# How does fractal geometry work

## Introduction to fractal geometry

### Table of Contents

1 Introduction

1.1. Preface: relevance of the topic and justification for the choice of topic

1.2. Desired goal of the skilled work

1.3. Outline of the technical work

2. What are fractals?

2.1. Definition of terms: " *Fractal* “

2.2. General characteristics of fractals

2.2.1. Creation through iteration

2.2.2. Self-likeness

2.2.3. Fractal dimension

2.2.4. complexity

2.3. Properties explained at the *Koch curve*

3. Historical digression: How were fractals discovered?

3.1. Why were fractals discovered so late?

3.2. Who discovered fractal geometry and how?

3.3. Importance of computers

3.3.1. Advancement in computer technology

4. Generation of fractals

4.1. Required knowledge of mathematical understanding

4.1.1. The imaginary unit ()

4.1.2. The complex plane ()

4.1.3. Iterations in the complex level

4.2. The amount of Mandelbrot

4.2.1. Definition via recursion

4.2.2. Generation in Python

5. Fractals in nature and high technology

5.1. Fractals in nature

5.1.1. biology

5.1.2. geology

5.1.3. The universe

5.2. Fractals in high technology

5.2.1. Fractals as antennas

5.2.2. Application in the representation of natural structures

6. Comparison with classical mathematics and conclusion

7. Appendix

7.1. Bibliography and sources

7.2. Media directory

7.3. Other resources and notes

### 1 Introduction

### 1.1. Preface: relevance of the topic and justification for the choice of topic

Fractals are an integral part of nature. They were invisible to man for centuries, even millennia, although he is surrounded by them everywhere. We owe it to the courage of a few scientists to take a different direction, that we can benefit from this knowledge today. I chose this topic because it creates a connection to nature in me. Fractals are immanent in nature and thus confirm an essential idea that the ancient high cultures already had on our planet, namely "as above, so below", keyword *Self-likeness*. This is a cornerstone of the esoteric worldview, which says that people are one with the world, that everything is one with one another. Ultimately, nature always expresses itself in the language of mathematics and is often puzzling, but the greatest philosophers in the history of the earth, who were also skilled mathematicians at the time, have already recognized that mathematics is the universal language of the universe because they are is generally valid.

In my thesis I dedicate myself to fractal geometry and how it can be expressed in mathematics. It is a fundamental element of the *sacred geometry* and, together with other aspects, forms a whole, which one later realizes that it is basically the same principle over and over again. In the past, I was always fascinated when I intuitively discovered connections between mathematical phenomena and these were later confirmed, such as the connection between the Fibonacci sequence and fractals. Reference should also be made to the golden ratio and the so-called flower of life, which are also other aspects of sacred geometry. The fractal geometry has an extremely positive influence on all aspects of our life, it only makes the universe possible if you will, but more on that later. Finally, it remains to be said that there were many reasons for me to choose this topic, I have been dealing with the topic for a long time and see this specialist work as an opportunity to condense my own knowledge about it.

### 1.2. Desired goal of the skilled work

In this thesis I have decided to give the reader a first impression of fractal geometry, it should be understood as an introduction to fractal geometry. I want to explain what fractals are, how they are created, when and how they were discovered. I would like to try to present clear calculation examples and to put difficult facts in simple words in other places, but to remain exact. My goal is for everyone who reads this paper to understand what fractals are and why it is so important to deal with them.

### 1.3. Outline of the technical work

After much deliberation, I decided to divide my specialist work into five parts. Of these five parts, two parts should be dealt with particularly intensively, namely the part " *Generation of fractals* " and " *Fractals in nature and high technology* ". But first we want to clarify what fractals are and how they were discovered. The last part is a short comparison to classical mathematics, in which I would like to refer to the fact that the mathematicians at the time of the discovery of fractals did not consider them to be equivalent to classical Euclidean mathematics. Finally, I draw up a summary of my work and ultimately try to summarize my results. For this reason I try to present the technical work in such a way that ultimately *the circle closes*.

### 2. What are fractals?

### 2.1. Definition of the term: "fractal"

According to Duden, a fractal is a " *complex geometric structure, similar to that found in nature (e.g. the vein network of the lungs)* “ ^{[i]}. The term "fractal" was coined by Benoît B. Mandelbrot, who broke it from the Latin fractus "broken"^{[ii]}derived. The reason for this is that fractal geometry does not deal with classical Euclidean structures such as circles, straight lines, cubes and others.^{[iii]}but with complex structures that appear broken. B. B. Mandelbrot begins his book “The Fractal Geometry of Nature” by stating that clouds and mountains do not consist of Euclidean bodies.

### 2.2. General characteristics of fractals

Among the general properties of fractals, self-similarity is probably the most frequently mentioned. However, there are other properties that make up a fractal and are directly causally related to self-similarity. In the following, the relationship to iteration, complexity and the fractal dimension is presented or these terms are defined in order to create a uniform definition in the further course of the technical work.

#### 2.2.1. Creation through iteration

Fractals are created by so-called "iterations". Iterating in mathematics means that one repeats the same process on results that have already been obtained.^{[iv]} Usually, iterating processes refer to fairly simple calculation rules that are repeated over and over again. An example:^{[v]}

This formula is intended to show the behavior of 0 under iteration of. In this case I substitute for the number 1, we will come back to later, and for the number 0. The result of the respective function is then substituted for and this infinitely times.

Figure not included in this excerpt

#### 2.2.2. Self-likeness

Fractals are self-similar structures. Something is self-similar when it keeps its shape, i.e. always looks the same or similar when it is enlarged or reduced, as above so below. This important property of fractals is directly related to the iterative creation of fractals. The same arithmetic rule is applied infinitely or several times, which often results in the same or similar result, even when viewing a larger or smaller section. Self-similarity will always play a major role in the following.

#### 2.2.3. Fractal dimension

The definition of the term dimension is not an easy one, so I will now refer first to the topological, straight dimension. In mathematics, a topological dimension is a concept that shows the number of degrees of freedom of a movement in a certain space. A point has the dimension 0 because it cannot expand in any direction. Consequently, the straight line has dimension 1 because it expands in one direction. When looking at a plane it is clear that it spreads in two directions, so you need two coordinates to locate a point on a plane, so the plane has dimension 2. Now you can take the next step and look at a body , It expands in three directions, so you need three coordinates to locate a point in its space, it has dimension 3.

Fractals do not have such a straight dimension, rather they extend over other dimensions, so it can happen that a fractal has the dimension 2.35. Felix Hausdorff laid the foundation stone for the understanding of fractal dimensions, it should indicate how “broken” a plane or a body is. The fractal dimension is between dimension 1 and dimension 3, between straight lines and bodies. If you “break” a figure that is close to dimension 1, i.e. a straight line, more and more - breaks infinitely often - it finally becomes a plane. The same thing happens between the second and third dimensions, the figure merges into space, into the body. It should be noted that the fact that it was impossible to construct natural structures such as clouds or mountains with Euclidean geometry prompted B. B. Mandelbrot to think about the concept of dimension ^{[vi]} and calculated, among other things, the average dimension of clouds: 2.35.

#### 2.2.4. complexity

The complexity of a fractal can hardly be grasped by humans because it is infinite. But its complexity has nothing to do with intricacy. So a fractal is infinitely complex, but what does that mean for us? In contrast to Euclidean geometry, a fractal always remains the same complex and does not adapt, like the circle, which approaches a straight line with constant enlargement. Here again the connection to iteration is given, because in order to generate an ideal fractal it must be iterated infinitely, consequently it is complex, not to say iterative, on a small as well as on a large scale. A consequent phenomenon, which we will deal with later in other contexts, is the infinite length of the fractal, because with increasing iteration, the length of almost every fractal itself also increases. Finally, it should be noted that the complexity is particularly evident when you change the starting conditions for the generation of fractals. Because the results are amazing. A tiny change in the starting conditions leads to an unexpected change in the overall picture, i.e. the fractal. This effect, also known as the “butterfly effect”, is an important aspect to link fractal geometry with chaos theory **(** DYS **)**.

### 2.3. Properties explained on the Koch curve

Figure not included in this excerpt

The Koch curve was introduced in 1904 by the Swedish mathematician Helge Koch and is thus one of the first formally presented fractals, although not named as such. It is a good illustration of all the properties described above.

You start with a straight line on which you draw a triangle and remove the part of the straight line that is below the triangle. Now, instead of one straight line, you have four straight lines. That was the first iteration. Next you do the same thing again, with every straight line, infinitely many times. After each iteration it is longer than before. In the end we have a curve the length of which cannot be measured, it appears finite to the eye, but mathematically it is infinitely long. We will come back to this phenomenon later (see points 5.1.2. And 5.2.1.), Because now it becomes clear that the infinite length is related to the fractal dimension. Also, when looking at **M1** clearly, it is at least implied that the Koch curve always looks the same when enlarged, so it is self-similar due to the always the same iteration. This property becomes clearer with each iteration and can be proven by the fact that you cannot create a tangent at any point on the Koch curve. So it remains to be stated that the Koch curve is a complex geometrical object which fulfills all the properties of a fractal. At the time of its discovery it was called the "pathological curve" because it could not be explained with Euclidean geometry (NOVA).

### 3. Historical digression: How were fractals discovered?

### 3.1. Why were fractals discovered so late?

Fractals have always been there, they are inherent in nature. To say that fractals were discovered late is perhaps a bit of a stretch. Like humans, until a few centuries ago they still lived in full harmony with nature and naturally perceived this fundamental property of nature. The Japanese painter Katsushika Hokusai has also drawn fractals **(M2)**. For example, he drew clouds or waves on which there were waves again. But why did this fundamental understanding only return to our consciousness in the early 20th century?

Figure not included in this excerpt

**M 2**: The Great Wave off Kanagawa (ca.1830)

According to Newton, all bodies were, so to speak, clamped into a clockwork in which everything could be explained with the existing laws of physics, mathematics and the other sciences. Smooth surfaces, straight lines, pyramids, icosahedra, basically smooth structures, as Euclidean geometry knew it, belonged in this worldview. It was only at the beginning of the 20th that this ingrained thinking was slowly broken open. At the beginning of the last century mathematicians had problems with so-called "mathematical monsters". The German-born Georg Cantor (1845 - 1918), who was born in Saint Petersburg, presented the so-called Cantor crowd, which represents such a "mathematical monster". You take a straight line, remove the middle third and do the same with the resulting straight lines, you perform this iteration infinitely times and you should think that nothing remains in the end, but in the end you actually have an infinite number of straight lines.

In the period that followed, more and more mathematicians were concerned with these "mathematical monsters" because computer technology made this possible. B. B. Mandelbrot was a pioneer in understanding fractal geometry. And he was also attacked, it was claimed that fractal geometry was not real mathematics and was just stupid things from a stupid adding machine. But the opposite was quickly proven, Mandelbrot brought with it a whole new understanding of the world from which we benefit in many ways. Many took on Mandelbrot's ideas and to this day a lot of useful inventions have been realized thanks to fractal geometry (NOVA).

**[...]**

^{[i]} www.duden.de/rechtschreibung/Fraktal, as of January 10, 2015

^{[ii]} www.frag-caesar.de/lateinwoerterbuch/fractus-uebersetzung.html

^{[iii]} Benoît B. Mandelbrot: The fractal geometry of nature, Birkhäuser 1987

^{[iv]} www.duden.de/rechtschreibung/Iteration, as of January 10, 2015

^{[v]} www.youtube.com/watch?v=NGMRB4O922I, Dr Holly Krieger from MIT

^{[vi]} www.wissensnavigator.com/interface4/management/endo-management/buch/hab233.pdf

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