- Email: [email protected]
Categorization of new fractal carpets
Chaos, Solitons and Fractals 41 (2009) 1020–1026
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Categorization of new fractal carpets Mamta Rani a,*, Saurabh Goel b a b
Department of Computer Science and Applications, Galgotias College of Engineering and Technology, 1, Knowledge Park II, Greater Noida, Uttar Pradesh, India Department of Computer Science and Applications, Galgotias Institute of Management and Technology, 1, Knowledge Park II, Greater Noida, Uttar Pradesh, India
a r t i c l e
i n f o
Article history: Accepted 24 April 2008
a b s t r a c t Sierpinski carpet is one of the very beautiful fractals from the historic gallery of classical fractals. Carpet designing is not only a fascinating activity in computer graphics, but it has real applications in carpet industry as well. One may find illusionary delighted carpets designed here, which are useful in real designing of carpets. In this paper, we attempt to systematize their generation and put them into categories. Each next category leads to a more generalized form of the fractal carpet. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Fractal is a set, which is self-similar under magnification. Polish mathematician Waclaw Sierpinski (1882–1969) presented the Sierpinski carpet in 1916. Besides being credited for being one of the most brilliant creations among the historical fractals, it has enjoyed a great topological importance as well. For a fundamental study of the Sierpinski carpet, one may refer [7,8], and for an excellent work on Brownian motion in the Sierpinski carpet, refer [1]. Indeed, this is a universal object, which incorporates all two dimensional objects in a plane or their topological equivalents in it. Recently, El Naschie has done some work on related universality problem (see, for instance, [4,5] and several references thereof). Self-similarity dimension of the Sierpinski carpet is 1.89 approximately. In 1994, Simpelaere [12,13] decomposed the same into multi-fractals and calculated its correlation dimension. Recently, using convex hull and projection, Min [3] obtained the exact value of Hausdorff dimension of the Sierpinski carpet. Much work has been done on different aspects of fractal carpets, e.g., Wojcik et al. [14] have given a universal relationship between dimension of space cross sections and time cross sections in fractal quantum carpet. Further, Dai and Tian [2] have studied properties related to the intersection of Sierpinski carpet with its rational translate, and Niu and Xi [6] have studied its singularity. In 2003, Rani [10] (see also Rani and Kumar [11]) came up with new ideas of developing fractal carpets. They obtained many new carpets and their algorithms. For details of these algorithms, one may refer [9]. In this paper, we present a variety of selected new fractal carpets and attempt to put them into six categories. Each next category leads to a more generalized form of the fractal carpet. 2. Generalization of fractal carpets We have written our program in C++ and generated several new fractal carpets. Now, there is a pool of carpets and at this point, we feel the need to put them into categories and give a generalization of the carpets. So, we put the carpets into six categories. Each next category is inherited from the previous one, i.e., each next category is an incremental category. Thus, in generation of fractal carpets, we move from specialization to generalization. * Corresponding author. Tel.: +91 135 2431624. E-mail addresses: [email protected] (M. Rani), [email protected] (S. Goel). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.04.056
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2.1. Category 1: fractal carpets obtained by dropping a square In order to design a carpet in this category, we divide the initiator square into nine equal parts, so that we have a grid of 3 3. Choose a square in an arbitrary manner and drop it. In this category, we have designed Stair carpet (see Fig. 1), where
Fig. 1. Stair carpet and its blueprint. Divide the square into 9 equal parts and drop the square at position (2, 3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Sky window carpet and its blueprint. Divide the square into 16 equal parts and drop the square at position on (2, 3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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we divide the initiator into nine equal parts and drop the square at location (2, 3) of the grid. Now, repeat the same process with the remaining eight squares. Self-similarity dimension of Stair carpet is same as that of the Sierpinski carpet, which itself falls under Category 1. Another example of this category is Black and Red carpet discussed in [9] (see also [11]). 2.2. Category 2: fractal carpets of order n n and dropping of a square To design a carpet in this category, divide the initiator into n n equal squares and drop any one of them. See Sky Window carpet in Fig. 2 as an example of order n carpet. In order to obtain the same, the whole initial square has been divided into 4 4 ¼ 16 equal squares. Now, drop the square at (2, 3) location and repeat the same process with remaining 15 squares. Self-similarity dimension of Sky Window carpet is close to 1.95. Layered carpet [10] is another example of this category. 2.3. Category 3: fractal carpets of order n n and dropping of more than one square To design the carpet in this category, divide the initiator into n n squares and drop more than one square before the subsequent iteration. See Rainbow and Floral carpets (Figs. 3 and 4) as examples of this category. Rainbow carpet has been designed by dividing the initiator into 4 4 equal squares and dropping squares at positions (2, 2), (2, 3), (3, 2) and (3, 3) in each iteration. The self-similarity dimension of Rainbow carpet is approximately 1.29. For Floral carpet, divide the initial square into 6 6 equal squares and drop squares at positions (1, 2), (1, 5), (2, 1), (2, 6), (5, 1), (5, 6), (6, 2) and (6, 5) in each iteration. The self-similarity dimension of Floral carpet is approximately 1.86. We remark that Dhan, Grid and Lair carpets studied in [9–11] also fall under Category 3. 2.4. Category 4: unequal sizing of squares in fractal carpets In this category, carpets are designed by dividing the initiator into squares of unequal sizes and dropping some of them before the subsequent iteration. For example, divide the initial square into 10 squares out of which one is 4/5th of the initiator and rest are 1/5th, and drop the small squares at positions (4, 1) and (5, 2) to get the Spark carpet (Fig. 5) and at (2, 1) and (5, 4) to get the Ocean carpet (Fig. 6). Self-similarity dimension of these two carpets is 1.95 approximately. There are a few other creations in this category. For example, Kangri, Vinod, Krishna, Fern and Aero carpets belong to the Category 4. For details of these carpets see [9–11]. Now we introduce the completely new concepts of designing of fractal carpets.
Fig. 3. Rainbow carpet and its blueprint. Divide the square into 16 equal parts and drop the squares at position (2, 2), (2, 3), (3, 2), (3, 3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 4. Floral carpet and its blueprint. Divide the square into 6 6 equal parts and drop the squares at position (1,2_), (1, 5), (2, 1), (2, 6), (5, 1), (5, 6), (6, 2), (6, 5). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Spark carpet and its blueprint. The largest square is 4/5 and rest are 1/5 of the whole square. Drop the squares at position (4, 1) and (5, 2). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. Ocean carpet and its blueprint. The largest square is 4/5 and rest are 1/5 of the whole square. Drop the squares at position (2, 1) and (5, 4). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
2.5. Category 5: overlapping of two carpets To obtain a carpet in Category 5, choose generators (or blueprints) of two different carpets and apply them at even and odd iterations repeatedly. The first carpet (at odd iterations) in Galgotia carpet (Fig. 7) is designed by dividing the initial square into seven equal parts, which are 1/4 of the whole square and one larger block, which is 3/4 of the initiator, and drop
Fig. 7. Galgotia carpet: Blueprint 1. The largest square is 3/4 and rest are 1/4 of the whole square. Drop the left bottom corner square. Blueprint 2. Divide the square into 16 equal parts and drop the square at position (1, 4). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Rangoli carpet.
Fig. 9. Block carpet: Blueprint 1. Divide the square into 16 equal parts and drop the square at position (2, 2). Blueprint 2. Divide the square into 16 equal parts and drop the square at position (2, 3). Blueprint 3. Divide the square into 16 equal parts and drop the square at position (3, 2). Blueprint 4. Divide the square into 16 equal parts and drop the square at position (3, 3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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the left bottom corner square. The second carpet (at even iterations) in Galgotia carpet is designed by dividing the initiator into 4 4 equal squares and dropping right top corner square. Self-similarity dimension of Galgotia carpet 1.95. Rangoli carpet (Fig. 8) is another example of this category, which has been created by overlapping of Dhan [9] and Rainbow carpets at odd and even iterations respectively. Self-similarity dimension of Rangoli carpet 1.79. 2.6. Category 6: overlapping of more than two carpets In this category, carpets may be generated by overlapping of n number of carpets at different iterations. In order to design Block carpet, divide the whole square into 16 equal parts and drop the square at position (2, 2) at iteration 1, (2, 3) at iteration 2, (3, 2) at iteration 3 and (3, 3) at iteration 4. Start the same whole process with the left out squares at iteration 5, then at iteration 9 and so on. To see the Block carpet and its blueprints of four overlapped carpets, see Fig. 9. Self-similarity dimension of Block carpet 1.95, which is same as of Galgotia carpet. 3. Conclusion Depending upon the scheme of the blue prints, we have put the gallery of fractal carpets into six categories. Each next category is inherited from the previous one. The first category gives the most specialized form among the carpets, while the sixth pertains to the most generalized form. Thus we may further generate numerous fractal carpets as we move from specialization (Category 1) to generalization (Category 6). Acknowledgement The authors thank Professor S. L. Singh and Mr. Yash Kumar for their perspicacious comments and suggestions for the work. References [1] Barlow Martin T, Bass Richard F. Local times for Brownian motion on the Sierpinski carpet. Probability theory and related fields, vol. 85. Berlin/ Heidelberg: Springer; 1990. no. 1. [2] Dai Meifeng, Tian Lixin. Intersection of the Sierpinski carpet with its rational translate. Chaos, Solitons & Fractals 2007;31(1):179–87. [3] Min Wu. The Hausdorff measure of some Sierpinski carpets. Chaos, Solitons & Fractals 2005;24(3):717–31. [4] El Naschie MS. Cantorian distance statistical mechanics and universal behaviour of multi-dimensional triadic sets. Chaos, Solitons & Fractals 1993;17(6):47–53. [5] El Naschie MS. On the universality class of all universality classes and E-infinity spacetime physics. Chaos, Solitons & Fractals 2007;32(3):927–36. [6] Niu M, Xi LF. Singularity of a class of self similar measures. Chaos, Solitons & Fractals 2007;34(2):376–82. [7] Peitgen HO, Jürgens H, Saupe D. Fractals for the classroom, part I: introduction to Fractals and Chaos. New York: Springer-Verlag, Inc.; 1992. [8] Peitgen HO, Jurgens H, Saupe D. Chaos and Fractals. New York: Springer-Verlag, Inc.; 1992. [9] Rani M. Iterative procedures in Fractal and Chaos, Ph.D. Thesis, Department of Computer Science, Gurukula Kangri Vishwavidyalaya, Hardwar; 2002. [10] Rani M. Fractals in vedic heritage and fractal carpets. In: Proceedings of National Seminar on History, Heritage and Development of Mathematical Science, Govt. Girls College, Allahabad, October 18–20; 2003. p. 110–21. [11] Rani M, Kumar V. New fractal carpets. Arab J Sci Eng Sect C: Theme Issues 2004;29(2):125–34. MR2126593. [12] Simpelaere Dominique. The correlation dimension of the Sierpinski carpet. Chaos, Solitons & Fractals 1994;4(12):2223–35. [13] Simpelaere Dominique. Multifractal decomposition of Sierpinski carpet. Chaos, Solitons & Fractals 1995;5(11):2153–70. [14] Wojcik Daniel, Iwo-Birula Bialynieki, Zyczkowski Karol. Time evolution of quantum fractals. Am Phys Soc 2000;85(24):5022–5.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Categorization of new fractal carpets Mamta Rani a,*, Saurabh Goel b a b
Department of Computer Science and Applications, Galgotias College of Engineering and Technology, 1, Knowledge Park II, Greater Noida, Uttar Pradesh, India Department of Computer Science and Applications, Galgotias Institute of Management and Technology, 1, Knowledge Park II, Greater Noida, Uttar Pradesh, India
a r t i c l e
i n f o
Article history: Accepted 24 April 2008
a b s t r a c t Sierpinski carpet is one of the very beautiful fractals from the historic gallery of classical fractals. Carpet designing is not only a fascinating activity in computer graphics, but it has real applications in carpet industry as well. One may find illusionary delighted carpets designed here, which are useful in real designing of carpets. In this paper, we attempt to systematize their generation and put them into categories. Each next category leads to a more generalized form of the fractal carpet. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Fractal is a set, which is self-similar under magnification. Polish mathematician Waclaw Sierpinski (1882–1969) presented the Sierpinski carpet in 1916. Besides being credited for being one of the most brilliant creations among the historical fractals, it has enjoyed a great topological importance as well. For a fundamental study of the Sierpinski carpet, one may refer [7,8], and for an excellent work on Brownian motion in the Sierpinski carpet, refer [1]. Indeed, this is a universal object, which incorporates all two dimensional objects in a plane or their topological equivalents in it. Recently, El Naschie has done some work on related universality problem (see, for instance, [4,5] and several references thereof). Self-similarity dimension of the Sierpinski carpet is 1.89 approximately. In 1994, Simpelaere [12,13] decomposed the same into multi-fractals and calculated its correlation dimension. Recently, using convex hull and projection, Min [3] obtained the exact value of Hausdorff dimension of the Sierpinski carpet. Much work has been done on different aspects of fractal carpets, e.g., Wojcik et al. [14] have given a universal relationship between dimension of space cross sections and time cross sections in fractal quantum carpet. Further, Dai and Tian [2] have studied properties related to the intersection of Sierpinski carpet with its rational translate, and Niu and Xi [6] have studied its singularity. In 2003, Rani [10] (see also Rani and Kumar [11]) came up with new ideas of developing fractal carpets. They obtained many new carpets and their algorithms. For details of these algorithms, one may refer [9]. In this paper, we present a variety of selected new fractal carpets and attempt to put them into six categories. Each next category leads to a more generalized form of the fractal carpet. 2. Generalization of fractal carpets We have written our program in C++ and generated several new fractal carpets. Now, there is a pool of carpets and at this point, we feel the need to put them into categories and give a generalization of the carpets. So, we put the carpets into six categories. Each next category is inherited from the previous one, i.e., each next category is an incremental category. Thus, in generation of fractal carpets, we move from specialization to generalization. * Corresponding author. Tel.: +91 135 2431624. E-mail addresses: [email protected] (M. Rani), [email protected] (S. Goel). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.04.056
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2.1. Category 1: fractal carpets obtained by dropping a square In order to design a carpet in this category, we divide the initiator square into nine equal parts, so that we have a grid of 3 3. Choose a square in an arbitrary manner and drop it. In this category, we have designed Stair carpet (see Fig. 1), where
Fig. 1. Stair carpet and its blueprint. Divide the square into 9 equal parts and drop the square at position (2, 3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Sky window carpet and its blueprint. Divide the square into 16 equal parts and drop the square at position on (2, 3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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we divide the initiator into nine equal parts and drop the square at location (2, 3) of the grid. Now, repeat the same process with the remaining eight squares. Self-similarity dimension of Stair carpet is same as that of the Sierpinski carpet, which itself falls under Category 1. Another example of this category is Black and Red carpet discussed in [9] (see also [11]). 2.2. Category 2: fractal carpets of order n n and dropping of a square To design a carpet in this category, divide the initiator into n n equal squares and drop any one of them. See Sky Window carpet in Fig. 2 as an example of order n carpet. In order to obtain the same, the whole initial square has been divided into 4 4 ¼ 16 equal squares. Now, drop the square at (2, 3) location and repeat the same process with remaining 15 squares. Self-similarity dimension of Sky Window carpet is close to 1.95. Layered carpet [10] is another example of this category. 2.3. Category 3: fractal carpets of order n n and dropping of more than one square To design the carpet in this category, divide the initiator into n n squares and drop more than one square before the subsequent iteration. See Rainbow and Floral carpets (Figs. 3 and 4) as examples of this category. Rainbow carpet has been designed by dividing the initiator into 4 4 equal squares and dropping squares at positions (2, 2), (2, 3), (3, 2) and (3, 3) in each iteration. The self-similarity dimension of Rainbow carpet is approximately 1.29. For Floral carpet, divide the initial square into 6 6 equal squares and drop squares at positions (1, 2), (1, 5), (2, 1), (2, 6), (5, 1), (5, 6), (6, 2) and (6, 5) in each iteration. The self-similarity dimension of Floral carpet is approximately 1.86. We remark that Dhan, Grid and Lair carpets studied in [9–11] also fall under Category 3. 2.4. Category 4: unequal sizing of squares in fractal carpets In this category, carpets are designed by dividing the initiator into squares of unequal sizes and dropping some of them before the subsequent iteration. For example, divide the initial square into 10 squares out of which one is 4/5th of the initiator and rest are 1/5th, and drop the small squares at positions (4, 1) and (5, 2) to get the Spark carpet (Fig. 5) and at (2, 1) and (5, 4) to get the Ocean carpet (Fig. 6). Self-similarity dimension of these two carpets is 1.95 approximately. There are a few other creations in this category. For example, Kangri, Vinod, Krishna, Fern and Aero carpets belong to the Category 4. For details of these carpets see [9–11]. Now we introduce the completely new concepts of designing of fractal carpets.
Fig. 3. Rainbow carpet and its blueprint. Divide the square into 16 equal parts and drop the squares at position (2, 2), (2, 3), (3, 2), (3, 3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 4. Floral carpet and its blueprint. Divide the square into 6 6 equal parts and drop the squares at position (1,2_), (1, 5), (2, 1), (2, 6), (5, 1), (5, 6), (6, 2), (6, 5). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Spark carpet and its blueprint. The largest square is 4/5 and rest are 1/5 of the whole square. Drop the squares at position (4, 1) and (5, 2). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. Ocean carpet and its blueprint. The largest square is 4/5 and rest are 1/5 of the whole square. Drop the squares at position (2, 1) and (5, 4). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
2.5. Category 5: overlapping of two carpets To obtain a carpet in Category 5, choose generators (or blueprints) of two different carpets and apply them at even and odd iterations repeatedly. The first carpet (at odd iterations) in Galgotia carpet (Fig. 7) is designed by dividing the initial square into seven equal parts, which are 1/4 of the whole square and one larger block, which is 3/4 of the initiator, and drop
Fig. 7. Galgotia carpet: Blueprint 1. The largest square is 3/4 and rest are 1/4 of the whole square. Drop the left bottom corner square. Blueprint 2. Divide the square into 16 equal parts and drop the square at position (1, 4). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Rangoli carpet.
Fig. 9. Block carpet: Blueprint 1. Divide the square into 16 equal parts and drop the square at position (2, 2). Blueprint 2. Divide the square into 16 equal parts and drop the square at position (2, 3). Blueprint 3. Divide the square into 16 equal parts and drop the square at position (3, 2). Blueprint 4. Divide the square into 16 equal parts and drop the square at position (3, 3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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the left bottom corner square. The second carpet (at even iterations) in Galgotia carpet is designed by dividing the initiator into 4 4 equal squares and dropping right top corner square. Self-similarity dimension of Galgotia carpet 1.95. Rangoli carpet (Fig. 8) is another example of this category, which has been created by overlapping of Dhan [9] and Rainbow carpets at odd and even iterations respectively. Self-similarity dimension of Rangoli carpet 1.79. 2.6. Category 6: overlapping of more than two carpets In this category, carpets may be generated by overlapping of n number of carpets at different iterations. In order to design Block carpet, divide the whole square into 16 equal parts and drop the square at position (2, 2) at iteration 1, (2, 3) at iteration 2, (3, 2) at iteration 3 and (3, 3) at iteration 4. Start the same whole process with the left out squares at iteration 5, then at iteration 9 and so on. To see the Block carpet and its blueprints of four overlapped carpets, see Fig. 9. Self-similarity dimension of Block carpet 1.95, which is same as of Galgotia carpet. 3. Conclusion Depending upon the scheme of the blue prints, we have put the gallery of fractal carpets into six categories. Each next category is inherited from the previous one. The first category gives the most specialized form among the carpets, while the sixth pertains to the most generalized form. Thus we may further generate numerous fractal carpets as we move from specialization (Category 1) to generalization (Category 6). Acknowledgement The authors thank Professor S. L. Singh and Mr. Yash Kumar for their perspicacious comments and suggestions for the work. References [1] Barlow Martin T, Bass Richard F. Local times for Brownian motion on the Sierpinski carpet. Probability theory and related fields, vol. 85. Berlin/ Heidelberg: Springer; 1990. no. 1. [2] Dai Meifeng, Tian Lixin. Intersection of the Sierpinski carpet with its rational translate. Chaos, Solitons & Fractals 2007;31(1):179–87. [3] Min Wu. The Hausdorff measure of some Sierpinski carpets. Chaos, Solitons & Fractals 2005;24(3):717–31. [4] El Naschie MS. Cantorian distance statistical mechanics and universal behaviour of multi-dimensional triadic sets. Chaos, Solitons & Fractals 1993;17(6):47–53. [5] El Naschie MS. On the universality class of all universality classes and E-infinity spacetime physics. Chaos, Solitons & Fractals 2007;32(3):927–36. [6] Niu M, Xi LF. Singularity of a class of self similar measures. Chaos, Solitons & Fractals 2007;34(2):376–82. [7] Peitgen HO, Jürgens H, Saupe D. Fractals for the classroom, part I: introduction to Fractals and Chaos. New York: Springer-Verlag, Inc.; 1992. [8] Peitgen HO, Jurgens H, Saupe D. Chaos and Fractals. New York: Springer-Verlag, Inc.; 1992. [9] Rani M. Iterative procedures in Fractal and Chaos, Ph.D. Thesis, Department of Computer Science, Gurukula Kangri Vishwavidyalaya, Hardwar; 2002. [10] Rani M. Fractals in vedic heritage and fractal carpets. In: Proceedings of National Seminar on History, Heritage and Development of Mathematical Science, Govt. Girls College, Allahabad, October 18–20; 2003. p. 110–21. [11] Rani M, Kumar V. New fractal carpets. Arab J Sci Eng Sect C: Theme Issues 2004;29(2):125–34. MR2126593. [12] Simpelaere Dominique. The correlation dimension of the Sierpinski carpet. Chaos, Solitons & Fractals 1994;4(12):2223–35. [13] Simpelaere Dominique. Multifractal decomposition of Sierpinski carpet. Chaos, Solitons & Fractals 1995;5(11):2153–70. [14] Wojcik Daniel, Iwo-Birula Bialynieki, Zyczkowski Karol. Time evolution of quantum fractals. Am Phys Soc 2000;85(24):5022–5.