!x!y)23J, we write F(u)" u#f (u) where f (u)"( f , f , f )2 and f (i"1, 2, 3) are G scalar functions of x and y. From the equivariant condition that F(u)"F(u), we have 1 (3 f (u)" f (u)! f (u), 2 2
(8a)
(3 1 f (u)# f (u), f (u)" 2 2
(8b)
(12a) where a, b3R. f can be obtained from (8a) as 1 2 f (u)" f (u)! f (u). (3 (3 Since u"F(u)"1, we have (x#f )#(y#f )#((1!x!y#f )"1 Nf "(1!(x#f )!(y#f )!(1!x!y. (12c) From the equivariant condition that F(u)"F(u) and (12a), the arbitrary function g(u) should be periodic in the x- and y-directions with periods 1 and (3, respectively, and satis"es the additional condition
g(u)"g u#
1 (3 2 , ,0 . 2 2
(8c)
f (u)!f (u)#f (u)"0 for i"1, 2. (9) G G G From the equivariant condition that F(u)"F(u), we have
(13)
We next consider the construction of F on ;. Without loss of generality, a mapping F : ;PH can be expressed as F(u)"u#f (u)#H(u),
f (u)"f (u). It follows from (8a) and (8b) that
(12b)
(14)
where f (u) is determined by (12) and (13) and H : ;PR should satisfy the following conditions due to equivariancy: (i) H(u)"H(u), (ii) H(u)"H(u), and (iii) H(u)"0 for u3J.
x#f (u) , f (u)"x! F(u)
To satisfy the above conditions, H can have the following general form:
y#f (u) , f (u)"!y# F(u)
H(u)"l(u)c[h(u)!h(u)]#d[h(u)!h(u)]
z#f (u) f (u)"!z# F(u)
#(d!c)[h(u)!h(u)], for z"(1!x!y.
(10)
(15)
where c, d3R; l(u)"0 for u3J (i.e. u"1); h(u) is a periodic function in both x- and y-directions with
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Fig. 3. P6 wallpaper patterns in the upper half-space: x)0.92, y)1.2. (a) 0(z)1. (b) 0(z)0.5.
Fig. 4. P6 wallpaper patterns in the upper half-space: x#y)1. (a) 0(z)0.81. (b) 0(z)0.41.
periods 1 and (3, respectively, and satis"es the condition
3.3. Determination of color
1 (3 2 h(u)"h u# , ,0 . 2 2
(16)
l(u) may be any simple function such as l(u)"1}1/u. For u3H;, F(u) can be obtained from the equivariancy property as F(u)" }F( u) where u3;. The construction of a -equivariant mapping F can be summarized in the following theorem. Theorem 3. Let be the symmetric group consisting of the generators dexned in (2), (3) and F : HPH be a mapping dexned by
F(u)"
u#f (u)#H(u) for u"(x, y, z)23;, \F( u)
for u,;, but u3;,
where f (u) and H(u) are given, respectively, in (12), (13), (15) and (16). Then, F(u) is a -equivariant mapping in H.
We consider the hyperbolic distance between consecutive points of the orbit u 3Hn*0. Let L u"(x, y, z)2 and u "(x , y , z )2, the hyperbolic distance d(u, u ) is given by [2] cosh d(u, u )" (u,u ), where (x!x )#(y!y )#z#z (u,u )" . 2zz
For an orbit with initial point u , we de"ne G(u ) as G(u )" (FG\(u ), FG(u )). For a given positive integer i and a constant c'0, we compute G (u )"W c G(u X ) which is used to determine the color at u . An image created by means of this color scheme re#ects the convergence rate of various orbits after a speci"c number of iterations. Furthermore, as the
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Fig. 7. P3 wallpaper patterns in the upper half-space. The cross-section is the curved surface of a right circular cone with the vertex at (0, 0, 1) and base radius 1.
Fig. 5. P6 wallpaper patterns in the upper half-space: x#y#z)1, z'0 and, for (b), x)0.2.
Fig. 6. P4 wallpaper patterns in the upper half-space: x#y#z)1, z'0. The vertical cross-section is at x"!0.2.
Fig. 8. P3 wallpaper patterns in the PoincareH model: x#y#z(1, x)!0.8 and z)0.
K.W. Chung et al. / Computers & Graphics 25 (2001) 333}341
function is invariant under , i.e. (u, u )" ( u, u ) for 3, the symmetrical points u and u will have the same color. It follows that the generated image also exhibits the symmetry of . This coloring algorithm is a modi"cation of the popular convergence algorithm. Its outstanding features in creating aesthetic patterns are described in [4,7]. Computer-generated images with various wallpaper symmetries including P3, P4 and P6 are shown in Figs. 3}8 at di!erent cross-sections. Tessellation in the hyperbolic 3-space can also be expressed elegantly in the PoincareH model that contains all the points of the open unit sphere S"u3R u"(x, y, z)2 with u(1. The upper half-space is related to the PoincareH model by the conformal mapping as 1 v" (u)" x#y#(z#1)
2x 2y
,
339
4. Conclusion In the hyperbolic 3-space H, we investigate the symmetry associated with the group which contains the three generators , and de"ned, respectively, in (2) and (3). Regarding the tessellations associated with , a vertical cross-section containing the x-axis exhibits the symmetry of the modular group while a horizontal crosssection exhibits the P6 wallpaper symmetry. With a slight modi"cation of to allow for other rotational symmetries and the inclusion of re#ection or glide re#ection, a horizontal cross-section will exhibit one of the 17 wallpaper symmetries. -equivariant mappings are constructed for the computer generation of aesthetic patterns with such symmetries. A modi"ed convergence color scheme is described to enhance the artistic appeal of the generated images.
Acknowledgements
x#y#z!1
where u3H and v3S. Fig. 8 shows the generated patterns in the PoincareH model.
Bennan Wang was supported by the SRG Grant 7000937 for working at the Department of Mathematics, City University of Hong Kong.
Appendix A. Algorithm for generating colored patterns in the hyperbolic 3-space ALGORITHM Title
3DhyperTessP6(i, j, Code, MaxIter) Display tessellation with p6 wallpaper symmetry and the symmetry of the modular group in the upper half-space
Arguments
i, j Code MaxIter X, >, Z, Z , X ,> ,Z , X ,> ,Z Xres, >res Start}x, End}x, Step}x, Start}y, End}y, Step}y,
Globals
Variables
Integers"grid point (i, j) of screen Integer"color code Integer"Maximum iteration
Real Numbers Resolution in X, > directions Real Numbers
BEGIN Z "0.5 MaxIter"5 Start}x"!2 End}x"2 Start}y"!1.5 End}y"1.5 Step}x " : (End}x!Start}x)/Xres Step}y " : (End}y!Start}y)/>res FOR i:"0 TO Xres DO X:"Start}x#i*Step}x FOR j " : 0 TO >res DO >" : Start}y#j*Step}y Z" : Z Rem: From grid of screen to a point in the upper half-space. MapToFund(X, >, Z) Rem: The output P(X, >, Z) is in the fundamental region. X " : X > " : > Z " : Z
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FOR k:"1 TO MaxIter DO X " : X > " : > Z " : Z IterFunc(X , > , Z ) Rem: Outputs are dX, d> and dZ X " : X #dX > " : > #d> Z " : Z #dZ END FOR Dist:"(dX#d>#Z #Z )/(2Z Z ) Rem: cosh[d(p, p1)] is used as the distance function. Code:"Color}Code(Dist) Rem: Code function generates a color code. In our examples, Code"Inc* (Dist-1) is used, where Inc is an integer constant, e.g. Inc"100 or so. SetColorPoint(i, j, Code) END FOR END FOR END ALGORITHM Title
MapToFund(x, y, z) Routine of mapping a point in the upper half-space onto The fundamental region
Arugments Globals
x, y, z X, >, Z
P(x, y, z) in the upper half-space P(X, >, Z) in the fundamental region
BEGIN d" : x*x#y*y#z*z WHILE (x'0.5 OR arctan(y/x)'/6 OR d(1) DO IF (x'0.5) THEN x " : x!1 ELSE IF (arctan(y/x)'/6) OR (arctan(y/x)(!/6) THEN Rotate(1, x, y) ELSE IF (d(1) THEN x" : !x/d y" : y/d z" : z/d d" : x*x#y*y#z*z END IF END WHILE END DEFINE}MAPPING Rotate(k, x, y) Title Routine of rotation x:"cos(2k/6)*x!sin(2k/6)*y y:"sin(2k/6)*x#cos(2k/6)*y DEFINE}FUNCTION Function}f 1(x, y) Title Routine rotating a point Global f1 Real number, OUTPUT Function g(a1, a2, a3, a4, a5, a6, u, v)"(a1#a2*sin(2u)#a3*cos(2v))* (a4#a5*cos(2u)#a6*sin(2v)) BEGIN g1"g(a1, a2, a3, a4, a5, a6, x, y) g2"g(a1, a2, a3, a4, a5, a6, !x, !y) g3"g(a1, a2, a3, a4, a5, a6, !x, y) g4"g(a1, a2, a3, a4, a5, a6, x, !y) f 1:"c1*(g1}g2}g3#g4)
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Rotate(1, x, y) g1"g(a1, a2, a3, a4, a5, a6, x, y) g2"g(a1, a2, a3, a4, a5, a6, !x, !y) Rotate(1, !x, y) g3"g(a1, a2, a3, a4, a5, a6, x, y) g4"g(a1, a2, a3, a4, a5, a6, !x, !y) f1 " : f1#c2*(g1!g2!g3#g4) Rotate(2, x, y) g1"g(a1, a2, a3, a4, a5, a6, x, y) g2"g(a1, a2, a3, a4, a5, a6, !x, !y) Rotate(2, !x, y) g3"g(a1, a2, a3, a4, a5, a6, x, y) g4"g(a1, a2, a3, a4, a5, a6, !x, !y) f1:"f1#(c2!c1)*(g1!g2!g3#g4) END ALGORITHM Title Arguments Globals
x, y, z h1, h2, h3, ¸z dX, d>, dZ
IterFunc(x, y, z) Routine for Iteration Input Real numbers Output
BEGIN Function}f1(x, y) Rem: The OUTPUT is f 1. dX " : f1 Rotate(1, x, y) Function}f1(x, y) Rem: The OUTPUT is f 1. d> " : 2*(0.5*dX}f1)/SQRT(3) dZ " : SQRT[1}(x#dX)*(x#dX)!(y#d>)*(y#d>)]!SQRT(1!x*x!y*y) ¸z " : x*x#y*y#z*z ¸z " : 1!1/¸z dX " : dX#¸z*h1 d> " : d>#¸z*h2 dZ " : dZ#¸z*h3 END Example: The following parameters are used for generating the P6 wallpaper patterns in Figs. 3}5. a1"1.0, a2"1.0, a3"1.0 a4"1.0, a5"1.0, a6"1.0 c1"0.01, c2"0.005 h1"0.1, h2"0, h3"0
References [1] Trudeau RJ. The non-Euclidean revolution. Boston: BirkhaK user, 1987. [2] Elstrodt J, Grunewald F, Mennicke J. Groups acting on hyperbolic space. Heidelberg: Springer, 1998. [3] Thurston WP. Three-dimensional geometry and topology. Princeton: Princeton University Press, 1997. [4] Chung KW, Chan HSY, Wang BN. Tessellations in three-dimensional hyperbolic space from dynamics
and the quaternions. Chaos, Solitons and Fractals, in press. [5] Armstrong MA. Groups and symmetry, New York: Springer, 1988. [6] Chung KW, Chan HSY, Wang BN. Tessellations with the modular group from dynamics. Computers & Graphics 1997;21(4):523}34. [7] Chung KW, Chan HSY, Wang BN. Spiral tilings with color symmetry from dynamics. Computers & Graphics 1999;23(3):439}48.
Chaos and Graphics
Tessellations with symmetries of the wallpaper groups and the modular group in the hyperbolic 3-space from dynamics K.W. Chung *, H.S.Y. Chan , B.N. Wang Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Department of Systems Ecology, Academia Sinica, Beijing, People's Republic of China
Abstract Automatic generation of colored patterns in the hyperbolic 3-space is considered from a dynamical system's point of view. The symmetries of various two-dimensional cross-sections of a tessellation are related to the 17 wallpaper groups and the modular group. The computer images generated in this paper reveal the exotic structures of the tessellations in the hyperbolic 3-space. A color scheme is described, which re#ects the rate of convergence of various orbits and, at the same time, enhances the artistic appeal of a generated image. This method can be used to create a great variety of aesthetic symmetrical patterns. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Dynamical systems; Hyperbolic 3-space; Wallpaper groups; Modular group
1. Introduction The studies of hyperbolic spaces appeared "rst in the work of Lobachevski in the "rst half of the 19th century [1]. The geometries underlying the hyperbolic spaces are of fundamental importance since they do not satisfy the axiom of parallels. The axiom of parallels states that, given a line and a point not on the line, there is exactly one parallel to the line through the point. This axiom holds in a Euclidean space. In the classical works, the hyperbolic 3-space was mainly studied by means of various concrete models such as the upper half-space model, the unit ball model, etc. [2]. In the upper half-space model, a point in the hyperbolic 3-space H is represented by a point u"(x, y, z)2 in the three-dimensional Euclidean space with z'0, i.e. H"u3Ru"(x, y, z)2 with z'0.
(1)
A hyperbolic line is represented by an upper half-circle or half-line which is orthogonal to the x}y plane. In the unit
* Corresponding author. Tel.: #852-2788-8671; fax: #8522788-8561. E-mail address: [email protected] (K.W. Chung).
ball model, a point in the hyperbolic 3-space is represented by a point inside an open unit sphere S. A hyperbolic line is represented by the segment in S of a circle or a straight line intersecting the boundary of S orthogonally. In the last 20 years, the subject has risen to particular importance due to the work of the mathematician Thurston on three-dimensional manifolds [3]. Tilings in the hyperbolic 3-space look rather complicated, but are well structured. For instance, beautiful images of tiling by dodecahedrals can be found at the web site of the Geometry Center at the University of Minnesota (http://www.geom.umn.edu/). Visual presentations of three-dimensional hyperbolic patterns reveal the exotic structure of this non-Euclidean geometry. In [4], the tessellations associated with the Picard group were considered. is the group of fractional linear transformation in H de"ned as w"z"(az#b)(cz#d )\, where w, z3H; 3 ; a, b, c, d3Z#Zi"Z and ad!bc"1. The multiplication on the right-hand side of the above equation obeys that of the quaternions. The parameters a, b, c and d are complex numbers of which both the real and imaginary parts are integers. Thus, the Picard group contains the well-known modular group as
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a subgroup. Various cross-sections of a tessellation associated with exhibit the symmetry of the modular group and also the P2 wallpaper symmetry. In [4], we considered the group action of on H and found that the symmetry of a horizontal crosssection of a tessellation was restricted to the P2 wallpaper symmetry only. In this paper, we consider the possible rotational group generators so that various horizontal cross-sections of the tessellations will exhibit the 17 wallpaper symmetries. A detailed description of the wallpaper groups is given in [5]. The approach considered in this paper is di!erent from that of [4] and the result is more general. In Section 2, we will describe the group generators for the creation of three-dimensional hyperbolic tessellations. The construction of equivariant mappings is considered in Section 3. A pseudo-code is given in Appendix A for the interested readers to create their own patterns.
The hyperbolic 3-space has various concrete models which all have certain advantages. We will consider in particular the upper half-space model. The upper halfspace gives a convenient model of the hyperbolic 3-space since its properties resemble closely the well-known upper half-plane which is a model for the plane hyperbolic geometry. Its relationship with the unit ball model will be outlined in Section 3. The tessellations considered in this paper will have the following symmetry properties: (i) an upper half-plane containing the x-axis exhibits the symmetry of the modular group; and (ii) a horizontal plane exhibits one of the 17 wallpaper symmetries. To exhibit the symmetry of the modular group in an upper half-plane containing the x-axis, the symmetric group under consideration should contain the two generators and given by
y " z
x#1 y z
1 2 x (3 y " 2 z 0
x
1 and y " x#y#z z
(3 ! 2 1 2
0
x
0
y
(3)
z
0
1
which is a counterclockwise rotation of /3 about the z-axis. Figs. 1 and 2 show the tessellations of at di!erent cross-sections. Consider next the fundamental region ; of H under . A fundamental region is a connected set whose transformed copies under cover the entire space without overlapping except at the boundaries. Let R; denote the boundaries of ;. For a P6 wallpaper pattern with periods 1 and (3 in the x- and y-directions, respectively, a fundamental region is given by 1 x) 2
2. Group generators
x
contains the generator
x and y) . (3
The following theorem describes a fundamental region ; of H under . Theorem 1. The region
1 x and u*1 ;" u"(x, y, z)23H x) , y) 2 (3
(4) is a fundamental region under . Proof. Assume that u is a point outside ;, i.e., u3H;. First of all we show that there exists 3 and v3; such that v" u. Let ; , ; and ; be three regions in
!x y
. (2)
z
When the actions of and are restricted to the x}z plane, these elements are the generators of the modular group as de"ned in [6]. To exhibit wallpaper symmetry in a horizontal plane, should contain other generators such as a rotation about the z-axis, a re#ection or a glide re#ection about a vertical plane. In the subsequent investigation, we will consider the P6 symmetry as an illustration. That is, a horizontal plane will exhibit a sixfold rotational symmetry about the origin. Thus, besides the two generators and de"ned in (2), also
Fig. 1. Upper half-space: x)0.92, y)1.2, z'0. The horizontal cross-sections are at z"1, 0.5 and 0.25.
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3. Colored tessellations from dynamics In this section, we investigate the construction of dynamical systems whose orbits exhibit the symmetry of and discuss the automatic generation of the associated colored tessellations. 3.1. Equivariancy of a dynamical system We restrict our consideration to iterative mappings F in H, i.e.
Fig. 2. Upper half-space: x#y#z)1, z'0. The vertical cross-sections are at x"0.1 and 0.4.
u "F(u ) where u 3H for i"0, 1, 2,2 . (5) G> G G An orbit of F is de"ned as the iterated sequence of points u , u , u ,2. As is shown in [6], the necessary and su$ cient condition for the orbits of F to exhibit the symmetry of a group is that F commutes with (or is equivariant), i.e. F " F for all 3.
H de"ned as, respectively,
1 ; " u"(x, y, z)23H x) 2
x and y) , (3
if F 50, (F , F , !F )2 if F (0. R In fact, the head operator ` a commutes with , and de"ned in (2) and (3), respectively. It follows that FK (u) is also a -equivariant function. As a result, it can be assumed that a -equivariant function always maps H to itself. FK (u)"
and
Furthermore, it is su$cient to check only whether the generators of satisfy (6). In general, F(u) may not lie in H. If F(u)"(F , F , F )2 where F (i"1, 2, 3) are scalar functions of x, y and z, we de"ne
; "u"(x, y, z)23H x)
(6)
x ; " u"(x, y, z)23H y) . (3 We note that ;L; "; ; . To determine , u is "rst translated in the x-direction to u 3; by as u"Lu"(x!W x# X, y, z)2, where !n"W x#X is the largest integer smaller than or equal to x#. If u ,; , then u is rotated to u 3; by . If u ,; , then u is translated in the x-direction again by to a point inside ; . Each time when a hori zontal translation is applied, the resulting point gets closer to the z-axis. Eventually, the image u will fall into ; . If u *1, then u 3;; otherwise u is transformed by . u will be transformed again by translations and rotations into ; . Each time when is applied, the resulting point will get higher and, eventually, the "nal image v will fall into ;. In this process, only the group elements , and are used in bringing any point u3H into ;. Therefore, for any point u outside ;, there exists 3 such that v" u3;. Furthermore, v is unique if it is inside ; and may have two values if it is on R;. In fact, under the generators , and , a point in ;R; is transformed to another point outside ;. Therefore, two points in ;R; cannot be related by any elements of . This completes the proof. 䊐
F(u)
3.2. Construction of -equivariant functions Let J be the part of the upper hemisphere in R;, i.e.
1 x J" u"(x, y, z)23H x) , y) and u"1 2 (3
(7) and K be a set of four points in J de"ned as
K" (0, 0, 1)2,
1 (3 2 , , 0, 2 2
1 1 2 2 , , ,( 2 2(3 3
1 1 2 2 ,! ,( . 2 3 2(3
The following theorem describes a basic property of K for a -equivariant mapping. Theorem 2. Let F be a -equivariant mapping on H. Then, the points in K are all xxed points for F.
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Proof. Consider the point u"(1/2, 1/2(3, (2/3)23K. Since u"u, we have, from the equivariant property, that F(u)"F(u)"F(u). Let F(u)"(F , F , F )2. From the above equation,
1 (3 ! F ! F #1 2 2
F (3 1 F " F ! F 2 2 F F
1 NF " 2
1 and F " . 2(3
To simplify the construction, we assume that F(u)"1. Eq. (10) becomes, f (u)"!f (u), (11a) f (u)"f (u), (11b) f (u)"f (u), (11c) If f (u) is expressed as a linear combination of g(Gu) and g(Gu) (i"0, 1,2, 5) where g(u) is an arbitrary scalar function of x and y, we obtain from (9) and (11a) that f (u)"a[g(u)!g(u)!g(u)#g(u)] #b[g(u)!g(u)!g(u)#g(u)] #(b!a)[g(u)!g(u)!g(u)#g(u)],
Furthermore, as u"u, we obtain F(u)"1. Therefore, u"(1/2, 1/2(3, (2/3)2 is a "xed point for F. Similarly, the other points in K can be shown to be "xed points for F from the following facts: (i) if u"(0, 0, 1)2, then u"u"u; (ii) if u"(1/2, 0, (3/2)2, then u"u"u; and (iii) if u"(1/2, !1/2(3, (2/3)2, then u"\u"u. 䊐 If a horizontal plane is tiled by regular hexagons, it follows from Theorem 2 that, for each hexagon, the center, its vertices and the midpoints of the edges are all "xed points for a -equivariant mapping. We "rst consider explicitly the construction of F on J. For u"(x, y, (1!x!y)23J, we write F(u)" u#f (u) where f (u)"( f , f , f )2 and f (i"1, 2, 3) are G scalar functions of x and y. From the equivariant condition that F(u)"F(u), we have 1 (3 f (u)" f (u)! f (u), 2 2
(8a)
(3 1 f (u)# f (u), f (u)" 2 2
(8b)
(12a) where a, b3R. f can be obtained from (8a) as 1 2 f (u)" f (u)! f (u). (3 (3 Since u"F(u)"1, we have (x#f )#(y#f )#((1!x!y#f )"1 Nf "(1!(x#f )!(y#f )!(1!x!y. (12c) From the equivariant condition that F(u)"F(u) and (12a), the arbitrary function g(u) should be periodic in the x- and y-directions with periods 1 and (3, respectively, and satis"es the additional condition
g(u)"g u#
1 (3 2 , ,0 . 2 2
(8c)
f (u)!f (u)#f (u)"0 for i"1, 2. (9) G G G From the equivariant condition that F(u)"F(u), we have
(13)
We next consider the construction of F on ;. Without loss of generality, a mapping F : ;PH can be expressed as F(u)"u#f (u)#H(u),
f (u)"f (u). It follows from (8a) and (8b) that
(12b)
(14)
where f (u) is determined by (12) and (13) and H : ;PR should satisfy the following conditions due to equivariancy: (i) H(u)"H(u), (ii) H(u)"H(u), and (iii) H(u)"0 for u3J.
x#f (u) , f (u)"x! F(u)
To satisfy the above conditions, H can have the following general form:
y#f (u) , f (u)"!y# F(u)
H(u)"l(u)c[h(u)!h(u)]#d[h(u)!h(u)]
z#f (u) f (u)"!z# F(u)
#(d!c)[h(u)!h(u)], for z"(1!x!y.
(10)
(15)
where c, d3R; l(u)"0 for u3J (i.e. u"1); h(u) is a periodic function in both x- and y-directions with
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Fig. 3. P6 wallpaper patterns in the upper half-space: x)0.92, y)1.2. (a) 0(z)1. (b) 0(z)0.5.
Fig. 4. P6 wallpaper patterns in the upper half-space: x#y)1. (a) 0(z)0.81. (b) 0(z)0.41.
periods 1 and (3, respectively, and satis"es the condition
3.3. Determination of color
1 (3 2 h(u)"h u# , ,0 . 2 2
(16)
l(u) may be any simple function such as l(u)"1}1/u. For u3H;, F(u) can be obtained from the equivariancy property as F(u)" }F( u) where u3;. The construction of a -equivariant mapping F can be summarized in the following theorem. Theorem 3. Let be the symmetric group consisting of the generators dexned in (2), (3) and F : HPH be a mapping dexned by
F(u)"
u#f (u)#H(u) for u"(x, y, z)23;, \F( u)
for u,;, but u3;,
where f (u) and H(u) are given, respectively, in (12), (13), (15) and (16). Then, F(u) is a -equivariant mapping in H.
We consider the hyperbolic distance between consecutive points of the orbit u 3Hn*0. Let L u"(x, y, z)2 and u "(x , y , z )2, the hyperbolic distance d(u, u ) is given by [2] cosh d(u, u )" (u,u ), where (x!x )#(y!y )#z#z (u,u )" . 2zz
For an orbit with initial point u , we de"ne G(u ) as G(u )" (FG\(u ), FG(u )). For a given positive integer i and a constant c'0, we compute G (u )"W c G(u X ) which is used to determine the color at u . An image created by means of this color scheme re#ects the convergence rate of various orbits after a speci"c number of iterations. Furthermore, as the
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Fig. 7. P3 wallpaper patterns in the upper half-space. The cross-section is the curved surface of a right circular cone with the vertex at (0, 0, 1) and base radius 1.
Fig. 5. P6 wallpaper patterns in the upper half-space: x#y#z)1, z'0 and, for (b), x)0.2.
Fig. 6. P4 wallpaper patterns in the upper half-space: x#y#z)1, z'0. The vertical cross-section is at x"!0.2.
Fig. 8. P3 wallpaper patterns in the PoincareH model: x#y#z(1, x)!0.8 and z)0.
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function is invariant under , i.e. (u, u )" ( u, u ) for 3, the symmetrical points u and u will have the same color. It follows that the generated image also exhibits the symmetry of . This coloring algorithm is a modi"cation of the popular convergence algorithm. Its outstanding features in creating aesthetic patterns are described in [4,7]. Computer-generated images with various wallpaper symmetries including P3, P4 and P6 are shown in Figs. 3}8 at di!erent cross-sections. Tessellation in the hyperbolic 3-space can also be expressed elegantly in the PoincareH model that contains all the points of the open unit sphere S"u3R u"(x, y, z)2 with u(1. The upper half-space is related to the PoincareH model by the conformal mapping as 1 v" (u)" x#y#(z#1)
2x 2y
,
339
4. Conclusion In the hyperbolic 3-space H, we investigate the symmetry associated with the group which contains the three generators , and de"ned, respectively, in (2) and (3). Regarding the tessellations associated with , a vertical cross-section containing the x-axis exhibits the symmetry of the modular group while a horizontal crosssection exhibits the P6 wallpaper symmetry. With a slight modi"cation of to allow for other rotational symmetries and the inclusion of re#ection or glide re#ection, a horizontal cross-section will exhibit one of the 17 wallpaper symmetries. -equivariant mappings are constructed for the computer generation of aesthetic patterns with such symmetries. A modi"ed convergence color scheme is described to enhance the artistic appeal of the generated images.
Acknowledgements
x#y#z!1
where u3H and v3S. Fig. 8 shows the generated patterns in the PoincareH model.
Bennan Wang was supported by the SRG Grant 7000937 for working at the Department of Mathematics, City University of Hong Kong.
Appendix A. Algorithm for generating colored patterns in the hyperbolic 3-space ALGORITHM Title
3DhyperTessP6(i, j, Code, MaxIter) Display tessellation with p6 wallpaper symmetry and the symmetry of the modular group in the upper half-space
Arguments
i, j Code MaxIter X, >, Z, Z , X ,> ,Z , X ,> ,Z Xres, >res Start}x, End}x, Step}x, Start}y, End}y, Step}y,
Globals
Variables
Integers"grid point (i, j) of screen Integer"color code Integer"Maximum iteration
Real Numbers Resolution in X, > directions Real Numbers
BEGIN Z "0.5 MaxIter"5 Start}x"!2 End}x"2 Start}y"!1.5 End}y"1.5 Step}x " : (End}x!Start}x)/Xres Step}y " : (End}y!Start}y)/>res FOR i:"0 TO Xres DO X:"Start}x#i*Step}x FOR j " : 0 TO >res DO >" : Start}y#j*Step}y Z" : Z Rem: From grid of screen to a point in the upper half-space. MapToFund(X, >, Z) Rem: The output P(X, >, Z) is in the fundamental region. X " : X > " : > Z " : Z
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FOR k:"1 TO MaxIter DO X " : X > " : > Z " : Z IterFunc(X , > , Z ) Rem: Outputs are dX, d> and dZ X " : X #dX > " : > #d> Z " : Z #dZ END FOR Dist:"(dX#d>#Z #Z )/(2Z Z ) Rem: cosh[d(p, p1)] is used as the distance function. Code:"Color}Code(Dist) Rem: Code function generates a color code. In our examples, Code"Inc* (Dist-1) is used, where Inc is an integer constant, e.g. Inc"100 or so. SetColorPoint(i, j, Code) END FOR END FOR END ALGORITHM Title
MapToFund(x, y, z) Routine of mapping a point in the upper half-space onto The fundamental region
Arugments Globals
x, y, z X, >, Z
P(x, y, z) in the upper half-space P(X, >, Z) in the fundamental region
BEGIN d" : x*x#y*y#z*z WHILE (x'0.5 OR arctan(y/x)'/6 OR d(1) DO IF (x'0.5) THEN x " : x!1 ELSE IF (arctan(y/x)'/6) OR (arctan(y/x)(!/6) THEN Rotate(1, x, y) ELSE IF (d(1) THEN x" : !x/d y" : y/d z" : z/d d" : x*x#y*y#z*z END IF END WHILE END DEFINE}MAPPING Rotate(k, x, y) Title Routine of rotation x:"cos(2k/6)*x!sin(2k/6)*y y:"sin(2k/6)*x#cos(2k/6)*y DEFINE}FUNCTION Function}f 1(x, y) Title Routine rotating a point Global f1 Real number, OUTPUT Function g(a1, a2, a3, a4, a5, a6, u, v)"(a1#a2*sin(2u)#a3*cos(2v))* (a4#a5*cos(2u)#a6*sin(2v)) BEGIN g1"g(a1, a2, a3, a4, a5, a6, x, y) g2"g(a1, a2, a3, a4, a5, a6, !x, !y) g3"g(a1, a2, a3, a4, a5, a6, !x, y) g4"g(a1, a2, a3, a4, a5, a6, x, !y) f 1:"c1*(g1}g2}g3#g4)
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Rotate(1, x, y) g1"g(a1, a2, a3, a4, a5, a6, x, y) g2"g(a1, a2, a3, a4, a5, a6, !x, !y) Rotate(1, !x, y) g3"g(a1, a2, a3, a4, a5, a6, x, y) g4"g(a1, a2, a3, a4, a5, a6, !x, !y) f1 " : f1#c2*(g1!g2!g3#g4) Rotate(2, x, y) g1"g(a1, a2, a3, a4, a5, a6, x, y) g2"g(a1, a2, a3, a4, a5, a6, !x, !y) Rotate(2, !x, y) g3"g(a1, a2, a3, a4, a5, a6, x, y) g4"g(a1, a2, a3, a4, a5, a6, !x, !y) f1:"f1#(c2!c1)*(g1!g2!g3#g4) END ALGORITHM Title Arguments Globals
x, y, z h1, h2, h3, ¸z dX, d>, dZ
IterFunc(x, y, z) Routine for Iteration Input Real numbers Output
BEGIN Function}f1(x, y) Rem: The OUTPUT is f 1. dX " : f1 Rotate(1, x, y) Function}f1(x, y) Rem: The OUTPUT is f 1. d> " : 2*(0.5*dX}f1)/SQRT(3) dZ " : SQRT[1}(x#dX)*(x#dX)!(y#d>)*(y#d>)]!SQRT(1!x*x!y*y) ¸z " : x*x#y*y#z*z ¸z " : 1!1/¸z dX " : dX#¸z*h1 d> " : d>#¸z*h2 dZ " : dZ#¸z*h3 END Example: The following parameters are used for generating the P6 wallpaper patterns in Figs. 3}5. a1"1.0, a2"1.0, a3"1.0 a4"1.0, a5"1.0, a6"1.0 c1"0.01, c2"0.005 h1"0.1, h2"0, h3"0
References [1] Trudeau RJ. The non-Euclidean revolution. Boston: BirkhaK user, 1987. [2] Elstrodt J, Grunewald F, Mennicke J. Groups acting on hyperbolic space. Heidelberg: Springer, 1998. [3] Thurston WP. Three-dimensional geometry and topology. Princeton: Princeton University Press, 1997. [4] Chung KW, Chan HSY, Wang BN. Tessellations in three-dimensional hyperbolic space from dynamics
and the quaternions. Chaos, Solitons and Fractals, in press. [5] Armstrong MA. Groups and symmetry, New York: Springer, 1988. [6] Chung KW, Chan HSY, Wang BN. Tessellations with the modular group from dynamics. Computers & Graphics 1997;21(4):523}34. [7] Chung KW, Chan HSY, Wang BN. Spiral tilings with color symmetry from dynamics. Computers & Graphics 1999;23(3):439}48.