Fractal

Fractal is a term coined by the mathematician Benoît Mandelbrot term (Latin fractus broken ', frangere from the Latin, ' ( in pieces decompose ) break '), referred to the specific natural or artificial structures or geometric patterns. These structures or patterns have in general not an integer Hausdorff dimension ( a mathematical term, the well-known supplies integer values ​​in many common geometric cases ), but a broken - hence the name - and also have a high degree of scale invariance or self-similarity. This is for example the case when an object from several smaller copies of itself. Geometric objects of this type differ in material respects from ordinary smooth figures.

Concept and environment

The term fractal can be used both as a noun and an adjective. The field of mathematics, are examined in the fractals and its laws, is called fractal geometry and extends into several other realms, such as function theory, computability theory and dynamical systems. As the name implies, the classical notion of Euclidean geometry is expanded, which is also reflected in the broken and not natural dimensions of many fractals. In addition to Mandelbrot include Waclaw Sierpiński and Gaston Maurice Julia to the eponymous mathematicians.

Fractal dimension; self-similarity

In traditional geometry a line a surface is one-dimensional, two-dimensional and three-dimensional spatial structure. For the fractal sets can be the dimensionality specify not directly: If we introduce, for example, an arithmetic operation for a fractal line pattern thousands of times gone, so filled with time, the whole drawing area approaches with lines (such as the computer screen ), and the one-dimensional structure is a two-dimensional.

Mandelbrot used the term of the generalized dimension by Hausdorff and found that fractal structures usually have a non-integer dimension. It is also called fractal dimension. Therefore, he introduced the following definition:

Lots of non -integer dimension is therefore a fractal. The converse is not true, fractals can also have integer dimension, such as the Peano curve or the Sierpinski pyramid.

If there is a fractal of a certain number of scaled copies of itself and this reduction factor for all copies of the same, you use the similarity dimension, which corresponds in such simple cases of demonstrative calculation of the Hausdorff dimension.

However, the self-similarity can exist only in a statistical sense. One then speaks of Zufallsfraktalen.

Something is considered abstract this dimension, if we introduce the following quantities:

Self-similarity, so-called " fractal growth," possibly in a statistical sense, and therefore associated fractal dimensions characterize a fractal system or in growth processes (eg, diffusion -limited growth).

Examples

The simplest examples of self-similar objects are lines, parallelograms ( and Others squares) and die because they can parallel to its sides cuts of their own are decomposed into reduced copies. However, these are not fractals, because their similarity dimension and their Lebesgue'sche coverage dimension match. An example of a self-similar fractal is the Sierpinski triangle, which in itself is constructed of three reduced to half copies. It thus has the dimension of similarity, while Lebesgue'sche overlap dimension is equal to 1.

The self-similarity does not have to be perfect, as the successful application of the methods of fractal geometry shows on natural features such as trees, clouds, coastlines, etc.. The objects mentioned are present in greater or lesser degree self-similar structure (a tree branch looks something like this like the little tree ), but the similarity is not strict, but stochastic. Unlike forms of Euclidean geometry, which are at a magnification often shallow, and thus easier ( as a circle ), can appear in fractals increasingly complex and new details.

Fractal patterns are often generated by recursive operations. Even simple production rules already give complex patterns after a few recursion steps.

This can be seen for example at the Pythagoras Tree. Such a tree is a fractal, which is composed of squares, which are so arranged as in the Pythagorean theorem defined.

Another fractal is the Newton fractal generated by the Newton's method is used for the zero point calculation.

Examples of fractals in three-dimensional space are the Menger sponge and the Sierpinski Pyramid.

Applications

Through its wealth of forms and the aesthetic appeal associated they play a role in digital art and there have spawned the genre of fractal art. They are further used in the computer-aided simulation form rich structures, such as realistic landscapes. To receive in radio technology different frequency ranges, fractal antennas are used.

Fractals in Nature

Fractal forms are also 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 Romanesco and breeding in the ferns. Also, the cauliflower has a fractal structure, where you often do not consider this carbon at first glance. But there are always some cauliflower heads who see the Romanesco in the fractal structure very similar.

Widely used are fractal structures without strict, but with statistical self-similarity. These include, for example, trees, blood vessels, river systems and coastlines. In the case of the coast line results as a consequence of the impossibility of an exact determination of the Coastline: The more accurately you measure the subtleties of the coastal course, the greater the length obtained. 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 oscillators ( standard example Belousov -Zhabotinsky reaction ) can be one hand to use as a basic diagram, but on the other hand, also explain as fractals. Also found are fractal structures in crystal growth and in the formation of mixtures, such as when one drop of dye solution in a glass of water.

The unraveling of Bast can be explained on the fractal geometry of Naturfaserfibrillen. In particular, the flax fiber is a fractal fiber.

A process for the production of fractals

Fractals can be created in many different ways, but all methods include a recursive procedure:

• The iteration of functions is the easiest and most popular way to generate fractals; the Mandelbrot set is created that way. A special form of this method are IFS fractals ( Iterated Function Systems ), in which multiple functions are combined. This allows to create natural formations.
• Dynamic systems generate fractal structures, so-called strange attractors.
• L- systems, which are based on repeated text replacement, are well suited to modeling of natural structures such as plants and cell structures.

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

" Simple and regular ' fractals

F → R or R → L R → R - L L → R L- Gosper curve F → R or R → L R → R L L -R - RR -L L → R LL L R - R -L Hilbert curve X X → YF XFX FY- Y → XF YFY -FX Koch Snowflake F - F - F F → F F - F F Peano curve X X → XFYFX F F YFXFY XFYFX Y → YFXFY -F F XFYFX YFXFY Peano curve F F → F-F F F F-F -F -F F Penta PLEXITY F F F F F F → R F F | F -F F arrowhead F → R or R → L R → L R L- L → L R -R Sierpinski triangle FXF - FF - FF X → - FXF FXF FXF - F → FF Sierpinski triangle, second variant F - F - F F → F - F - F - ff f → ff Sierpinski carpet F F → F F -F -FF -F -F - fF f → fff Lévy C curve F → F F - F Statement of the L- system:

The optional, not necessary F is generally used as a route, which is replaced by a sequence of statements. How the F are also other letters capitalized as R and L for a section that is being replaced. And - represent a specific angle, which runs in a clockwise or counterclockwise. The | symbol means a volte-face of the pen, ie a rotation of 180 °. If appropriate, one for one, a corresponding multiple of the rotation angle.

• Example dragon curve:

F → R R → R - L L → R L- F is a simple path between two points. F → R is said that the distance R is replaced by R. This step is necessary because it has two recursive substitutions R and L, which contain each other. In addition, is replaced as follows:

R R - L ( R - L ) - ( R L ) ( ( R - L ) - ( R L ) ) - (- ( R - L ) ( L -R - ) - ) . . . At a certain section of this replacement process must be stopped to get a graphic:

( ( r - l ) - ( r l ) ) - (- ( R - L ) ( r l ) - ) Where r and l represent respectively a fixed predetermined distance

Zufallsfraktale

Besides playing in nature also " Zufallsfraktale " a major role. These are generated by probabilistic rules. This can be done for example by growth processes, where, for example, diffusion- limited growth ( Witten and Sander ) and " tumor growth " is different. In the first case arise tree-like structures, in the latter case structures with round shape, depending on the manner in which it attaches the newly added particles to the already existing aggregates. If the fractal exponents are not constant, but for example depend on the distance from a central point of the aggregate, we speak of so-called multifractal.