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Automatic generation of aesthetic patterns on fractal tilings by means of dynamical systems
Chaos, Solitons and Fractals 24 (2005) 1145–1158 www.elsevier.com/locate/chaos
Automatic generation of aesthetic patterns on fractal tilings by means of dynamical systems K.W. Chung *, H.M. Ma Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Accepted 15 September 2004
Abstract A fractal tiling or f-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. In this paper, we investigate the classification of fractal tilings with kite-shaped and dart-shaped prototiles from which three new f-tilings are found. Invariant mappings are constructed for the creation of aesthetic patterns on such tilings. A modified convergence time scheme is described, which reflects the rate of convergence of various orbits and at the same time, enhances the artistic appeal of a generated image. A scheme based on the frequency of visit at a pixel is used to generate chaotic attractors. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Fractal tiling; Dynamical system; Invariant mapping
1. Introduction The art of tilings and patterns can be found in the history of all ancient civilizations. However, the mathematical theory of tilings is comparatively recent and many aspects of the subject are still unexplored [1]. New constructions of tilings are discovered from time to time. For instance, in the 1960Õs, sets of prototiles were discovered which admit infinitely many tilings of the plane but those tilings were all nonperiodic, i.e. they did not possess translational symmetry [2]. The Penrose tilings are remarkable examples which have local five-fold rotational symmetry but with no translational repetition [3]. Regarded as models for quasicrystals, they are generated from kite- and dart-shaped polygons by using the method of substitution [4–8]. Recently, Fathauer constructed new families of tilings with fractal characteristics [9,10]. Such tilings are called fractal tilings or f-tilings which possess self-similarity and fractal boundaries. The tilings described in [9] were constructed by the repetition of similar copies of either a kite- or a dart-shaped polygon which looks somewhat like the prototiles of the Penrose tilings whereas those described in [10] were based on v-shaped prototiles. These tilings were obtained by Fathauer through extensive trial and error, but no formal proof of the classification was given. In fact, the construction of patterns by similar tiles can be dated back to the 1950Õs. The Dutch artist M.C. Escher created the print Smaller and Smaller in which tiles are reduced in size towards the center [11]. In the prints Circle Limit
*
Corresponding author. Tel.: +852 2788 8671; fax: +852 2788 8561. E-mail address: [email protected] (K.W. Chung).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.115
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I–IV, hyperbolic geometry is used to depict an infinite tiling in a finite area. Although these designs possess self-similarity, none of the boundaries of these tilings are fractals. In 1985, P. Raedschelders created the print Butterflies I in which tiles are reduced in size at both the center and the boundary which is a fractal (p. 108 of [12]). In this paper, we consider both the classification of f-tilings based on the matching rules described in [9] and the creation of aesthetic patterns on such tilings by means of dynamical systems. Fathauer found only six well-behaved kite and dart f-tilings. In Section 2, we give a formal proof of the classification from which three more new f-tilings are found. An algorithm similar to that discussed in [13,14] is considered in Section 3 for the construction of invariant mappings. Two colour schemes of generating aesthetic patterns on f-tilings are discussed in Section 4.
2. Fractal tilings from kite- or dart-shaped prototiles In this section, we consider f-tilings constructed by using either a kite- or dart-shaped prototile. As defined in [9], these f-tilings share the following properties: (1) Each f-tilings is constructed from a single prototile, to which all the tiles are similar. (2) The prototiles have edges of two lengths, denoted ‘‘long’’ and ‘‘short’’. (3) The f-tilings are by design edge to edge. As a consequence of this choice, from a given generation of tiles to the next smaller generation, the tiles are scaled by the ratio of the short to long edges of the prototile. (4) The f-tilings are bounded in the Euclidean plane. Through extensive trial and error, Fathauer found six such f-tilings which are divided into two distinct branches with their own matching rules. The six prototiles for these f-tilings are shown in Fig. 1. We will prove that these are the only prototiles from which f-tilings with the above properties can be constructed. However, three more f-tilings are found from the classification. For a prototile satisfying property (2) above, we denote the tail, side vertex and tip for the vertex joining two long edges, a long edge and a short edge, and two short edges, respectively. A f-tiling is constructed in a recursive way. To start with, the first generation tiles are all joined at the tails to form a polygon with rotational symmetry. Then, arrange the second generation tiles around the first generation tiles according to some simple set of rules. If the f-tilings are to be edge to edge, the only choices for subsequent generations are to join the tail of a tile to either the tip or a side vertex of a previous generation tile. Fig. 2 shows the first two generations of some f-tilings. While the tiles in the first generation meet only along long edges, those of the second generation may also meet along short edges. A f-tiling is obtained after an infinite number of iterations. In the recursive construction, low points at the expanding boundary are created which are vertices closer to the center than other boundary vertices of the same generation. As more and more such low points are created at each subsequent generation, the boundary exhibits fractal characteristics. In the following subsections, we investigate three matching rules from which different families f-tilings are derived. In some cases, rhombus-shaped regions exist within the boundary. If a rhombus-shaped polygon is allowed as the second prototile, more f-tilings are obtained.
Fig. 1. Six kite- and dart-shaped prototiles that allow well-behaved f-tilings.
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Fig. 2. The first two generations of some f-tilings.
2.1. Tail-to-tip configuration We first consider the matching rule that, for all the generations, the tail of a tile is joined to the tip of a previous generation tile. If some of the second generation tiles meet along short edges, then each side vertex must be met by exactly four tiles from two consecutive generations. Therefore, the angle of a side vertex is p/2 and the prototile must be a kite-shaped polygon. Theorem 1. Assume that a f-tilings with kite-shaped prototile satisfies the following matching rules: (a) The tails of at most n tiles are joined to the tip of a previous generation tile. (b) Some of the second generation tiles meet along short edges. Then, there exist only three f-tilings with the configurations (m, n)I 2 {(6, 4)I,(8, 5)I,(12, 7)I}. Proof. A kite-shaped prototile has a tail angle and a tip angle of 2p/m and p 2p/m, respectively. Consider the tiles joined to the tip of a prototile. It follows from rule (a) that 2p 2p m þn ¼ 2p ) n ¼ þ 1: ð1Þ p m m 2 We note that m must be even. At the second generation, the short edges joining to a low point (which is a tip) form a concave part of the expanding boundary. The external angle at a low point is 4p/m. At the ith generation, a concave part consists of i sides where neighbouring sides intersect at an angle of 4p/m. As i increases, a concave part eventually closes up and forms a regular polygon. Let j be the number of sides of the polygon. By considering the internal angles, we have 4p 4j j ¼ ðj 2Þp ) m ¼ : m j2
ð2Þ
Since j P 3, it follows from (2) that
j m
3 12
4 8
6 6
10 5
As m must be even, we reject the case of j = 10 and m = 5. The 3 f-tilings with configurations (6, 4)I, (8, 5)I and (12, 7)I follow from (1). If rule (b) is modified in such a way that, for i > 2, some ith generation tiles meet along short edges, then some degenerate cases are found. For instance, if i = 3, there is only one degenerate f-tiling with configuration (6, 3)I in which the prototile is a congruent triangle and the boundary of the f-tiling is a hexagon. u Next, we consider dart-shaped prototile instead in Theorem 1. Then, rule (b) is not required as the short edges of two dart-shaped tiles never join together. Let the angle of a side vertex be pa with a < 12 m1 . From the internal angles of the prototile, we have
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n ¼ ma þ 1:
ð3Þ
Assume that 2i tiles from the first i generations meet at a side vertex (see vertex a of Fig. 3a). To form a f-tiling with no gaps and none overlapping, it is necessary that the two ith generation tiles also meet each other at the second side vertex (see vertex b of Fig. 3a). Considering all the angles at vertex a, we obtain 1 1 mþ2 2iðpaÞ þ 2p a ¼ 2p ) i ¼ þ 1: ð4Þ 2 m 2ma Assume that 2j tiles from the ith to (i + j 2)th generation meet at vertex b. Note that the two (i + j 2)th generation tiles also meet each other at the second side vertex. By considering all the angles at vertex b, we have 1 1 2 ¼ 2p ) j ¼ þ 2: ð5Þ 2jðpaÞ þ 4p a 2 m ma From (5), ma must be either 1 or 2. For the first case of ma = 1, it follows from (3)–(5) that a ¼ m1 , n = 2, i ¼ m2 þ 2 and j = 4. We note that w must be even. For the second case of ma = 2, we have a ¼ m2 , n = 3, i ¼ m4 þ 32 and j = 3. It follows that m + 2 must be a multiple of 4. In both cases, there exist hollow rhombus-shaped regions which remain uncovered within the boundary, such as the rhombus with vertices a and b in Fig. 3a. To cover these regions with similar tiles, the tail of a tile has to be joined to a side vertex of a previous generation tile (see Fig. 3b). Then, the angle of vertex a of the rhombus must be equal to that of the tail, i.e. 2pð12 a m1 Þ ¼ 2p . It turns out that only the configuration m (6, 2)I of the first case satisfies this condition. It is a new f-tiling compared with those found in [9]. This result is summarized in the following theorem. Theorem 2. Assume that a f-tiling satisfies the following matching rules: (a) The tails of at most n tiles are joined to the tip of a previous generation tile provided that they are not in a rhombusshaped region. (b) Inside a rhombus-shaped region, the tail of a tile is joined to a side vertex of a previous generation tile. Then, there exists only one f-tiling with the configuration (6, 2)I. We note that the rhombus-shaped regions considered above are similar in shape. If we allow a rhombus-shaped polygon as the second prototile in addition to the dart-shaped prototile, then the above two cases (ma = 1 or 2) form a new family of f-tilings which consists of two prototiles. Theorem 3. Assume that a f-tiling consists of dart-shaped and rhombus-shaped prototiles, satisfying the following matching rules:
Fig. 3. (a) Part of the f-tiling (6, 2)I. (b) Partition of the rhombus-shaped region.
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(a) The tails of at most n tiles are joined to the tip of a previous generation tile provided that they are not in the rhombusshaped regions. (b) The rhombus-shaped regions are covered by rhombus-shaped polygons. Then, the configurations of the f-tilings are either (m, 2)a where m is even or (m, 3)a where m + 2 is a multiple of 4. 2.2. Tail-to-side-vertex configuration In this section, we consider the configuration that the tail of a tile is joined to a side vertex of a previous generation tile. Theorem 4. Assume that a f-tiling with dart-shaped prototile satisfies the following matching rules: (a) The tails of at most n tiles are joined to a side vertex of a previous generation tile. (b) The two second generation tiles joining to the short edges of the same first generation tile meet each other at both side vertices. Then, there exist only three f-tilings with the configuration (m, n)II 2 {(6, 5)II, (8, 6)II, (12, 8)II}. Proof. Let the angle of a side vertex be pa with a < 12 m1 . From the tiles joining to a side vertex of a prototile and rule (a), we have 2p 2pa þ n ¼ 2p ) n ¼ mð1 aÞ: ð6Þ m From rule (b), a prototile together with the two second generation tiles joined to its short edges form a kite-shaped polygon (see Fig. 4). By considering its internal angles, we have 2p 1 2 4pa þ 4 ¼ 2p ) a : ð7Þ m 2 m Consider vertex b which is met by second and third generation tiles. Assume that the tails of i third generation tiles are joined to such vertex. It follows from (7) that i = 4. On the expanding boundary of the jth generation, a concave part consists of 2j short edges from the jth generation tiles. Neighbouring edges of a concave part intersect at an angle of either 3(2p/m) or p 2p/m. As j increases, a concave part eventually closes up and forms a 2j-sided regular polygon. Considering the internal angles of such a polygon, we have 2p 2p 2m j 3 þ p ¼ ð2j 2Þp ) j ¼ : ð8Þ m m m4 It follows that m j
5 10
6 6
Fig. 4. Part of the f-tiling (6, 5)II.
8 4
12 3
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However, from (6) and (7), m must be even. Therefore, the case of m = 5 and j = 10 is rejected. This completes the proof. h Next, we consider kite-shaped prototile instead in Theorem 4. Similar to Theorems 2 and 3, there are rhombusshaped regions to be covered within the boundary. However, such regions cannot be covered by kite-shaped polygons with no gaps and none overlapping. If rhombus-shaped polygon is included as the second prototile, another new family of f-tilings is obtained. The proof of the following theorem can be found in [15]. Theorem 5. Assume that a f-tiling consists of kite-shaped and rhombus-shaped prototiles, satisfying the following matching rules: (a) The tails of at most n tiles are joined to a side vertex of a previous generation tile provided that they are not in the rhombus-shaped regions. (b) The rhombus-shaped regions are covered by rhombus-shaped polygons. Then, the f-tiling has a configuration of (2n, n)b with n P 3. It follows from the above theorem and (6) that the angle of a side vertex is p/2 2.3. Combined configuration In this section, we allow both tail-to-tip and tail-to-side-vertex matching rules in the construction of a f-tiling. Theorem 6. Assume that a f-tiling with dart-shaped prototile satisfies the following matching rules: (a) The tails of at most n tiles of an even generation are joined to the tip of a tile of the previous odd generation provided that they are not in a rhombus-shaped region. (b) The tails of some tiles of an odd generation are joined to a side vertex of a tile of the previous even generation provided that they are not in a rhombus-shaped region. (c) Inside a rhombus-shaped region, the tail of a tile is joined to a side vertex of a previous generation tile. (d) The two third generation tiles joining to the short edges of a second generation tile meet each other at both side vertices. Then, there exists two f-tilings with the configurations (m, n)III 2 {(6, 2)III,(8, 3)III}. Proof. Let the angle of a side vertex be pa with a < 12 m1 . Consider the tiles joined to the tip of a prototile. It follows from rule (a) that n ¼ ma þ 1:
ð9Þ
Consider the kite-shaped polygon KP composed of a second generation tile, two third generation tiles and a rhombus-shaped region (see Fig. 5). It follows from rules (c) and (d) regarding the internal angles that
Fig. 5. Part of the f-tilings (8, 3)II.
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2p 1 2 4 þ 4ðpaÞ ¼ 2p ) a ¼ : m 2 m
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ð10Þ
Assume that the tails of i fifth generation tiles are joined to the tip KP (see vertex a of Fig. 5). From the matching rules (a)–(d) and (10), we have 2p 2p m þ 6pa þ j ¼ 2p ) j ¼ 5 : ð11Þ m m 2 Since j P 1 and m > 4, the theorem follows from (9) and (11). This completes the proof. h We note that the boundary of the f-tiling with configuration (6, 2)III is a hexagon while that with configuration (8, 3)III is a fractal.
3. Aesthetic patterns from invariant mappings In this section, we describe the construction of invariant mappings for the automatic generation of aesthetic patterns on f-tilings. For a tiling associated with a group C, the creation of aesthetic symmetrical patterns is based on the construction of equivariant mapping F[16–19]. A mapping F is said to be C-equivariant if it commutes with C, i.e. F c ¼ c F for all c 2 C: However, for a tiling not associated with any group, equivariant mapping is no longer appropriate for the creation of aesthetic symmetrical patterns. In [13,14], the transformations between congruent tiles in a nonperiodic/aperiodic tiling do not constitute a group. A new algorithm based on invariant mappings was developed for such tilings. It can also be applied to f-tilings. Let c be a transformation between two similar tiles in a f-tilings T and U the set of all transformations. A mapping is said to be invariant if it maps symmetrically placed points z and cz to the same image, i.e. F ðzÞ ¼ F ðczÞ for c 2 U
and
z 2 T:
ð12Þ
A tiling with tail-to-tip configuration (m, n)I is used as an illustration for the construction of invariant mappings. In Fig. 6, let U be a prototile in the first generation with 2p/m, (n 1)p/m and 2(m n)p/m as the angles of the tail (O), the p p side vertices (Q1, Q2) and the tip (Q). Without loss of generality, we let jOQ1j = 1, Q1 ¼ eim and Q2 ¼ eim . It follows from (3) that
Fig. 6. The symmetrically placed points on a kite prototile.
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Q¼
sin ðn1Þ p m : sin np m
ð13Þ
Let U1 and U2 be the second generation tiles adjacent to U. (Although kite-shaped tiles are shown in Fig. 6, the following consideration can also be applied to dart-shaped tiles.) For z = reih 2 U, the transformations ci(i = 1, 2) of U to U1 and U2 are given by, respectively, c1 z ¼ Q þ srei½
ðn1Þ m pþh
;
ð14aÞ
and c2 z ¼ Q þ srei½
ðn1Þ m pþh
;
ð14bÞ
and where s ¼ sin mp = sin np is the length of short edge of the prototile. m We first consider the condition that no seam appears in a generated pattern. Let R 0 be a point on the long edge OQ1 , and R,R and R0 be the images of R 0 under the transformations c1, c2 and a rotation of 2p/m clockwise, respectively, i.e. 0 0 2p R ¼ c1 R0 ; R ¼ c2 R0 and R ¼ R0 ei m . These four symmetrically placed points R0 ; R; R and R are all on the edges of U and should have the same colour. Therefore, an invariant mapping F for the generation of aesthetic patterns with no seam should satisfy the following boundary condition: n o 2p F ðR0 Þ ¼F ðcR0 Þ for c 2 c1 ; c2 ; ei m ; n o ¼F ðSÞ for S 2 R; R; R0 : ð15Þ Now, we considered the construction of F in DOQQ1. Let z = reih 2 DOQQ1 and R be the intersection of the radiant line Oz and the short edge QQ1 . Then, we have R¼
sin ðn1Þp m z: r sin np h m
ð16Þ
If z ¼ reih 2 QQ1 , then z coincides with R, giving np
reih ¼ Q þ tei m ; where 0 6 t 6 sð¼
ð17Þ
sin mp = sin np Þ. m
By eliminating t, we obtain
ðn 1Þp np r sin h ¼ 0: sin m m
Now, we define n1 ðzÞ ¼ sin
np
ðn 1Þp r sin h : m m
ð18Þ
Note that n1(z) = 0 if z 2 QQ1 . For z = reih 2 DOQQ1, we consider invariant mapping of the form F ðzÞ ¼ H ðzÞu½n1 ðzÞ þ KðRÞv½n1 ðzÞ ;
ð19Þ
where R ¼ Oz \ QQ1 , u and v are functions of n1 with the restrictions that u(0) = 0 and v(0) 5 1, and H : U ! C, K : QQ1 ! C are functions of z. If z 2 QQ1 , then z = R and F(R) = H(R)u(0) + K(R)v(0) = K(R)v(0). If z ¼ R0 2 OQ1 , it follows from the boundary condition of F(R) = F(R 0 ) that KðRÞ ¼
H ðR0 Þu½n1 ðR0 Þ þ kv½n1 ðR0 Þ ; vð0Þ
ð20Þ
where k = K(Q1). We observe from (20) that the function K depends actually on the functions H, u, v and the constant k. Further, as z tends to the origin O, we obtain, from (19) and the symmetrically placed points O, Q that H ðOÞu½n1 ðOÞ þ KðRÞv½n1 ðOÞ ¼ KðQÞvð0Þ: Since R can be any point on QQ1 , v½n1 ðOÞ must be zero. It follows from (18) that ðn 1Þp ¼ 0: v sin m
ð21Þ
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Next, we consider the construction of F in DOQQ2. Similar to (18), we define np
ðn 1Þp r sin þh n2 ðzÞ ¼ sin m m
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ð22Þ
and note that n2(z) = 0 if z 2 OQ2 . For z2DOQQ2, we consider invariant mapping of the form F ðzÞ ¼ H ðzÞu½n2 ðzÞ þ KðRÞv½n2 ðzÞ ; 0
c1 1 R 0
ð23Þ 0
i2p m
where R ¼ Oz \ QQ2 . Since R ¼ 2 QQ1 and R ¼ Re 2 QQ2 , are symmetrically placed points, it follows from the p p boundary condition of F ðR0 Þ ¼ F ðR Þ that H ðreim Þ ¼ H ðreim Þ. Let H(z) be expressed as H(r, h). Then, H(r, h) is a periodic function of h with period 2p/m and H(0, h) is independent of h. The construction of invariant mapping F on U is summarized in the following theorem. Theorem 7. Let z = reih be a point on the prototile U shown in Fig. 6 and F : U ! C be mapping defined as H ðzÞu½n1 ðzÞ þ KðRÞv½n1 ðzÞ if z 2 DOQQ1 ; F ðzÞ ¼ H ðzÞu½n2 ðzÞ þ KðRÞv½n2 ðzÞ if z 2 DOQQ2 ; where, H(r, p/m) = H(r, p/m) and H(0, h) is independent of h; n1 ; n2 : C ! R are defined in (18) and (22), respectively; u; v : R ! R satisfy the conditions u(0) = 0, v(0) 5 1 and vðsin ðn1Þp Þ ¼ 0; R 2 Oz \ QQ1 , K(R) is defined in (20) and m KðRÞ ¼ KðRÞ. Then, F satisfies the boundary condition (15). For z 2 TnU, there exists a symmetrically placed point z1 = cz in U. From the invariant property, the mapping of z can be defined as F ðzÞ ¼ F ðc1 z1 Þ ¼ F ðz1 Þ: For the aesthetic patterns in Figs. 7–14, we choose H ðr; hÞ ¼ aðrÞ þ bðrÞ cosðmhÞ þ cðrÞ sinðmhÞ;
Fig. 7. f-tiling (6, 2)I: a(r) = 60r4sin(1.8r) + i(1.6r2sin(r)), b(r) = 1.2r + i(0.9r), c(r) = 1.2r + i(0.56r), u(n) = 0.75n, v(n) = 0.5(1 n/n0).
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Fig. 8. f-tiling (6, 3)b: a(r) = 1.6r4sin(1.25r) + i(1.5r3), b(r) = 1.25r2) + ir, c(r) = 1.6r2 + i(0.56r), u(n) = 0.32n, v(n) = 0.66(1 n/n0).
where a, b, c are complex functions of r with b(0) = c(0) = 0. Furthermore, we simply choose linear functions for u and v, and k = H(Q1)/v(0). In Figs. 7–14, n0 = n1(O) = sin[(n 1)p/m].
4. Colour schemes In this section, we describe two colour schemes of generating aesthetic patterns by means of invariant mappings. 4.1. Modified convergence time scheme We consider the colouring of a point z0 on U. If F(z0) 2 U, there exists a transformation c 2 U such that cF(z0) 2 U. Thus, F induces a mapping Fu:U ! U such that F u ðz0 Þ ¼ cF ðz0 Þ 2 U
with
c 2 U:
Let {zn 2 Ujn P 0} be an orbit under Fu and define the distance dn(zo) as d n ðzoÞ ¼ jzn F ðzn Þj: For a given positive integer n and a constant c > 0, we compute dn,c(z0) = bcdn(z0)c which is the largest integer smaller than or equal to cdn(z0). The value of dn,c(z0) is used to determine the colour at z0. For n P 1, dn,c(z0) for z0 fR; R; R0 ; R0 g gives the same value and the aesthetic patterns thus generated have no seam. If z0 62 U, it will be given the color of the point cz0 2 U. Outstanding features of this colour scheme are described in [18]. Figs. 7–12 are generated from this scheme. 4.2. Generation of chaotic attractors Figs. 13 and 14 show the chaotic attractors generated from invariant mappings constructed in Section 3. This colour scheme is related to the frequency of visit at pixels. Consider the orbit fF nu ðz0 Þj0 6 n 6 Ng with an arbitrary initial point
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Fig. 9. f-tiling (8, 2)a: a(r) = 68r2sin(r) + ir, b(r) = 0.12r + i(0.9r), c(r) = 1.2 + i(0.56r), u(n) = 0.13n, v(n) = 0.5(1 n/n0).
Fig. 10. f-tiling (8, 4)b: a(r) = r4sinh(r) + i(1.5r2 sin(r)), b(r) = 0.32r + i(0.56r), c(r) = 1.2r3 + i(0.56r), u(n) = 0.08n, v(n) = 0.65(1 n/n0).
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Fig. 11. f-tiling (8, 5)I: a(r) = r4sin(r) + i(1.2 sin(r)), b(r) = 1.25r + ir, c(r) = r + i(0.56r), u(n) = 0.94n, v(n) = 1 n/n0.
Fig. 12. f-tiling (12, 7)I: a(r) = 1.65r4sin(1.6r) + i(1.2r), b(r) = 1.25r + ir, c(r) = 1.2r + i(0.56r), u(n) = 1.39n, v(n) = 0.66(1 n/n0).
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Fig. 13. f-tiling (6, 4)I: a(r) = 0.5r3 + i(0.96r3), b(r) = 0.8r + 0.25r, c(r) = 1.2r2 + 1.4r, u(n) = 0.75n, v(n) = 0.76(1 n/n0).
Fig. 14. f-tiling (8, 5)I: a(r) = r3sin(r) + i(0.24r), b(r) = 0.78r + i(0.25r), c(r) = 1.2rsin(r) + i(1.4r), u(n) = 1.01n, v(n) = 0.45(1 n/n0).
z0 2 U. (In Figs. 13 and 14, we choose N = 50,000.) The colour of a pixel is determined by the number of times it is hit. The colour of a point outside U is assigned to that of a symmetrically placed point on U.
5. Conclusions In this paper, we have investigated the classification of fractal tilings. For the case of a single prototile which is either a kite-shaped or dart-shaped polygon, nine f-tilings are obtained, of which six have been described in [9] and the other
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three are new. If a rhombus is included as the second prototile, new families of f-tilings are found. As the transformations between congruent tiles do not constitute a group, equivariant mapping is no longer appropriate for the creation of aesthetic patterns on a f-tiling. We presented a fast algorithm by using invariant mappings. A modified convergence time scheme is described to enhance the artistic appeal of a generated image. The colour scheme of frequency of visit at pixels is applied to generate chaotic attractors.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Gru¨nbaum B, Shephard GS. Tilings and patterns. New York: Freeman; 1986. Berger R. The undecidability of the domino problem. Memoirs Am Math Soc 1966;66:72. Penrose R. Pentaplexity. Eureka 1978;39:16-22. Reprint from Mathematical Intelligencer 1979;2:32-37. Levine D, Steinhardt PJ. Quasicrystals. I. Definition and structure. Phys Rev B 1986;34(2):596–616. Socolar JES, Steinhardt PJ. Quasicrystals. II. Unit-cell configurations. Phys Rev B 1986;34(2):617–47. Senechal M. Quasicrystals and geometry. Cambridge University Press; 1995. Goodman-Strauss C. Matching rules and substitute tilings. Ann Math 1998;147:181–223. Radin C. Miles of tiles. Am Math Soc; 1999. Fathauer RW. Fractal tilings based on kite- and dart-shaped prototiles. Comput Graph 2001;25(2):323–31. Fathauer RW. Fractal tilings based on v-shaped prototiles. Comput Graph 2002;26(4):635–43. Escher MC. Escher on Escher: exploring the infinity. New York: Harry N. Abrams; 1989. Pickover CA. Surfing through hyperspace: understanding higher universes in six easy lessons. New York: Oxford University Press; 1999. Chung KW, Chan HSY, Wang BN. Automatic generation of nonperiodic patterns from dynamical systems. Chaos, Solitons & Fractals 2004;19(5):1177–87. Chung KW, Wang BN. Automatic generation of aesthetic patterns on aperiodic tilings by means of dynamical systems. Int J Bifurcat Chaos, in press. Ma HM. Automatic generation of aesthetic patterns on fractal tilings. MPhil thesis, City University of Hong Kong; 2003. Field M, Golubitsky M. Symmetry in chaos. New York: Oxford University Press; 1992. Dumont JP, Heiss FJ, Jones KC, Reiter CA, Vislocky LM. n-Dimensional chaotic attractors with crystallographic symmetry. Chaos, Solitons & Fractals 2001;12(4):761–84. Chung KW, Chan HSY, Wang BN. Tessellations in three-dimensional hyperbolic space from dynamics and the quaternions. Chaos, Solitons & Fractals 2001;12(10):1941–5. Chung KW, Chan HSY, Wang BN. Efficient generation of hyperbolic symmetries from dynamics. Chaos, Solitons & Fractals 2002;13(6):1175–90.
Automatic generation of aesthetic patterns on fractal tilings by means of dynamical systems K.W. Chung *, H.M. Ma Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Accepted 15 September 2004
Abstract A fractal tiling or f-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. In this paper, we investigate the classification of fractal tilings with kite-shaped and dart-shaped prototiles from which three new f-tilings are found. Invariant mappings are constructed for the creation of aesthetic patterns on such tilings. A modified convergence time scheme is described, which reflects the rate of convergence of various orbits and at the same time, enhances the artistic appeal of a generated image. A scheme based on the frequency of visit at a pixel is used to generate chaotic attractors. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Fractal tiling; Dynamical system; Invariant mapping
1. Introduction The art of tilings and patterns can be found in the history of all ancient civilizations. However, the mathematical theory of tilings is comparatively recent and many aspects of the subject are still unexplored [1]. New constructions of tilings are discovered from time to time. For instance, in the 1960Õs, sets of prototiles were discovered which admit infinitely many tilings of the plane but those tilings were all nonperiodic, i.e. they did not possess translational symmetry [2]. The Penrose tilings are remarkable examples which have local five-fold rotational symmetry but with no translational repetition [3]. Regarded as models for quasicrystals, they are generated from kite- and dart-shaped polygons by using the method of substitution [4–8]. Recently, Fathauer constructed new families of tilings with fractal characteristics [9,10]. Such tilings are called fractal tilings or f-tilings which possess self-similarity and fractal boundaries. The tilings described in [9] were constructed by the repetition of similar copies of either a kite- or a dart-shaped polygon which looks somewhat like the prototiles of the Penrose tilings whereas those described in [10] were based on v-shaped prototiles. These tilings were obtained by Fathauer through extensive trial and error, but no formal proof of the classification was given. In fact, the construction of patterns by similar tiles can be dated back to the 1950Õs. The Dutch artist M.C. Escher created the print Smaller and Smaller in which tiles are reduced in size towards the center [11]. In the prints Circle Limit
*
Corresponding author. Tel.: +852 2788 8671; fax: +852 2788 8561. E-mail address: [email protected] (K.W. Chung).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.115
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I–IV, hyperbolic geometry is used to depict an infinite tiling in a finite area. Although these designs possess self-similarity, none of the boundaries of these tilings are fractals. In 1985, P. Raedschelders created the print Butterflies I in which tiles are reduced in size at both the center and the boundary which is a fractal (p. 108 of [12]). In this paper, we consider both the classification of f-tilings based on the matching rules described in [9] and the creation of aesthetic patterns on such tilings by means of dynamical systems. Fathauer found only six well-behaved kite and dart f-tilings. In Section 2, we give a formal proof of the classification from which three more new f-tilings are found. An algorithm similar to that discussed in [13,14] is considered in Section 3 for the construction of invariant mappings. Two colour schemes of generating aesthetic patterns on f-tilings are discussed in Section 4.
2. Fractal tilings from kite- or dart-shaped prototiles In this section, we consider f-tilings constructed by using either a kite- or dart-shaped prototile. As defined in [9], these f-tilings share the following properties: (1) Each f-tilings is constructed from a single prototile, to which all the tiles are similar. (2) The prototiles have edges of two lengths, denoted ‘‘long’’ and ‘‘short’’. (3) The f-tilings are by design edge to edge. As a consequence of this choice, from a given generation of tiles to the next smaller generation, the tiles are scaled by the ratio of the short to long edges of the prototile. (4) The f-tilings are bounded in the Euclidean plane. Through extensive trial and error, Fathauer found six such f-tilings which are divided into two distinct branches with their own matching rules. The six prototiles for these f-tilings are shown in Fig. 1. We will prove that these are the only prototiles from which f-tilings with the above properties can be constructed. However, three more f-tilings are found from the classification. For a prototile satisfying property (2) above, we denote the tail, side vertex and tip for the vertex joining two long edges, a long edge and a short edge, and two short edges, respectively. A f-tiling is constructed in a recursive way. To start with, the first generation tiles are all joined at the tails to form a polygon with rotational symmetry. Then, arrange the second generation tiles around the first generation tiles according to some simple set of rules. If the f-tilings are to be edge to edge, the only choices for subsequent generations are to join the tail of a tile to either the tip or a side vertex of a previous generation tile. Fig. 2 shows the first two generations of some f-tilings. While the tiles in the first generation meet only along long edges, those of the second generation may also meet along short edges. A f-tiling is obtained after an infinite number of iterations. In the recursive construction, low points at the expanding boundary are created which are vertices closer to the center than other boundary vertices of the same generation. As more and more such low points are created at each subsequent generation, the boundary exhibits fractal characteristics. In the following subsections, we investigate three matching rules from which different families f-tilings are derived. In some cases, rhombus-shaped regions exist within the boundary. If a rhombus-shaped polygon is allowed as the second prototile, more f-tilings are obtained.
Fig. 1. Six kite- and dart-shaped prototiles that allow well-behaved f-tilings.
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Fig. 2. The first two generations of some f-tilings.
2.1. Tail-to-tip configuration We first consider the matching rule that, for all the generations, the tail of a tile is joined to the tip of a previous generation tile. If some of the second generation tiles meet along short edges, then each side vertex must be met by exactly four tiles from two consecutive generations. Therefore, the angle of a side vertex is p/2 and the prototile must be a kite-shaped polygon. Theorem 1. Assume that a f-tilings with kite-shaped prototile satisfies the following matching rules: (a) The tails of at most n tiles are joined to the tip of a previous generation tile. (b) Some of the second generation tiles meet along short edges. Then, there exist only three f-tilings with the configurations (m, n)I 2 {(6, 4)I,(8, 5)I,(12, 7)I}. Proof. A kite-shaped prototile has a tail angle and a tip angle of 2p/m and p 2p/m, respectively. Consider the tiles joined to the tip of a prototile. It follows from rule (a) that 2p 2p m þn ¼ 2p ) n ¼ þ 1: ð1Þ p m m 2 We note that m must be even. At the second generation, the short edges joining to a low point (which is a tip) form a concave part of the expanding boundary. The external angle at a low point is 4p/m. At the ith generation, a concave part consists of i sides where neighbouring sides intersect at an angle of 4p/m. As i increases, a concave part eventually closes up and forms a regular polygon. Let j be the number of sides of the polygon. By considering the internal angles, we have 4p 4j j ¼ ðj 2Þp ) m ¼ : m j2
ð2Þ
Since j P 3, it follows from (2) that
j m
3 12
4 8
6 6
10 5
As m must be even, we reject the case of j = 10 and m = 5. The 3 f-tilings with configurations (6, 4)I, (8, 5)I and (12, 7)I follow from (1). If rule (b) is modified in such a way that, for i > 2, some ith generation tiles meet along short edges, then some degenerate cases are found. For instance, if i = 3, there is only one degenerate f-tiling with configuration (6, 3)I in which the prototile is a congruent triangle and the boundary of the f-tiling is a hexagon. u Next, we consider dart-shaped prototile instead in Theorem 1. Then, rule (b) is not required as the short edges of two dart-shaped tiles never join together. Let the angle of a side vertex be pa with a < 12 m1 . From the internal angles of the prototile, we have
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n ¼ ma þ 1:
ð3Þ
Assume that 2i tiles from the first i generations meet at a side vertex (see vertex a of Fig. 3a). To form a f-tiling with no gaps and none overlapping, it is necessary that the two ith generation tiles also meet each other at the second side vertex (see vertex b of Fig. 3a). Considering all the angles at vertex a, we obtain 1 1 mþ2 2iðpaÞ þ 2p a ¼ 2p ) i ¼ þ 1: ð4Þ 2 m 2ma Assume that 2j tiles from the ith to (i + j 2)th generation meet at vertex b. Note that the two (i + j 2)th generation tiles also meet each other at the second side vertex. By considering all the angles at vertex b, we have 1 1 2 ¼ 2p ) j ¼ þ 2: ð5Þ 2jðpaÞ þ 4p a 2 m ma From (5), ma must be either 1 or 2. For the first case of ma = 1, it follows from (3)–(5) that a ¼ m1 , n = 2, i ¼ m2 þ 2 and j = 4. We note that w must be even. For the second case of ma = 2, we have a ¼ m2 , n = 3, i ¼ m4 þ 32 and j = 3. It follows that m + 2 must be a multiple of 4. In both cases, there exist hollow rhombus-shaped regions which remain uncovered within the boundary, such as the rhombus with vertices a and b in Fig. 3a. To cover these regions with similar tiles, the tail of a tile has to be joined to a side vertex of a previous generation tile (see Fig. 3b). Then, the angle of vertex a of the rhombus must be equal to that of the tail, i.e. 2pð12 a m1 Þ ¼ 2p . It turns out that only the configuration m (6, 2)I of the first case satisfies this condition. It is a new f-tiling compared with those found in [9]. This result is summarized in the following theorem. Theorem 2. Assume that a f-tiling satisfies the following matching rules: (a) The tails of at most n tiles are joined to the tip of a previous generation tile provided that they are not in a rhombusshaped region. (b) Inside a rhombus-shaped region, the tail of a tile is joined to a side vertex of a previous generation tile. Then, there exists only one f-tiling with the configuration (6, 2)I. We note that the rhombus-shaped regions considered above are similar in shape. If we allow a rhombus-shaped polygon as the second prototile in addition to the dart-shaped prototile, then the above two cases (ma = 1 or 2) form a new family of f-tilings which consists of two prototiles. Theorem 3. Assume that a f-tiling consists of dart-shaped and rhombus-shaped prototiles, satisfying the following matching rules:
Fig. 3. (a) Part of the f-tiling (6, 2)I. (b) Partition of the rhombus-shaped region.
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(a) The tails of at most n tiles are joined to the tip of a previous generation tile provided that they are not in the rhombusshaped regions. (b) The rhombus-shaped regions are covered by rhombus-shaped polygons. Then, the configurations of the f-tilings are either (m, 2)a where m is even or (m, 3)a where m + 2 is a multiple of 4. 2.2. Tail-to-side-vertex configuration In this section, we consider the configuration that the tail of a tile is joined to a side vertex of a previous generation tile. Theorem 4. Assume that a f-tiling with dart-shaped prototile satisfies the following matching rules: (a) The tails of at most n tiles are joined to a side vertex of a previous generation tile. (b) The two second generation tiles joining to the short edges of the same first generation tile meet each other at both side vertices. Then, there exist only three f-tilings with the configuration (m, n)II 2 {(6, 5)II, (8, 6)II, (12, 8)II}. Proof. Let the angle of a side vertex be pa with a < 12 m1 . From the tiles joining to a side vertex of a prototile and rule (a), we have 2p 2pa þ n ¼ 2p ) n ¼ mð1 aÞ: ð6Þ m From rule (b), a prototile together with the two second generation tiles joined to its short edges form a kite-shaped polygon (see Fig. 4). By considering its internal angles, we have 2p 1 2 4pa þ 4 ¼ 2p ) a : ð7Þ m 2 m Consider vertex b which is met by second and third generation tiles. Assume that the tails of i third generation tiles are joined to such vertex. It follows from (7) that i = 4. On the expanding boundary of the jth generation, a concave part consists of 2j short edges from the jth generation tiles. Neighbouring edges of a concave part intersect at an angle of either 3(2p/m) or p 2p/m. As j increases, a concave part eventually closes up and forms a 2j-sided regular polygon. Considering the internal angles of such a polygon, we have 2p 2p 2m j 3 þ p ¼ ð2j 2Þp ) j ¼ : ð8Þ m m m4 It follows that m j
5 10
6 6
Fig. 4. Part of the f-tiling (6, 5)II.
8 4
12 3
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However, from (6) and (7), m must be even. Therefore, the case of m = 5 and j = 10 is rejected. This completes the proof. h Next, we consider kite-shaped prototile instead in Theorem 4. Similar to Theorems 2 and 3, there are rhombusshaped regions to be covered within the boundary. However, such regions cannot be covered by kite-shaped polygons with no gaps and none overlapping. If rhombus-shaped polygon is included as the second prototile, another new family of f-tilings is obtained. The proof of the following theorem can be found in [15]. Theorem 5. Assume that a f-tiling consists of kite-shaped and rhombus-shaped prototiles, satisfying the following matching rules: (a) The tails of at most n tiles are joined to a side vertex of a previous generation tile provided that they are not in the rhombus-shaped regions. (b) The rhombus-shaped regions are covered by rhombus-shaped polygons. Then, the f-tiling has a configuration of (2n, n)b with n P 3. It follows from the above theorem and (6) that the angle of a side vertex is p/2 2.3. Combined configuration In this section, we allow both tail-to-tip and tail-to-side-vertex matching rules in the construction of a f-tiling. Theorem 6. Assume that a f-tiling with dart-shaped prototile satisfies the following matching rules: (a) The tails of at most n tiles of an even generation are joined to the tip of a tile of the previous odd generation provided that they are not in a rhombus-shaped region. (b) The tails of some tiles of an odd generation are joined to a side vertex of a tile of the previous even generation provided that they are not in a rhombus-shaped region. (c) Inside a rhombus-shaped region, the tail of a tile is joined to a side vertex of a previous generation tile. (d) The two third generation tiles joining to the short edges of a second generation tile meet each other at both side vertices. Then, there exists two f-tilings with the configurations (m, n)III 2 {(6, 2)III,(8, 3)III}. Proof. Let the angle of a side vertex be pa with a < 12 m1 . Consider the tiles joined to the tip of a prototile. It follows from rule (a) that n ¼ ma þ 1:
ð9Þ
Consider the kite-shaped polygon KP composed of a second generation tile, two third generation tiles and a rhombus-shaped region (see Fig. 5). It follows from rules (c) and (d) regarding the internal angles that
Fig. 5. Part of the f-tilings (8, 3)II.
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2p 1 2 4 þ 4ðpaÞ ¼ 2p ) a ¼ : m 2 m
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ð10Þ
Assume that the tails of i fifth generation tiles are joined to the tip KP (see vertex a of Fig. 5). From the matching rules (a)–(d) and (10), we have 2p 2p m þ 6pa þ j ¼ 2p ) j ¼ 5 : ð11Þ m m 2 Since j P 1 and m > 4, the theorem follows from (9) and (11). This completes the proof. h We note that the boundary of the f-tiling with configuration (6, 2)III is a hexagon while that with configuration (8, 3)III is a fractal.
3. Aesthetic patterns from invariant mappings In this section, we describe the construction of invariant mappings for the automatic generation of aesthetic patterns on f-tilings. For a tiling associated with a group C, the creation of aesthetic symmetrical patterns is based on the construction of equivariant mapping F[16–19]. A mapping F is said to be C-equivariant if it commutes with C, i.e. F c ¼ c F for all c 2 C: However, for a tiling not associated with any group, equivariant mapping is no longer appropriate for the creation of aesthetic symmetrical patterns. In [13,14], the transformations between congruent tiles in a nonperiodic/aperiodic tiling do not constitute a group. A new algorithm based on invariant mappings was developed for such tilings. It can also be applied to f-tilings. Let c be a transformation between two similar tiles in a f-tilings T and U the set of all transformations. A mapping is said to be invariant if it maps symmetrically placed points z and cz to the same image, i.e. F ðzÞ ¼ F ðczÞ for c 2 U
and
z 2 T:
ð12Þ
A tiling with tail-to-tip configuration (m, n)I is used as an illustration for the construction of invariant mappings. In Fig. 6, let U be a prototile in the first generation with 2p/m, (n 1)p/m and 2(m n)p/m as the angles of the tail (O), the p p side vertices (Q1, Q2) and the tip (Q). Without loss of generality, we let jOQ1j = 1, Q1 ¼ eim and Q2 ¼ eim . It follows from (3) that
Fig. 6. The symmetrically placed points on a kite prototile.
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Q¼
sin ðn1Þ p m : sin np m
ð13Þ
Let U1 and U2 be the second generation tiles adjacent to U. (Although kite-shaped tiles are shown in Fig. 6, the following consideration can also be applied to dart-shaped tiles.) For z = reih 2 U, the transformations ci(i = 1, 2) of U to U1 and U2 are given by, respectively, c1 z ¼ Q þ srei½
ðn1Þ m pþh
;
ð14aÞ
and c2 z ¼ Q þ srei½
ðn1Þ m pþh
;
ð14bÞ
and where s ¼ sin mp = sin np is the length of short edge of the prototile. m We first consider the condition that no seam appears in a generated pattern. Let R 0 be a point on the long edge OQ1 , and R,R and R0 be the images of R 0 under the transformations c1, c2 and a rotation of 2p/m clockwise, respectively, i.e. 0 0 2p R ¼ c1 R0 ; R ¼ c2 R0 and R ¼ R0 ei m . These four symmetrically placed points R0 ; R; R and R are all on the edges of U and should have the same colour. Therefore, an invariant mapping F for the generation of aesthetic patterns with no seam should satisfy the following boundary condition: n o 2p F ðR0 Þ ¼F ðcR0 Þ for c 2 c1 ; c2 ; ei m ; n o ¼F ðSÞ for S 2 R; R; R0 : ð15Þ Now, we considered the construction of F in DOQQ1. Let z = reih 2 DOQQ1 and R be the intersection of the radiant line Oz and the short edge QQ1 . Then, we have R¼
sin ðn1Þp m z: r sin np h m
ð16Þ
If z ¼ reih 2 QQ1 , then z coincides with R, giving np
reih ¼ Q þ tei m ; where 0 6 t 6 sð¼
ð17Þ
sin mp = sin np Þ. m
By eliminating t, we obtain
ðn 1Þp np r sin h ¼ 0: sin m m
Now, we define n1 ðzÞ ¼ sin
np
ðn 1Þp r sin h : m m
ð18Þ
Note that n1(z) = 0 if z 2 QQ1 . For z = reih 2 DOQQ1, we consider invariant mapping of the form F ðzÞ ¼ H ðzÞu½n1 ðzÞ þ KðRÞv½n1 ðzÞ ;
ð19Þ
where R ¼ Oz \ QQ1 , u and v are functions of n1 with the restrictions that u(0) = 0 and v(0) 5 1, and H : U ! C, K : QQ1 ! C are functions of z. If z 2 QQ1 , then z = R and F(R) = H(R)u(0) + K(R)v(0) = K(R)v(0). If z ¼ R0 2 OQ1 , it follows from the boundary condition of F(R) = F(R 0 ) that KðRÞ ¼
H ðR0 Þu½n1 ðR0 Þ þ kv½n1 ðR0 Þ ; vð0Þ
ð20Þ
where k = K(Q1). We observe from (20) that the function K depends actually on the functions H, u, v and the constant k. Further, as z tends to the origin O, we obtain, from (19) and the symmetrically placed points O, Q that H ðOÞu½n1 ðOÞ þ KðRÞv½n1 ðOÞ ¼ KðQÞvð0Þ: Since R can be any point on QQ1 , v½n1 ðOÞ must be zero. It follows from (18) that ðn 1Þp ¼ 0: v sin m
ð21Þ
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Next, we consider the construction of F in DOQQ2. Similar to (18), we define np
ðn 1Þp r sin þh n2 ðzÞ ¼ sin m m
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ð22Þ
and note that n2(z) = 0 if z 2 OQ2 . For z2DOQQ2, we consider invariant mapping of the form F ðzÞ ¼ H ðzÞu½n2 ðzÞ þ KðRÞv½n2 ðzÞ ; 0
c1 1 R 0
ð23Þ 0
i2p m
where R ¼ Oz \ QQ2 . Since R ¼ 2 QQ1 and R ¼ Re 2 QQ2 , are symmetrically placed points, it follows from the p p boundary condition of F ðR0 Þ ¼ F ðR Þ that H ðreim Þ ¼ H ðreim Þ. Let H(z) be expressed as H(r, h). Then, H(r, h) is a periodic function of h with period 2p/m and H(0, h) is independent of h. The construction of invariant mapping F on U is summarized in the following theorem. Theorem 7. Let z = reih be a point on the prototile U shown in Fig. 6 and F : U ! C be mapping defined as H ðzÞu½n1 ðzÞ þ KðRÞv½n1 ðzÞ if z 2 DOQQ1 ; F ðzÞ ¼ H ðzÞu½n2 ðzÞ þ KðRÞv½n2 ðzÞ if z 2 DOQQ2 ; where, H(r, p/m) = H(r, p/m) and H(0, h) is independent of h; n1 ; n2 : C ! R are defined in (18) and (22), respectively; u; v : R ! R satisfy the conditions u(0) = 0, v(0) 5 1 and vðsin ðn1Þp Þ ¼ 0; R 2 Oz \ QQ1 , K(R) is defined in (20) and m KðRÞ ¼ KðRÞ. Then, F satisfies the boundary condition (15). For z 2 TnU, there exists a symmetrically placed point z1 = cz in U. From the invariant property, the mapping of z can be defined as F ðzÞ ¼ F ðc1 z1 Þ ¼ F ðz1 Þ: For the aesthetic patterns in Figs. 7–14, we choose H ðr; hÞ ¼ aðrÞ þ bðrÞ cosðmhÞ þ cðrÞ sinðmhÞ;
Fig. 7. f-tiling (6, 2)I: a(r) = 60r4sin(1.8r) + i(1.6r2sin(r)), b(r) = 1.2r + i(0.9r), c(r) = 1.2r + i(0.56r), u(n) = 0.75n, v(n) = 0.5(1 n/n0).
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Fig. 8. f-tiling (6, 3)b: a(r) = 1.6r4sin(1.25r) + i(1.5r3), b(r) = 1.25r2) + ir, c(r) = 1.6r2 + i(0.56r), u(n) = 0.32n, v(n) = 0.66(1 n/n0).
where a, b, c are complex functions of r with b(0) = c(0) = 0. Furthermore, we simply choose linear functions for u and v, and k = H(Q1)/v(0). In Figs. 7–14, n0 = n1(O) = sin[(n 1)p/m].
4. Colour schemes In this section, we describe two colour schemes of generating aesthetic patterns by means of invariant mappings. 4.1. Modified convergence time scheme We consider the colouring of a point z0 on U. If F(z0) 2 U, there exists a transformation c 2 U such that cF(z0) 2 U. Thus, F induces a mapping Fu:U ! U such that F u ðz0 Þ ¼ cF ðz0 Þ 2 U
with
c 2 U:
Let {zn 2 Ujn P 0} be an orbit under Fu and define the distance dn(zo) as d n ðzoÞ ¼ jzn F ðzn Þj: For a given positive integer n and a constant c > 0, we compute dn,c(z0) = bcdn(z0)c which is the largest integer smaller than or equal to cdn(z0). The value of dn,c(z0) is used to determine the colour at z0. For n P 1, dn,c(z0) for z0 fR; R; R0 ; R0 g gives the same value and the aesthetic patterns thus generated have no seam. If z0 62 U, it will be given the color of the point cz0 2 U. Outstanding features of this colour scheme are described in [18]. Figs. 7–12 are generated from this scheme. 4.2. Generation of chaotic attractors Figs. 13 and 14 show the chaotic attractors generated from invariant mappings constructed in Section 3. This colour scheme is related to the frequency of visit at pixels. Consider the orbit fF nu ðz0 Þj0 6 n 6 Ng with an arbitrary initial point
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Fig. 9. f-tiling (8, 2)a: a(r) = 68r2sin(r) + ir, b(r) = 0.12r + i(0.9r), c(r) = 1.2 + i(0.56r), u(n) = 0.13n, v(n) = 0.5(1 n/n0).
Fig. 10. f-tiling (8, 4)b: a(r) = r4sinh(r) + i(1.5r2 sin(r)), b(r) = 0.32r + i(0.56r), c(r) = 1.2r3 + i(0.56r), u(n) = 0.08n, v(n) = 0.65(1 n/n0).
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Fig. 11. f-tiling (8, 5)I: a(r) = r4sin(r) + i(1.2 sin(r)), b(r) = 1.25r + ir, c(r) = r + i(0.56r), u(n) = 0.94n, v(n) = 1 n/n0.
Fig. 12. f-tiling (12, 7)I: a(r) = 1.65r4sin(1.6r) + i(1.2r), b(r) = 1.25r + ir, c(r) = 1.2r + i(0.56r), u(n) = 1.39n, v(n) = 0.66(1 n/n0).
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Fig. 13. f-tiling (6, 4)I: a(r) = 0.5r3 + i(0.96r3), b(r) = 0.8r + 0.25r, c(r) = 1.2r2 + 1.4r, u(n) = 0.75n, v(n) = 0.76(1 n/n0).
Fig. 14. f-tiling (8, 5)I: a(r) = r3sin(r) + i(0.24r), b(r) = 0.78r + i(0.25r), c(r) = 1.2rsin(r) + i(1.4r), u(n) = 1.01n, v(n) = 0.45(1 n/n0).
z0 2 U. (In Figs. 13 and 14, we choose N = 50,000.) The colour of a pixel is determined by the number of times it is hit. The colour of a point outside U is assigned to that of a symmetrically placed point on U.
5. Conclusions In this paper, we have investigated the classification of fractal tilings. For the case of a single prototile which is either a kite-shaped or dart-shaped polygon, nine f-tilings are obtained, of which six have been described in [9] and the other
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three are new. If a rhombus is included as the second prototile, new families of f-tilings are found. As the transformations between congruent tiles do not constitute a group, equivariant mapping is no longer appropriate for the creation of aesthetic patterns on a f-tiling. We presented a fast algorithm by using invariant mappings. A modified convergence time scheme is described to enhance the artistic appeal of a generated image. The colour scheme of frequency of visit at pixels is applied to generate chaotic attractors.
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