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Home , Fractal curve, Gosper curve, Hilbert curve

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 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 ...... 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- ...... 11

1.3.4 The ...... 14

1.3.5 TheL-System...... 15

1.4 Applications of Fractals ...... 16

1.4.1 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

θ (0, π) ...... 44 ∈ 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, who was the Swedish mathematician printed an article on continuous curve, at any 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 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. Hence, after n iterations, the total number of

Koch-edges will be 4n.

1.3.3 Other Example of Koch-Curve

Koch Snowflake

We can apply the Von Koch method on the three sides of an equilateral triangle, shows in the following Fig. 1.11 after the different iteration steps.

Fig. 1.11 Koch-curve on equilateral triangle after one, two and three iterations 12

Fig. 1.12 Koch-curve on equilateral after the many iterations.

The Koch curve on equilateral triangle shows in Fig. 1.12 is also called Snowflake curve.

Koch-Star

We can apply the same iteration process on any polygon instead of equilateral triangle.

As, we choose the hexagram as an example. This time round, we include the inner sides of the hexagram. At each iteration, new small hexagrams are generated at the six vertices of the original hexagram.

Fig. 1.13 Koch star 13

Fig. 1.14 Koch star after one iteration

Fig. 1.15 Koch star after two iterations

Fig. 1.16 Koch star after three iterations 14

Fig. 1.13 shows at zeroth level of Koch star curve, Fig. 1.14, Fig. 1.15 and Fig.

1.16 are shows resultant after one, two and three iterations of Koch star respectively.

1.3.4 The Hilbert Curve

David Hilbert who was first introduce the Hilbert Curve. This curve is known as a space-filling curve, as it will cover the whole space after certain number of iterations.

Construction of Hilbert Curve

1. Start with the basic staple-like shape as shows in Fig. 1.17.

Fig. 1.17 Hilbert curve after one iteration

2. Fig. 1.18 Get smaller the preceding curve to half its size. Simultaneously, reduce the grid size by the factor of two. Adjusting four copies of the curve on the grid. The lower two must be placed directly as they are. The upper two must be rotated a quarter turn - one left, another right. Finally, join the four pieces with little straight segments to gain the next step curve. Sometimes the joint segments are horizontal, and sometimes they are vertical. 15

Fig. 1.18 Hilbert curve after two iterations

3. All the rest of the curves are formed consecutively one from another using the

similar algorithm shows in Fig. 1.19

Fig. 1.19 Hilbert curve after three iterations.

1.3.5 The L-System

Unlike Von Koch Curve and the Sierpinski Triangle in the method of constructing the

Hilbert Curve, the identical staple-like figure gets shrink and convert into other places.

Some of them are rotated by 90 degree as well. This algorithm is called Lindenmayer

System (L-system). This is a string rewriting system that is particularly used to generate fractal with dimension between 1 and 2. After conversion, the curve has to be connected by introducing some line-segments. That is why, after certain iteration, the curve will finally cover up the whole surface. 16

Fig.1.20 Different L-system diagrams after different level of iteratons

Length of the Curve

After each iteration, the fractal becomes 2 time long. After n iterations the length of the curve will increase up to 2n time long.

1.4 Applications of Fractals

From several views, hundreds of applications of fractal are presented, such as produc- ing fractal music, generating computer aided mammography, creating realistic image and etc. Here, we are discuss some practical applications of fractal.

1.4.1 Tessellation by Von Koch Curves

Tessellation by Koch Snowflake curves is very nice tile style. Here, Koch Snowflake curves are use two different size in the ratio 1:3 to tile in the plane. By tile from these different size of Koch Snowflake, we get a nice image called "Mandelbrot" as shown in the Fig. 1.21. 17

Fig. 1.21 Mandelbrot image

By the similar rule, Koch star generate the new tessellation style. Now we consider the hexagon Koch Star and tessellate these hexagons on all over the surface which shows in Fig. 1.22.

Fig. 1.22 Tile by Hexagon Koch Star image

And this type of tessellation may be usually found in ancient Chinese window

lattice. Here, by using pattern shows in Fig. 1.22 and can generate an ideal Chinese

window lattice shows in Fig. 1.23 18

Fig. 1.23 Chinese window lattice

1.4.2 Space-Filling Fractal

The fractals continuously attempts to fill in the empty space any holes as it goes along its iteration. This type of fractal, call space filling fractals. The good example of space filling fractal is Hilbert Curve. is also space filling fractal. The

Hilbert curve fill a square shape plane and Peano curve fill a rhombus shape plane. In

Fig. 1.24 first two images are by Hilbert Curve and last two images by Peano Curve.

Fig. 1.24 Hilbert images and Peano images 19

Other examples of space filling fractal are Peano- and which shows in Fig. 1.25

Fig. 1.25 Peanno-Gosper image and Dragon image.

1.4.3 Reproduction of Realistic Images

Reproduce of natural images is a very important application of fractal like as moun- tains, trees, clouds and etc. The plants, are much complex and exhibit some self- similarity. The self-similarity properties and complexity of fractals allows to repro- duce a large set of real images of the world. Such as leaf of tree shows in Fig. 1.26 and Fig. 1.27

Fig. 1.26 Leaf of tree 20

Fig. 1.27 Another leaf

For further explore how random fractor affects the process of iteration. Here, by

Geometer’s Sketchped we did an experiment on fractal known as Binary Tree. The process goes like as shown in Fig. 1.28, Fig. 1.29, Fig. 1.30 and Fig. 1.31.

Fig. 1.28 At first level 21

Fig. 1.29 At second level

Fig. 1.30 At third level 22

Fig. 1.31 At fourth level

From the above Fig. 1.28, Fig. 1.29, Fig. 1.30 and Fig. 1.31, regularly grow the

Binary Tree. The ratio of bough BC and the ratio of stem AB remain constant at all iterations. The angle of bending of boughs is also remain same. Random binary tree can also be generated shown in Fig. 1.32 and more changing will generate the skeleton tree structure which like as real tree shows in Fig. 1.33

Fig. 1.32 Random Binary Tree 23

Fig. 1.33 Like as real tree image

1.5 Generation of Fractals

Now a days, there are many tools to generate the fractals. The computer aided geometric design is the one method to generate the fractals. Some complex valued function especially the solution of variational method of solving the ODE can also generate the fractals 50

1 π special case take a = 3 and θ = 2 .The sum of length of the all edges on each iteration in the following table

Iteration Sum of Length

1 1.7926

2 3.4251

3 5.6018

4 8.9948

5 14.0642

The above numerical results clearly shows that the sum of all small edges tends to infinity as the iterations tends to infinity.Hence curve generated by (3.1) is non- rectifiable.

3.4.2 Continuous but Nowhere Differentiable Property Theorem 9 Generated fractal-type curve by subdivision scheme (3.1) is continuous but nowhere differentiable.

Proof. By the theorem 6, we notice that the curve is continuous. Here, we shall show that the curve is nowhere differentiable. For this, we know that each curve generated by (3.1) makes number of corner points at the whole fractal-type curve.

Since function is not differentiable at the corner points. So generated fractal-type curve by scheme (3.1) is continuous but nowhere differentiable.

3.5 Conclusion

This chapter deals the two under consideration subdivision schemes to generate the fractal-type curve based on a vector which makes an angle 0 < θ < π as modified 51 scheme of [Hongchan ZHENG 2012]. These fractals are defined on the polygonal domain and depends on one and two parameters along with an angle θ which deter- mines the direction of the rotating vector. The fractals are continuous and nowhere differentiable. We also proved that these fractals are non-rectifiable.