ITERATIVE OPERATIONS IN CONSTRUCTION CIRCULAR AND SQUARE FRACTAL CARPETS

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1 ITERATIVE OPERATIONS IN CONSTRUCTION CIRCULAR AND SQUARE FRACTAL CARPETS Dr. Yusra Faisal Al-Irhaim, Marah Mohamed Taha University of Mosul, Iraq ABSTRACT: Carpet designing is not only a fascinating activity in computer graphics, but it has real applications in carpets industry as well. In this research, we design 2D models using fractal geometry, which is about apply of certain additions to a basic model (square or circle), enter in a certain number of iterations, which produces different patterns of fractal carpets. Keywords; Iterative Operations, 2D Models, Fractal Geometry 1- Introduction: A fractal is a rough or fragmented geometric shape that can be subdivided in part, each of which is a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale [4]. In 1975, Benoit Mandelbrot coined the term fractal when studying self-similarity. He also defined fractal dimension and provided fractal examples made with computer. Mandelbrot also defined a very wellknown fractal called Mandelbrot Set. The study of self-similar objects and similar functions began With Leibnitz in the 17th century and was intense at the end of the 19th century and beginning of 20th century by H. Koch (koch's curve), W.Sierpinski (Sierpinski triangle), G. Cantor (Cantor Set), H. Poincare (attractor and dynamical systems) and G. Julia (Julia Set), among others. M. Barnsley has developed during the last two decades applications of fractals to computer graphics [3]. In this research, we design many example of fractal carpets, by apply certain additions to the basic model (square or circle) with certain number of iteration. Many of the related researches in computer graphics relied on concepts of fractal geometry, following is some of what has reviewed the work of the researchers in the field fractal geometry with computer graphic: - In 2010, researcher Joelle Thollot presented project on "set manipulation of Fractal objects using Matrices of IFS". This work proposed some methods for increasing the modeling capabilities of fractal shape constructions and presented an extension of the IFS model based on the definition of matrices of IFS that provides a constructive approach of fractal shape [5]. - In 2011, each of researchers Fucheng You and Yeli Li presented project on "The E-Learning Environment Development in Design of Fractal Tree Graphics". this work presented a new learning environment of fractal tree graphics based on E- Learning is proposed, development of which is also introduced in detail. In this new learning environment, students can repeat drawing the colorful fractal tree graphics which may arouse the students' interests and attract their attention [6]. 2 -The previous works: 91

2 - In same year, researcher Joanpere Salvado presented thesis on how fractal geometry can be used in applications to computer graphics or to model natural objects [3]. - In same year, each of researcher Munesh Chandra, Sanjay M. Shah presented project on "Iterative Procedures in Generation Fractal Carpets", this work provided a new fractal carpets using iterations and made at attempt to put them into categories [1]. 3- Iteration Function System (IFS): Most of fractal models have self- similarity property, that is, each can be tiled with congruent tiles where the tiles can be mapped onto the original using similarities with the same scaling-factor or inversely, the original object can be mapped onto the individual tiles using similarities with a common scaling factor. In general, modeling such complicated objects require involved algorithms, but one can develop quite simple algorithms by studying the relations between parts of a fractal that allow us to use relative small sets of transformations [2]. The transformation function 'F' generates successive levels of details with calculation: A1=F {Z (A0)}, A2=F {Z(A1)}, A3=F{Z(A2)},., An=F {Z(An-1)}. Where A0 ={X0, Y0, Z0}, is selected initial point, the transformation function f (Z) can be defined in terms of geometric transformations (translation, rotation, scaling,) or it is can be set up with nonlinear coordinate transformations. Transformation function can be applied to the initial set of primitives, like lines, areas, volumes, curves, surfaces, and solids objects. As we know, the fractal objects have infinite details at each point, but the transformation function is used only for finite number of times. the amount of details included the final display of the image depends on the number of the iterations performed and the resolution of the display system [2]. 4- Construction of Fractal carpets: In this section we design many patterns of fractal carpets. By using a certain basic model (square, circle) and applying specific additions, that are entered in the recursion function with specific number of iterations to design two types of carpets (Circular and Square Fractal Carpets). 4-1 Square Fractal Carpets: The Square Fractal Carpets are designed by using the basic model (square) with certain Side length (S) and dividing this basic model to nine smaller squares with certain side length (S* 1/3). In every case of design, one of two or more smaller squares are deleted and then insert the remaining squares in the recursion function with specific number of iterations to design Square Fractal Carpet. Note: This idea was derived from the composition Sierpinski carpet [3]. Case 1: In this case the square in the middle of above squares is deleted and the remaining squares are inserted in recursion function. As shown in figure (1) the steps of design square fractal carpet (case 1) from left to right, up to bottom with specific number of iterations (N=5). 92

$Figure (1): square fractal carpet (case 1) Case 2: In this case the two squares in the left and$ $As shown in figure (2) the steps of design square fractal carpet (case 2) from left to right, up to$

3 Figure (1): square fractal carpet (case 1) Case 2: In this case the two squares in the left and right of above squares respectively are deleted and the remaining squares are inserted in recursion function. As shown in figure (2) the steps of design square fractal carpet (case 2) from left to right, up to bottom with specific number of iterations (N=5). Figure (2): square fractal carpet (case 2) 93

$As shown in figure (3) the steps of design square fractal carpet (case 3) from left to right, up to bottom with specific number of iterations (N=5).$ $Figure (3): square fractal carpet (case 3) Case 4: In this case the square in the left of above squares and the square in the right of bottom squares are deleted and the remaining$

4 Case 3: In this case the two squares in the left and right of center squares respectively are deleted and the remaining squares are inserted in recursion function. As shown in figure (3) the steps of design square fractal carpet (case 3) from left to right, up to bottom with specific number of iterations (N=5). Figure (3): square fractal carpet (case 3) Case 4: In this case the square in the left of above squares and the square in the right of bottom squares are deleted and the remaining squares are inserted in recursion function. As shown in figure (4) the steps of design square fractal carpet (case 4) from left to right, up to bottom with specific number of iterations (N=5) Figure (4): square fractal carpet (case 4) 94

$There are other examples of square fractal carpets, as shown in figure (5,a) the fractal carpet (case 5) is designed by deleting the square in middle of above squares and the square in the middle of$ $And the fractal carpet (case 6) in figure (5,b) is designed by deleting the square in middle of above squares and the square in the middle of bottom squares and the squares in the left and right of$

5 There are other examples of square fractal carpets, as shown in figure (5,a) the fractal carpet (case 5) is designed by deleting the square in middle of above squares and the square in the middle of bottom squares respectively. And the fractal carpet (case 6) in figure (5,b) is designed by deleting the square in middle of above squares and the square in the middle of bottom squares and the squares in the left and right of the center squares respectively. And the fractal carpet (case 7) in figure (5, c) is designed by deleting the square in the right of above squares and the square in the left of bottom squares respectively. And the fractal carpet (case 8) in figure (5,d) is designed by deleting the square in the middle of above squares and the squares in the left and right of center squares respectively. And figure (6) shows other examples of square fractal carpet. Figure (5): square fractal carpets (case 5, 6,7, and 8) Figure (6): other examples of square fractal carpets 95

6 4-2 Circular fractal carpets: The Circular Fractal Carpets are designed by using the basic model (circle) with certain radius (r) and position (0,0),and added certain number smaller circles with specific radius to the basic model, that are entered in recursion function with specific number of iterations. Case 1: In this case, two smaller circles is added to the basic model with specific radius (r* 0.5) and translation factor (tx, ty) respectively is (r, 0) for right circle, (-r, 0) for left circle, figure (7) shows the steps of design circular fractal carpet (case 1) with specific number of iterations (N=5). Figure (7): circular fractal carpets (case 1) Case 2: In this case, four smaller circles is added to the basic model with specific radius (r* 0.5) and translation factor (tx, ty) respectively is (r, 0) for right circle, (-r, 0) for left circle, (0,r) for above circle, (0,-r) for bottom circle. figure (8) shows the steps of design circular fractal carpet (case 2) with specific number of iterations (N=3). 96

$Figure (8): circular fractal carpets (case 2) Case 3: In this case, four smaller circles is added to the basic model with specific$ $figure (9) shows the steps of design circular fractal carpet (case 3) with specific number of iterations (N=3).$

7 Figure (8): circular fractal carpets (case 2) Case 3: In this case, four smaller circles is added to the basic model with specific radius (r* 0.5) and translation factor (tx, ty) respectively is (r+r*0.5, 0) for right circle, (-r-r*0.5, 0) for left circle, (0,r+r*0.5) for above circle, (0,-r-r*0.5) for bottom circle, and added Rhombic form in the center of each smaller circle. figure (9) shows the steps of design circular fractal carpet (case 3) with specific number of iterations (N=3). Figure (9): circular fractal carpets (case 3) 97

$circle, (-r,-r) for left bottom circle, figure (10) shows the steps of design circular fractal carpet (case 4) with specific number of$

8 Case 4: In this case, four smaller circles is added to the basic model with specific radius (r* 0.5) and translation factor (tx, ty) respectively is (r, r) for right above circle, (-r, r) for left above circle, (r,-r) for right bottom circle, (-r,-r) for left bottom circle, figure (10) shows the steps of design circular fractal carpet (case 4) with specific number of iterations (N=4). Figure (11) shows other examples of circular fractal carpets. Figure (10): circular fractal carpets (case 4) Figure (11): other examples of circular fractal carpets 98

9 5- Conclusion: In this research we design many pattern of fractal carpets, the process is a sample idea (added specific number of models (circles or squares) to basic models), that are entered in recursion function with specific number of iterations. this process depends on familiar mathematical laws. result of which complex compound models, that difficult to configure theirs in other ways. 6- References: [1] Chandra M., Shah S. M., 2011, "Iterative Procedures in Generation Fractal Carpets", International Conference on Computational Intelligence and Communication Systems, India, IEEE computer society, Pages [2] Kaur L., 2000, "Faster Generation of Algebraic Fractals", thesis of partial fulfillment of the requirements for the degree of master of engineering in computer Science, submitted to thapar institute of engineering and technology of deemed university. [3] Salvado M. J., 2011, "Fractals and Compute Graphics ", Applied Mathematics, Linkoping University Electronic Press, 2011, Meritxell Joanpere Salvado. [4] Sukumaran S., 2009, ''Mandel and Julia set Fractal image generation methods with Mathematica", International Journal of Engineering and Technology, VOL. 2 NO. 4, Eashwar Publications. [5] Thollot J., 2010, "Set manipulations of Fractal Objects Using Matrices IFS ", university Claude Bernard, HAL, inria , version 1, pp [6] Zhang Y., You F., 2011, " The E-Learning Environment Development of Triangle Fractal Graphics Programming ", Information & Mechanical Engineering School, Beijing Institute of Graphic Communication, Beijing, Page , China, 2011 IEEE 99

Fractals: Self-Similarity and Fractal Dimension Math 198, Spring 2013

Fractals: Self-Similarity and Fractal Dimension Math 198, Spring 2013 Background Fractal geometry is one of the most important developments in mathematics in the second half of the 20th century. Fractals