Introduction

Mandelbrot termed a new object as fractal (from the Latin word 'fractus'). The corresponding Latin concept frangere means to break, or to establish irregular fragments. In the English language the words 'fracture' and 'fraction' corespond to this. Mandelbrot formed a new word 'fractal', with noun and adjective functions in both English and French languages.

This is relatively appropriate for rendering the meaning of the word 'fractal' because in the concept" 'fragmented' "fractal can mean also "'irregular'". Examples of mathematical structures that are fractals: Sierpinski triangle, Koch flake, Peane curve, Mandelbrot set, Lorenz attractor. Using fractals we can also describe many objects in real world (e.g., clouds, mountains, turbulence, coastal outlines) that are not easily described by other simple geometric forms.

Benoit Mandelbrot gives a mathematical definition of a fractal as a geometric unit, that has a Husdorff-Besicovich dimension other than a topological integer dimension. The concept of 'topological dimension' can be understood through the following sentence: A point has topological dimension equal to zero; an abscissa has a topological dimension equal to one; a cube has a topological dimension equal to two. The dimension of these objects is an integral number.

Fractal geometry

Benoit B. Mandelbrot laid the foundations of this science. He designed one of the best known fractals: the Mandelbrot set. Whereas in the case of CAD model programs mere basic mathematical objects are enough for describing products and various objects, distinguished by geometric precision, in the case of describing a natural object this does not suffice, at all. This is due to their high irregularity and high ineffectiveness of describing through classical geometrical modeling. Let's take the example of clouds: how to describe them? As the product of a set of spheres, or parametric surfaces? This description would not be the best. Therefore, a method was sought of how to best to describe the given objects. This stimulated the arising of fractal geometry, which provides effective and relatively simple methods of modeling natural objects.

Fractal dimension

One of the basic questions in assessing a specific geometric unit is what size and dimension does this unit have. For instance: a point has dimension 0, a line has dimension 1, a surface has dimension 2, a cube has dimension 3. But there are several ways and methods of how to calculate fractal dimension. We can calculate it using a limit of the rate of recording of alternation of object size and recording of alteration of scale rate so that the rate of scaling converges to zero. Differences occur in what it means 'size of an object' and what it means 'rate of scaling', and how we arrive at an average number from many diverse parts of a geometric unit. The fractal dimension quantifies a static *geometric* object.

Hausdorff dimension

The Hausdorff dimension represents the dimension of a fractal. We can understand it from the following example: Let us have a curve with the length of one. Let us measure this curve using the abscissa d (equal to 1/n), so that we cover this curve by N abscissas d of the length 1/n. Then, for the length of the abscissa is valid:

L = lim N (d). d =1.

where the limit for d goes to zero :   lim N (d). d^d < Y,
is reasonable only if d=1.

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We know many artificially created geometric shapes, but also natural objects, which have the same shape or are ressemble in various sizes. They are named selfressembling shapes. One of such object is Koch flake(In 1904 was created by Sweden mathematician Helga von Koch as an example of continuous, space bounded, endless curve). Construction of this flake:

at the beginning we have triangel with each edge same equal 1. In the first step on each edge we put unit line. In the second step we devide the line into three parts and the middle part replace with two line. In each step we create the same in the each fourts curve.

Koch flake

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Applet: Part of Koch flake

Hausdorff dimenzion of Koch flake is :

   lim N (d). d^d = lim 4^i [(1/3)^i]^d < 00,

which has meaning only if d = log 4 / log 3 = 1.26.  

Other example of selfressembling shape is cloud. It looks same from each distance.

Dividing the fractal :

  • deterministic
  • linear
  • non linear
  • non deterministic

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Applet Koch Snow Author: Strizenec Michal

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Applet Snowflake Koch. Tento aplet zobrazuje kochovu vlocku roznych urovni. Pokliknuti sa "uroven vlocky" zvysuje. jednotlive hrany sa vypocitavaju tak ze sa strana rozdeli na tri rovnake casti, nakresli sa prva tretina a posledna tretina, medzi nimi sa vytvori treti vrchol. Author Jalcovik Jan

Source code for NetBeans
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