How do I get to fractal analytics

Fundamentals of fractal geometry with iterated function systems (IFS)


On this website you will find my treatise on "Fractal Geometry". This work was created several years ago during the 12th year of my upper school level as part of a so-called "special learning achievement". The aim was to provide an introduction to this extremely interesting subject that could also be understood by laypeople. Since the work was rated very positively, I decided to publish it here on my website. Only recently - after more than six years - did I have time to revise the work again. Some errors have been corrected and the text has been expanded in several places.

I hope you enjoy reading it and I hope that you will enjoy this very exciting topic.

Adrian Jablonski, July 2017

Here you can download the IFS generator software mentioned in the text

Special learning achievement in mathematics / computer science
Corrected and revised: July 15, 2017
Original version: March 28, 2011

Table of Contents

1 Introduction

Fractal geometry is a relatively new branch of mathematics. It deals with geometric objects, the so-called fractals, the properties of which differ fundamentally from those of "classical" geometry. The most important feature of fractals is the scale invariance, which means that you can see details at every level of magnification, regardless of how deeply you look into it Object penetrates. If, on the other hand, the edge of a "classic" object, such as the circle, is enlarged, then with increasing enlargement it resembles more and more a simple straight line. Such objects are therefore referred to as smooth. With a fractal, however, you will never be able to recognize a straight line, but rather more and more subtleties of the object. Hence the name "fractal", derived from the Latin "fractus" for "broken", i.e. littered with innumerable details. Such objects had been known since the beginning of the 20th century, but their fundamental importance was only recognized from around 1970. Previously, these objects were called "mathematical monsters" because, as will be explained below, they have paradoxical properties that appear more or less "incomprehensible" to the human mind. This only changed with the work of the mathematician Benoît Mandelbrot. He realized that one could do something completely new with fractals, something that was considered practically mathematically impossible up to that time: the modeling and description of "irregular" objects of nature, especially the animate ones, which one assumed could not be geometrical to be discribed.
In this special learning achievement, the "classic" fractals of the 20th century are first dealt with in order to explain the basic concepts of fractal geometry on the basis of these. Then the so-called iterated function systems (IFS) are presented, a powerful method for coding and generating fractals. The exact definition and its use for modeling and representing natural structures will be discussed. In order to be able to explain the theory of fractals clearly, this work was illustrated with numerous pictures, most of which were created by the author himself. While this increases the scope of the main part, it is almost impossible to explain geometry without using images.
As part of this work, a computer program was also created that implements the functionality of the IFS and makes it clearly understandable. In addition, there are elaborations in this work that were not taken from the literature, in particular the area formula of the Cesàro curve.

2 Terms and Definitions

In order to understand fractals and their underlying geometry, it is first necessary to explain some mathematical terms that were not taught in school mathematics. Without the knowledge of these terms, an in-depth examination of the methods and theories of fractal geometry is not possible. Unfortunately, however, the topics are so extensive that only the most important aspects can be dealt with.

2.1 rooms

The concept of space is very broad in mathematics. Therefore, in the following only those rooms are discussed that are already known from school mathematics or can be easily derived from it. Let's start with the following definition:

Definition 2.1A space is a set whose elements are points of space.

Some illustrations of such spaces are already known from analytical geometry, namely the two-dimensional and the three-dimensional coordinate system. But the real number line is also an illustration of a space, namely the space that corresponds to the set of real numbers. Accordingly, the two-dimensional space is referred to as and the three-dimensional space as (also called the Euclidean plane, after the geometry of the Euclid). The points of these respective spaces are represented by coordinates, the number of coordinates is equal to the dimension, i.e. the "spatial dimensions" of the space, which are reflected in the exponent after; so any point in space has the coordinates and in, where applies. The above definition implies (although not explicitly stated) that it defines how the points are to be arranged in space and how they are related to one another. A space can also be a completely different "type" of set than the examples presented here, e.g. the points of a space can be entire functions that have a certain property (e.g. the space of all differentiable functions, a point of this space would then be, for example ). The more detailed classification of rooms is discussed below.

2.1.1 Vector spaces

A vector space V (or linear space) is a space whose points are called vectors if:

  1. The addition of two points results in a point in space, formally: for all applies:.
  2. The scalar multiplication in turn results in a point in space, formally: for all applies:.

Thus, and are examples of vector spaces since both conditions are met. However, a vector space can also be any other set to which the above conditions apply.

at the room:

2.1.2 Metric spaces

Now we come to one type of space that is vital to understanding fractal geometry.
A metric space is a given space (e.g. the vector space) that is provided with a so-called metric. A metric is a function that expresses the distance between two given points. However, this function must not be arbitrary, but must have the following properties:

  1. The following must apply to the distance between two points (there must be a symmetry): for all
  2. The distance between two different points must not be infinitely large or 0, formally: for all if.
  3. The distance between identical points must be 0, formally: for all
  4. The triangle inequality (in any triangle: "The sum of the lengths of sides a and b is greater than or equal to the length of side c, i.e.") must be fulfilled:
    for all

Such a metric is already known from school mathematics without having explicitly understood it as such, namely the distance between two points as the amount of the vector from A to B:




This metric is known as the Euclidean metric. Another metric is e.g. the so-called grid metric or Manhattan metric:




Figure 1: Illustration of the grid metric

This metric no longer indicates the shortest direct distance, but the length of the path over the mesh of an imaginary grid, similar to the street layout in Manhattan (hence the name Manhattanmetric). There are innumerable types of metrics, but not all of them are as intuitive to grasp as the ones discussed above. So there is the so-called discrete metric, the definition of which is relatively abstract and offers no practical illustration (it can be applied to any space):


In the discrete metric, the distance between identical points is 0 and different points 1. This metric is more of a theoretical gimmick to show that even very abstract functions can fulfill the four properties of a metric.

2.2 Affine maps

A mapping is generally a function that maps a room to a room, formally, each point of the room is mapped to a point of the room. For the fractal geometry, a special type of mapping, so-called affine mapping, is absolutely necessary for understanding. An affine mapping maps a point of any vector space onto another point of the same space. We limit ourselves to the space. An affine mapping therefore has the general form


Affine images assign exactly one image point to an original image point; such an image is called one-to-one or bijective. A number of related images are already known from school geometry, including similarity images, such as centric stretching. The affine mappings include the translation (displacement), the scaling, the rotation (rotation), the mirroring and the transvection (shear). All of these images and their combinations can be represented by a matrix and a displacement vector:

An affine mapping can be summarized by the formula


describe. In order to calculate the coordinates of a point, one must first perform the matrix multiplication

To run. Then you add and get

for the two coordinates of the image point.
It will now be explained how the matrix display can be used to describe all of the above-mentioned figures.

Figure 2: A scaling and shearing affine mapping is applied to the square. The result is the hatched parallelogram.

2.2.1 Translation

The translation around a vector

is the displacement of an object by exactly this vector. Since the original coordinates are to be retained when the matrix is ​​multiplied, the mapping matrix must be the identity matrix. So we get: