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MANDELBROT and JULIA SET FRACTALS Bulusu Rama1, K Sai

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MANDELBROT and JULIA SET FRACTALS Bulusu Rama1, K Sai International Journal of Pure and Applied Mathematics Volume 119 No. 14 2018, 203-209 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu 3D DEVELOPMENT - MANDELBROT AND JULIA SET FRACTALS 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 Information Technology, Institute of Aeronautical Engineering, Hyderabad, India 5Department of CSE, Vardhaman College of Engineering, Hyderabad, India Sierpinski Gasket, Cantor Set or Koch Curve exhibit Abstract: exact self-similarity. Fractals are a means of giving us new concept/process This paper gives the creation of two best-known fractals for creation of 3D images of real-world objects making use of computational modeling algorithms which depend – the Mandelbrot Set and the Julia Set by making use of on the parameters of self-similarity, scale invariance and the deterministic method of IFS (Iterated Function dimensionality. Coastlines, clouds, mountains, random Systems) and affine transformations. The rendering was shapes containing curves, sets of curves etc. provide a performed in 3D, the program implemented in C++ and diversified spectrum of fractals utilization in areas the displayed output, taking into account multiple test varying from multi-colored, multi-patterned fractal cases varied by number of iterations along with a given landscapes of natural geographic entities, image parameter pertaining to a coefficient value, are compression extending to even modeling of molecular presented. Finally, the conclusion is given at the end. ecosystems. Fractal geometry gives the underlying support or foundation for modeling the infinite II.detail About Mandelbrot And Julia Sets available in nature. Fractals consist of their scale down, rotate and skew exact copies firmly placed in them. An abbreviated note of Mandelbrot and Julia Sets and their corresponding properties based on iterative Several kinds of fractals have been developed since their origin which are of public interest. This paper describes function systems is outlined in the sub-sections that the creation of two famous types of fractals, namely the follow. An elaborate account of the Mandelbrot and Julia Mandelbrot Set and Julia Set, the3D rendering of the Sets can be got in the books as in [2], [7] and in the same provides a real-world focus and touch in the world research project compilation as [4]. concept/processAbstractFractalsof fractal images. forare creationa means ofof 3Dgiving images us new of real-world A. The Mandelbrot Set Keywords: Fractals, 3-Dimensional Images, Mandelbrot Set, Julia Set, Recursion, Affine Transformations, 3D The Mandelbrot was formulated by the French Rendering, IFS mathematician Benoit Mandelbrot in 1979 when he was I. Introduction carrying out study on the simple equation z = z2 + c. In this equation, z and c denote complex numbers. A fractal is a rough geometric shape broken into pieces Accordingly, it can be said that, the Mandelbrot set is the truly be subdivided into parts of which is (at minimum set of all such complex numbers c, that iterating z = z2+ approximately) a smaller size copy of the whole i.e. self- c does not diverge. Thus, it is a connected set of points, similar analogous to the original shape. The term which is bounded. originated from the Latin word “Fractus” meaning “broken” or “fractional” invented by Benoit Mandelbrot An Iterated Function System (IFS) based on the total in 1975.Fractals exhibit the properties of self-similarity, number of iterations and a starting specified area scale invariance and common irregularity in shape indicating the lower and upper limits of the bounded thereby having a remarkable detail even after space, is iterated the number of times as the sum total of magnification - the greater the magnification, the more iterations. The accompanying set of points spread an the detail. Mostly, fractals can be created recursively by undetermined volume of space which is a function of the a repeating pattern. Natural fractals have characteristics total number of iterations. Then the collection of all of statistical self-similarity while regular fractals such as points for which the line spacing from the origin to that 203 International Journal of Pure and Applied Mathematics Special Issue point in every coordinate direction is zero comprises the image. This result is best visualized when rendered in Mandelbrot Set. Administering the set of affine 3D. transformations on the starting set of points, iterating 1) Relationship between Mandelbrot and Julia Sets: each step, the upcoming fractal is a self-similar shape resembling an approximation to the original image. This Mandelbrot set is a one-page dictionary of Julia sets. It result is best visualized when rendered in 3D. is so because if we enlarge the Mandelbrot set. Sufficiently at any given point we obtain something that In the process of creation of the Mandelbrot Set looks very much like the Julia set at that point. Let us see graphically, the computer screen turns out to be the what that means mathematically. Suppose we have a complex plane. All the points on the plane are tested into Julia set for which we set the value of c to w. Consider the equation z = z2+c. If the iterated z stayed inside the the point of the Julia set at which z0 is also w. given boundary forever, i.e., is convergent, the pointed is plotted black. Otherwise it is divergent and the point is We then have plotted with the color corresponding to how quickly it z0 = w escaped. The initial value of z is always zero when z1 = w2 + w trying to know if a point in a plane is part of the set. This Now let us look at the Mandelbrot set at the point where is in consequence with the hypothesis that zero is the c is w. We have critical point of the equation used to generate the set. Z0 = w B. The Julia Set Z1= w2 +w The Julia set was formulated by the French Thus, the result of iterating is the same for both the Julia mathematician Gaston Julia in 1918 when exploring the and Mandelbrot sets at this particular point. Now let us iteration of polynomials and relational functions. It is in move from this point by a distance of d, where d is also a close relation to the Mandelbrot set obtained by iterating complex number. For the Julia set, we then have the equation z = z2+ c. The principal distinction between the Julia set and the Mandelbrot set is the ways in which Z0 = w + d the function is iterated. The Mandelbrot set iterates as Z1 = (w+d)2 + w 2 per z = z + c with z always starting at 0 and varying the And for the Mandelbrot set, we then have 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 Z0 = w + d in the parameter space or the c- plane, whereas the Julia Z1 = (w+d)2 + w+d set is in the dynamical space or the z-plane. We see that if d gets smaller and smaller, the result of An Iterated Function System (IFS) devised on a iterating again and again is the same for the two sets, but coefficient c and the maximum number of iterations, is as d increases, the results diverge more and more. iterated the number of times as the sum total of Furthermore, d can be varied so that the result (w+d) iterations. The accompanying set of points spread an represents, in turn, every point on the screen. We then observe that at every point on the screen, the iteration undetermined volume of space which is a function of the begins in the same way for both the sets, but diverges total number of iterations. For any arbitrary point z from more and more as the distance d increases. Hence, for this set, its location can be either be inside or outside the very large magnifications, the pictures of the Mandelbrot generated area, depending on the value of c and the co- and Julia should be the same. ordinate-axes range. The Julia set consists of all such points z, each of which are outside the bounded space before the IFS was applied. Administering the set of affine transformations on the starting set of points, iterating each step, the upcoming fractal is a self-similar shape resembling an approximation to the original 204 International Journal of Pure and Applied Mathematics Special Issue III. Methods to generate MANDELBROT AND 5. The program runs as a Windows console application JULIA SETS with the 3D figure of the Mandelbrot set being rendered as a color-mapped picture projected onto the 3D plane, A step-by-step explanation the methods to generate the taking into use MATHLAB 3D Image Rendering Mandelbrot and Julia Sets are given in the sub-sections software. that follow. A. Method for creating Mandelbrot Set The utilization of OpenGL algorithms and the Standard Standard Graphics library makes the application of the The steps used in the development of the Mandelbrot Set routines for construction of the starting grid and the are as follows: upcoming Mandelbrot set as also the color-coding and 3D rendering very flexible and efficient. More 1.For a likely maximum quantity of iterations denoted by descriptions and other techniques to generate the 3D the constant maxit, and a co-efficient parameter denoted Mandelbrot Sets are available in the book as in [1] and by c, devise the set specified by the (min, max ) stretch the publications [3]- [6]. of values of every point in the area given by (Xmin, Xmax), (Ymin, Ymax).
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