Fractal

Famous fractal:
the Mandelbrot crowd (so-called "apple man")

Fractal is a term coined by the mathematician Benoît Mandelbrot in 1975 ( Latin fractus ' broken ', from Latin frangere ' (to pieces -) 'break') that describes certain natural or artificial structures or geometric patterns .

These structures or patterns generally do not have an integral Hausdorff dimension , but a broken one - hence the name - and also have a high degree of scale invariance or self-similarity . This is the case, for example, when an object consists of several scaled-down copies of itself. Geometric objects of this type differ in essential respects from ordinary smooth figures.

Concept and environment

The term fractal can be used both noun and adjectival . The field of mathematics in which fractals and their regularities are examined is called fractal geometry and extends into several other areas, such as function theory , computability theory and dynamic systems . As the name suggests, the classic concept of Euclidean geometry is expanded, which is also reflected in the broken and unnatural dimensions of many fractals. In addition to Mandelbrot, Wacław Sierpiński and Gaston Maurice Julia are among the mathematicians who give it its name.

Fractal dimension; Self-likeness

A small section from the edge of the almond bread . Here she shows resemblance to a Julia crowd

In traditional geometry, a line is one-dimensional, a surface is two-dimensional and a spatial structure is three-dimensional. The dimensionality of the fractal sets cannot be specified directly: if, for example, an arithmetic operation for a fractal line pattern is carried out thousands of times, the entire drawing surface (e.g. the computer screen) fills with lines and the one-dimensional structure approaches become a two-dimensional.

Mandelbrot used the concept of the generalized dimension according to Hausdorff and found that fractal structures usually have a non-integer dimension. It is also known as the fractal dimension . Therefore he introduced the following definition:

A fractal is a set whose Hausdorff dimension is larger than its Lebesgue cover dimension .

So any set with a non-integer dimension is a fractal. The converse does not apply, fractals can also have integer dimensions, for example the Sierpinski pyramid .

If a fractal consists of a certain number of reduced copies of itself and if this reduction factor is the same for all copies, then one uses the similarity dimension, which in such simple cases corresponds to the Hausdorff dimension. ${\ displaystyle D}$

${\ displaystyle D = {\ frac {\ log ({\ text {number of self-similar parts}})} {\ log ({\ text {reduction factor}})}}}$

But the self-similarity can only exist in the statistical sense. One then speaks of random fractals.

Self-similarity, possibly in a statistical sense, and associated fractal dimensions characterize a fractal system or, in the case of growth processes, so-called “fractal growth” (e.g. diffusion-limited growth ).

Examples

Menger sponge after the 4th iteration stage

The simplest examples of self-similar objects are lines, parallelograms (including squares) and cubes, because they can be broken down into reduced copies of themselves by making cuts parallel to their sides. However, these are not fractals because their similarity dimension and their Lebesgue coverage dimension match. An example of a self-similar fractal is the Sierpinski triangle , which is made up of three copies of itself reduced to half the length of the side. It thus has the similarity dimension, while the Lebesgue coverage dimension is equal to 1. ${\ displaystyle {\ tfrac {\ log 3} {\ log 2}} \ approx 1 {,} 585}$

The self-similarity does not have to be perfect, as the successful application of the methods of fractal geometry to natural structures such as trees, clouds, coastlines, etc. shows. The objects mentioned have a self-similar structure to a greater or lesser extent (a tree branch looks roughly like a scaled-down tree), but the similarity is not strict, but stochastic . In contrast to forms of Euclidean geometry , which often become flatter and therefore simpler when enlarged (e.g. a circle), more and more complex and new details can appear in fractals.

Fractal patterns are often created through recursive operations. Even simple generation rules result in complex patterns after a few recursion steps.

This can be seen , for example, in the Pythagoras tree . Such a tree is a fractal made up of squares arranged as defined in the Pythagorean Theorem .

Another fractal is the Newton fractal , generated using the Newton method used to calculate the zeros .

Examples of fractals in three-dimensional space are the Menger sponge and the Sierpinski pyramid based on the tetrahedron (just as the Sierpinski triangle is based on the equilateral triangle). Accordingly, fractals according to Sierpinski can also be formed in higher dimensions - e.g. based on the pentachoron in four-dimensional space.

Applications

Due to their wealth of shapes and the associated aesthetic appeal, they play a role in digital art and have created the genre of fractal art . They are also used in the computer-aided simulation of structures with many shapes, such as realistic landscapes. Fractal antennas are used to receive different frequency ranges in radio technology .

Fractals in nature

Fractal manifestations can also be found in nature. However, the number of levels of self-similar structures is limited and is often only three to five. Typical examples from biology are the fractal structures in the green cauliflower cultivation Romanesco and in ferns . The cauliflower also has a fractal structure, although this cabbage is often not seen at first glance. However, there are always some cauliflower heads that look very similar to Romanesco in their fractal structure.

Fractal structures without strict, but with statistical self-similarity, are widespread. These include trees, blood vessels, river systems and coastlines. In the case of the coastline, the consequence is that it is impossible to determine the exact length of the coast : the more precisely you measure the subtleties of the coastline, the greater the length you get. In the case of a mathematical fractal such as the Koch curve , it would be unlimited.

Fractals can also be found as explanatory models for chemical reactions. Systems such as the oscillators (standard example Belousov-Zhabotinsky reaction ) can on the one hand be used as a principle picture, on the other hand they can also be explained as fractals. One also finds fractal structures in crystal growth and in the formation of mixtures , e.g. B. if you put a drop of color solution in a glass of water. The Lichtenberg figure also shows a fractal structure.

The unraveling of bast can be explained by the fractal geometry of natural fiber fibrils. In particular, the flax fiber is a fractal fiber.

Method of creating fractals

Fractals can be created in many different ways, but all methods involve a recursive approach:

• The iteration of functions is to create the easiest and most popular way of fractals; the amount of Mandelbrot is created this way. IFS fractals ( Iterated Function Systems ), in which several functions are combined, are a special form of this procedure . This is how natural structures can be created.
• Dynamic systems create fractal structures, so-called strange attractors .
• L-systems based on repeated text replacement are well suited for modeling natural structures such as plants and cell structures.

There are ready-made programs, so-called fractal generators , with which computer users can display fractals even without knowledge of the mathematical principles and procedures.

"Simple and Regular" fractals

 Fractal L system angle Distance ratio Visualization Kite curve F → R oder F → L R → +R--L+ L → -R++L-  ${\ displaystyle 45 ^ {\ circ}}$ ${\ displaystyle 1: 1 / {\ sqrt {2}}}$ Gosper curve F → R oder F → L R → R+L++L-R--RR-L+ L → -R+LL++L+R--R-L  ${\ displaystyle 60 ^ {\ circ}}$ ${\ displaystyle 1: 1 / {\ sqrt {7}}}$ Hilbert curve X X → -YF+XFX+FY- Y → +XF-YFY-FX+  ${\ displaystyle 90 ^ {\ circ}}$ ${\ displaystyle 1: 1/2}$ Koch flake F--F--F F → F+F--F+F  ${\ displaystyle 60 ^ {\ circ}}$ ${\ displaystyle 1: 1/3}$ Peano curve X X → XFYFX+F+YFXFY-F-XFYFX Y → YFXFY-F-XFYFX+F+YFXFY  ${\ displaystyle 90 ^ {\ circ}}$ Peano curve F F → F-F+F+F+F-F-F-F+F  ${\ displaystyle 90 ^ {\ circ}}$ ${\ displaystyle 1: 1/3}$ Penta Plexity F++F++F++F++F F → F++F++F|F-F++F  ${\ displaystyle 36 ^ {\ circ}}$ ${\ displaystyle 1: 1 / \ left ({\ frac {{\ sqrt {5}} + 1} {2}} \ right) ^ {2}}$ Arrowhead F → R oder F → L R → -L+R+L- L → +R-L-R+  ${\ displaystyle 60 ^ {\ circ}}$ ${\ displaystyle 1: 1/2}$ Sierpinski triangle FXF--FF--FF X → --FXF++FXF++FXF-- F → FF  ${\ displaystyle 60 ^ {\ circ}}$ Sierpinski triangle , 2nd variant F--F--F F → F--F--F--ff f → ff  ${\ displaystyle 60 ^ {\ circ}}$ ${\ displaystyle 1: 1/3}$ Sierpinski carpet F F → F+F-F-FF-F-F-fF f → fff  ${\ displaystyle 90 ^ {\ circ}}$ ${\ displaystyle 1: 1/3}$ Lévy C curve F F → +F--F+  ${\ displaystyle 45 ^ {\ circ}}$ ${\ displaystyle 1: 1 / {\ sqrt {2}}}$
Explanation of the L-system

The optional , i.e. not necessary, F is generally used as a route that is replaced by a sequence of instructions. Like the F , other capitalized letters such as R and L stand for a section of the route that is being replaced. + and - stand for a specific angle that runs clockwise or counterclockwise. The symbol | refers to a U-turn of the pencil, i.e. a turn by 180 °. If necessary, a corresponding multiple of the angle of rotation is used for this.

Example kite curve
 F → R
R → +R--L+
L → -R++L-


F is a simple line between two points. F → R means that the segment F is replaced by R. This step is necessary because it has two recursive replacements R and L that contain each other. The following is replaced as follows:

 R
+R--L+
+(+R--L+)--(-R++L-)+
+(+(+R--L+)--(-R++L-)+)--(-(+R--L+)++(-R++L-)-)+
.
.
.


At a certain section, this replacement process must be canceled in order to get a graphic:

 +(+(+r--l+)--(-r++l-)+)--(-(+r--l+)++(-r++l-)-)+


Here r and l each represent a fixed route.

Random fractals

In addition, “random fractals” also play a major role in nature. These are generated according to probabilistic rules. This can happen through growth processes, for example, a distinction between diffusion-limited growth (Witten and Sander) and "tumor growth". In the first case tree-like structures arise, in the latter case structures with a round shape, depending on the way in which the newly added particles are attached to the existing aggregates. If the fractal exponents are not constant, but e.g. B. depend on the distance from a central point of the aggregate, one speaks of so-called. Multifractals .

Individual evidence

1. See e.g. B. Z. Eisler, J. Kertész: Multifractal model of asset returns with leverage effect . In: Physica A: Statistical Mechanics and its Applications . 343, November 2004, pp. 603-622. arxiv : cond-mat / 0403767 . doi : 10.1016 / j.physa.2004.05.061 .