Fractal geometry and its analysis By Muhammad Ajmal M. Phil. Thesis Session 2013-2015 DEPARTMENT OF MATHEMATICS The Islamia University of Bahawalpur Bahawalpur, Pakistan 2015 Fractal geometry and its analysis By Muhammad Ajmal Supervised by Dr. Muhammad Mustahsan DEPARTMENT OF MATHEMATICS The Islamia University of Bahawalpur Bahawalpur, Pakistan 2015 Abstract In this thesis, we used the two quaternary interpolating subdivision schemes which are modified using the rotation matrix in which the points are translated along the vector that rotates with the angle(0, ) . Using these modified schemes, we generate the fractal curves which are based on rotating vector. Moreover, we studied the convergence of the schemes. The resulting curves are non-rectifiable and curves are continuous but nowhere differentiable. Contents 1 Introduction 4 1.1 History of Fractals . 4 1.2 Definition of Fractal . 5 1.2.1 Fractal Properties . 6 1.3 Some Fractals in Nature . 6 1.3.1 The Sierpinski Triangle . 6 1.3.2 VonKochCurve ......................... 9 1.3.3 Other Example of Koch-Curve . 11 1.3.4 The Hilbert Curve . 14 1.3.5 TheL-System........................... 15 1.4 Applications of Fractals . 16 1.4.1 Tessellation by Von Koch Curves . 16 1.4.2 Space-Filling Fractal . 18 1.4.3 Reproduction of Realistic Images . 19 1.5 Generation of Fractals . 23 2 Literature Survey 24 1 2 2.1 Introduction................................ 24 2.2 A General p-ary Interpolatory Subdivision Scheme . 25 2.3 Different Fractal Curves Generate By Subdivision Scheme . 27 2.3.1 Generation of Fractal Curves by Binary Interpolatory Subdivi- sionScheme............................ 27 2.3.2 A Ternary Interpolatory Subdivision Scheme to Generate Frac- talCurves............................. 28 2.3.3 A Quaternary Subdivision Scheme to Generate Fractal Curves 30 2.3.4 Generation of Fractal Curves by Using the 6-ary Interpolatory Subdivision Scheme . 32 2.3.5 Generation of Curves by Using the 7-ary Interpolatory Subdi- visionScheme........................... 35 2.4 Fractal Properties . 37 2.4.1 Convergent Property . 37 2.4.2 Non-Rectifiable Property . 37 2.4.3 Continuous but Nowhere Differentiable Property . 37 3 Subdivision Scheme to Generate Fractal-Type Curves Based on Ro- tating Vector 38 3.1 Introduction................................ 38 3.2 Quaternary Interpolatory Subdivision Schemes Based On Rotating Vec- tor..................................... 39 3 3.2.1 A Quaternary Interpolatory Subdivision Scheme with Single Parameter............................. 39 3.2.2 A Quaternary Interpolatory Subdivision Scheme with More Than One Parameters Based on Arbitrary Rotating Vector at Angle (0, ) ............................. 44 2 3.3 Convergence Property . 46 3.4 Fractal Properties . 49 3.4.1 Non-Rectifiable Property . 49 3.4.2 Continuous but Nowhere Differentiable Property . 50 3.5 Conclusion................................. 50 Chapter 1 Introduction 1.1 History of Fractals When future a new ground, we investigate its history. The history of field give much information about the field, and specially right in mathematics field, in which all new consequences are elaborated in terms of previous form. The name fractal was invented by Mandelbrot [Barnsley, M. F. 1985] and derived from the Latin word fractus, which means broken in pieces. In the early years of nineteen century and twentieth century, the concentration on this field was extended and a large amount of literature was printed about the fractal geometry. The work of Benoît Mandelbrot’s so much in 1970s, in the starting the figures of fractal never possible to draw. As the progression of computers in science, generation of fractal figures are possible by computers. In 1904, Helge Von Koch who was the Swedish mathematician printed an article on continuous curve, at any point of this continuous curve tangent cannot be draw. 4 5 These continuous curve constructed by very simple geometry. [H. Koch. 1904] Cesàro instantaneously accepted this geometrical curve as being self-similar, and did a lot effort on the self-similar curves. The effort of Cesàro was taken further in a 1938 printed by Paul Lévy, he introduces innovative, more universal and self-similar curves. The fractal set’sproperties results proved by different authors in 1940’sand 1950’s. The work of Benoît Mandelbrot on theory of self-similar sets in 1960’s. Benoît Mandel- brot also says self similarity and fractional dimensions not used only in mathematics field, it also used in the further branches of science. In 1975, Mandelbrot introduced the word Fractal, [Benoît B. Mandelbrot 1975]. He also said that self similar things rarely found in nature, but that a numerical form of self-similarity is everywhere. By the use of computers complete it to probable to do remarkable bright image of the sets, in starting this theory existed only in articles or papers. In 1980’s and 1990’sstudy of self-similar in fractal geometry became major discipline of study. The field of geometry of fractal is young and flourishing, and we can safely spec- ulate that the field will keep on growing up for a lot of years, with new sub-field and discoveries 1.2 Definition of Fractal "Describe a fractal as a irregular or fragmented geometric figure that can be subdi- vided in parts, each of which is ( approximately same) a reduced size duplicate of the whole. Fractals are usually self-similar and free of scale". 6 1.2.1 Fractal Properties Fractal curves have the following properties. a) Its parts have the same form or construction as the entire, except that they are at a several scale and may be a little deformed. b) Its form is tremendously rough or fragmented, and remains so, whatever the scale of inspection. c) It contains "distinct elements" whose scales are very various and cover a huge range. d) Formation by iteration, the fractals are formed by the iterative means. From each progressive iterative, the fractals become more dense with multiple coppices of a single stencil. 1.3 Some Fractals in Nature In this section, we discuss about the some fractals in nature and their properties. 1.3.1 The Sierpinski Triangle The Polish mathematician Waclaw Sierpinski introduce the Sierpinski’sTriangle and also described some properties in 1916. By joining the center points of each side of triangle, generate new four triangles and vanish triangle which place at the cen- ter, these triangles are equilateral triangles. Sierpinski’striangle can be generate by repeating the same procedure infinite number of time. 7 Construction of Sierpinski Triangle Consider a equilateral triangle as an example, join the middle points of the sides of triangle and removing at the center. In this way there leave three triangles, again join the midpoints of the edges of the triangles and removing the middle one. We get the Sierpinski triangle. The procedure can elaborated in Fig. 1.1, Fig. 1.2, Fig. 1.3, Fig. 1.4 and Fig. 1.5. Fig. 1.1 A black equilateral triangle Fig. 1.2 At first level Fig. 1.3 At second level Fig. 1.4 After repeating the same procedure on three black triangles 8 Fig. 1.5 After certain number of iterations Area of Sierpinski Triangle As, removed the one quarter area of triangle at each iteration. After first iteration, three quarters area of the actual triangle is left.Thus the area of triangle after n iteration steps will be (0.75)n times the area of the actual triangle. Number of Triangles at Different Level After observing the number of triangles generated for different iterations level are tabulated as follows: Level of iteration Generated Triangles 1 1 2 4 3 13 4 40 5 121 From the above table, we can drive the general formula to calculate the number 9 of triangles being removed for any iteration. n 1 At nth iteration, the number of triangles being removed, N = 3i i=0 P 1.3.2 Von Koch Curve A very simple geometric method is used to generate the Von Koch curve. Von Koch curve generated by intersecting the line-segment into three equal size pieces and replace the center piece with two the equal size of line segment at angle 60 and 120. Very simple example of fractal curve is Von Koch curve, it follows the following construction rules. Construction of Koch Curve Taking a line-segment shows in Fig. 1.6, the line segment is divided into 3 equal parts, and the intermediate part is substitute by two linear edges at angles 60o and 120o Fig 1.7 shows the Von Koch curve after one iteration. The Von Koch curve after two, three and four iterations are shown in Fig. 1.8, Fig. 1.9 and Fig. 1.10 respectively. Fig. 1.6 A line-segment at zeroth level Fig. 1.7 Koch curve after one iteration 10 Fig. Koch curve after two iterations Fig. 1.9 Koch curve after three iterations Fig. 1.10 Koch curve after after four iterations Properties of Koch-Curve The Von Koch Curve shows the self-similarity of fractal. The similar pattern shows all over the place along the curve in different size, from visible to insignificant. In an ideal world the iteration method should go on indefinitely; On the other hand, in performance, the curve displayed on the screen no longer changes when the elementary part becomes less than the pitch. And the iteration can be stopped as well. Total Length of the Koch-Curve In each iteration each line segment would be divided into three equal edges of same 4 length. In each iteration the length of the size is multiplied by 3 times. By the rule 11 of iteration, at the nth iteration we can deduce a formula for the total length of the Koch Edges 4 n At nth iteration, the total length of Koch edges L = ( 3 ) Total Number of the Koch Edge After each iteration, we observed that every edge divided into four equal segments 1 with length 3 of its original length.