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The Development of 3D Fractal Shapes for Mandelbrot and Julia Sets

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The Development of 3D Fractal Shapes for Mandelbrot and Julia Sets International Journal of Pure and Applied Mathematics Volume 118 No. 15 2018, 233-239 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu THE DEVELOPMENT OF 3D FRACTAL SHAPES FOR MANDELBROT AND JULIA SETS Bulusu Rama1, K Sai Prasad2, N Chandrasekhar Reddy3,I Surya Prabha4,J Phani Prasad5 1,2,3Department of CSE, M L R Institute of Technology, Hyderabad, India 4Department of IT, Institute of Aeronautical Engineering, Hyderabad, India 5Department of CSE, Vardhaman College of Engineering, Hyderabad, India [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: Fractals are a means of giving us new Keywords: Fractals, 3-Dimensional Images, concept/process for creation of 3D images of real-world Mandelbrot Set, Julia Set, Recursion, Affine objects making use of computational modeling Transformations, 3D Rendering, IFS algorithms which depend on the parameters of self- similarity, scale invariance and dimensionality. 1. Introduction Coastlines, clouds, mountains, random shapes containing curves, sets of curves etc. provide a diversified spectrum A fractal is a rough geometric shape broken into pieces of fractals utilization in areas varying from multi- truly be subdivided into parts of which is (at minimum colored, multi-patterned fractal landscapes of natural approximately) a smaller size copy of the whole i.e. self- geographic entities, image compression extending to similar analogous to the original shape. The term even modeling of molecular ecosystems. Fractal originated from the Latin word “Fractus” meaning geometry gives the underlying support or foundation for “broken” or “fractional” invented by Benoit Mandelbrot modeling the infinite detail available in nature. Fractals in 1975.Fractals exhibit the properties of self-similarity, consist of their scale down, rotate and skew exact copies scale invariance and common irregularity in shape firmly placed in them. Several kinds of fractals have thereby having a remarkable detail even after been developed since their origin which are of public magnification - the greater the magnification, the more interest. This paper describes the creation of two famous the detail. Mostly, fractals can be created recursively by types of fractals, namely the Mandelbrot Set and Julia a repeating pattern. Natural fractals have characteristics Set, the3D rendering of the same provides a real-world of statistical self-similarity while regular fractals such as focus and touch in the world of fractal images. Sierpinski Gasket, Cantor Set or Koch Curve exhibit AbstractFractals are a means of giving us new exact self-similarity. concept/process for creation of 3D images of real-world objects making use of computational modeling This paper gives the creation of two best-known fractals algorithms which depend on the parameters of self- – the Mandelbrot Set and the Julia Set by making use of similarity, scale invariance and dimensionality. the deterministic method of IFS (Iterated Function Coastlines, clouds, mountains, random shapes containing Systems) and affine transformations. The rendering was curves, sets of curves etc. provide a diversified spectrum performed in 3D, the program implemented in C++ and of fractals utilization in areas varying from multi- the displayed output, taking into account multiple test coloured, multi-patterned fractal landscapes of natural cases varied by number of iterations along with a given geographic entities, image compression extending to parameter pertaining to a coefficient value, are even modelling of molecular ecosystems. Fractal presented. Finally, the conclusion is given at the end. geometry gives the underlying support or foundation for modeling the infinite detail available innature. FractalsI. About Mandelbrot And Julia Sets consist of their scale down, rotate and skew exact copies firmly placed in them.Several kinds of fractals have been An abbreviated note of Mandelbrot and Julia Sets and developed since their origin which are of public their correspondingproperties based on iterative function interest.This paper describes the creation of two famous systems is outlined in the sub-sections that follow. An types of fractals, namely the Mandelbrot Set andJulia elaborate account of the Mandelbrot and Julia Sets can Set, the3D rendering of the same provides a real-world be got in the books as in [2], [7] and in the research focus and touch in the world of fractal images. project compilation as [4]. 233 International Journal of Pure and Applied Mathematics Special Issue A. The Mandelbrot Set in the parameter space or the c- plane, whereas the Julia set is in the dynamical space or the z-plane. The Mandelbrot was formulated by the French mathematician Benoit Mandelbrot in 1979 when he was An Iterated Function System (IFS) devised on a carrying out study on the simple equation z = z2 + c. coefficient c and the maximum number of iterations, is iterated the number of times as the sum total of In this equation, z and c denote complex numbers. iterations. The accompanying set of points spread an Accordingly,it can be said that, the Mandelbrot set is the undetermined volume of space which is a function of the set of all such complex numbers c, that iterating z = z2+ total number of iterations. For any arbitrary point z from cdoes not diverge. Thus, it is a connected set of points, this set, its location can be either be inside or outside the which is bounded. generated area, depending on the value of c and the co- ordinate-axes range. The Julia set consists of all such An Iterated Function System (IFS) based on the total points z, each of which are outside the bounded space number of iterations and a starting specified area before the IFS was applied. Administering the set of indicating thelower and upper limits of the bounded affine transformations on the starting set of points, space, is iterated the number of times as the sum total of iterating each step, the upcoming fractal is a self-similar iterations. The accompanying set of points spread an shape resembling an approximation to the original undetermined volume of space which is a function of the image. This result is best visualized when rendered in total number of iterations. Then the collection of all 3D. points for which the line spacing from the origin to that point in every coordinate direction is zero comprises the 1) Relationship between Mandelbrot and Julia Sets: Mandelbrot Set. Administering the set of affine transformations on the starting set of points, iterating Mandelbrot set is a one-page dictionary of Julia sets. It is each step, the upcoming fractal is a self-similar shape so because if we enlarge the Mandelbrot set. resembling an approximation to the original image. This Sufficiently at any given point we obtain something that result is best visualized when rendered in 3D. looks very much like the Julia set at that point. Let us see what that means mathematically. Suppose we have a In the process of creation of the Mandelbrot Set Julia set for which we set the value of c to w. Consider graphically, the computer screen turns out to be the the point of the Julia set at which z0 is also w. We then complex plane. All the points on the plane are tested into have the equation z = z2+c. If the iterated z stayed inside the z0 = w given boundary forever, i.e., is convergent, the pointed is z1 = w2 + w plotted black.Otherwise it is divergent and the point is plotted with the color corresponding to how quickly it Now let us look at the Mandelbrot set at the point escaped.The initial value of z is always zero when trying where c is w. We have to know if a point in a plane is part of the set. This is in consequence with the hypothesis that zero is the critical Z0 = w point of the equation used to generate the set. Z1= w2 +w Thus, the result of iterating is the same for both the Julia B. The JuliaSet and Mandelbrot sets at this particular point. Now let us move from this point by a distance of d, where d is also a The Julia set was formulated by the French complex number. For the Julia set, we then have mathematician Gaston Julia in 1918 when exploring the iteration of polynomials and relational functions. It is in Z0 = w + d close relation to the Mandelbrot setobtained by iterating Z1 = (w+d)2 + w 2 the equationz = z + c. The principal distinction between And for the Mandelbrot set, we then have the Julia set and the Mandelbrot set is the ways in which the function is iterated. The Mandelbrot set iterates as Z0 = w + d 2 per z = z + c with z always starting at 0 and varying the Z1 = (w+d)2 + w+d 2 c value. The Julia set iterates as per z = z + c, where c is constant and z is variable. That is, the Mandelbrot set is We see that if d gets smaller and smaller, the result of iterating again and again is the same for the two sets, 234 International Journal of Pure and Applied Mathematics Special Issue but as d increases, the results diverge more and more. the value exceeding maxit, to make certain that the Furthermore, d can be varied so that the result (w+d) generated Mandelbrot set doesn’t blow up into an represents, in turn, every point on the screen. We then infinite space on the screen. observe that at every point on the screen, the iteration begins in the same way for both the sets, but diverges 5. The program runs as a Windows console application more and more as the distance d increases. Hence, for with the 3D figure of the Mandelbrot set being rendered very large magnifications, the pictures of the Mandelbrot as a color-mapped picture projected onto the 3D plane, and Julia should be the same.
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