Fractal

The word "fractal" has two related meanings. In colloquial usage, it denotes a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification and is therefore often referred to as "infinitely complex." In Mathematics a fractal is a Geometric object that satisfies a specific technical condition, namely having a Hausdorff Dimension greater than its Topological Dimension . The term ''fractal'' was coined in 1975 by Benoît Mandelbrot , from the Latin ''fractus,'' meaning "broken" or "fractured."

EXAMPLES

, a fractal related to the Mandelbrot set]]

A relatively simple class of examples is the Cantor Set s, in which short and then shorter (open) intervals are struck out of the Unit Interval 1 , leaving a set that might (or might not) actually be self-similar under enlargement, and might (or might not) have dimension ''d'' that has 0 < ''d'' < 1. A simple recipe, such as excluding the Digit ''7'' from Decimal Representation s, is self-similar under 10-fold Enlargement , and also has dimension log 9/log 10 (this value is the same, no matter what Logarithm ic base is chosen), showing the connection of the two concepts.

Additional examples of fractals include the Lyapunov Fractal , Sierpinski Triangle and Carpet , Menger Sponge , Dragon Curve , Space-filling Curve , limit sets of Kleinian Group s, and the Koch Curve . Fractals can be Deterministic or Stochastic (i.e. non-deterministic).

Chaotic Dynamical Systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (see Attractor ). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so has dimension 2 and is not fractal--but what is truly surprising is that the '' Boundary '' of the Mandelbrot set also has a Hausdorff dimension of 2. (M. Shishikura proved that in 1991.)

The fractional dimension of the boundary of the Koch snowflake

The following analysis of the Koch Snowflake suggests how self-similarity can be used to analyze fractal properties. This argument is only a sketch, but provides some of the flavor of the field.

The total length of a number, ''N'', of small steps, ''L'', is the product ''NL''. Applied to the boundary of the Koch snowflake this gives a boundless length as ''L'' approaches zero. But this distinction is not satisfactory, as different Koch snowflakes do have different sizes. A solution is to measure, not in meter, m, nor in square meter, m², but in some other power of a meter, m^''x''. Now 4''N''(''L''/3)^''x'' = ''NL''^''x'', because a three times shorter steplength requires four times as many steps, as is seen from the figure. Solving that equation gives ''x'' = (log 4)/(log 3) ≈ 1.26186. So the unit of measurement of the boundary of the Koch snowflake is approximately m^1.26186.

Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set.

Three common techniques for generating fractals are:

Iterated Function System s — These have a fixed geometric replacement rule. Cantor Set , Sierpinski Carpet , Sierpinski Gasket , Peano Curve , Koch Snowflake , Harter-Heighway Dragon Curve , T-Square , Menger Sponge , are some examples of such fractals.

Escape-time fractals — Fractals defined by a Recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot Set , the Burning Ship Fractal and the Lyapunov Fractal .

Random fractals, generated by stochastic rather than deterministic processes, for example, Fractal Landscapes , Lévy Flight and the Brownian Tree . The latter yields so-called mass- or dendritic fractals, for example, Diffusion Limited Aggregation or Reaction Limited Aggregation clusters.

Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:

Exact self-similarity — This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity.

Quasi-self-similarity — This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by Recurrence Relation s are usually quasi-self-similar but not exactly self-similar.

Statistical self-similarity — This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.

It should be noted that not all self-similar objects are fractals — e.g., the Real Line (a straight Euclidean line) is exactly self-similar, but since its Hausdorff dimension and topological dimension are both equal to one, it is not a fractal.

FRACTALS IN NATURE

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Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include Cloud s, Snow Flakes , Mountain s, River networks, and systems of Blood Vessel s.

Trees and ferns are fractal in nature and can be modeled on a computer using a Recursive Algorithm . This recursive nature is clear in these examples — a branch from a tree or a Frond from a fern is a miniature replica of the whole: not identical, but similar in nature.

The surface of a mountain can be modeled on a computer using a fractal: Start with a triangle in 3D space and connect the central points of each side by line segments, resulting in 4 triangles. The central points are then randomly moved up or down, within a defined range. The procedure is repeated, decreasing at each iteration the range by half. The recursive nature of the algorithm guarantees that the whole is statistically similar to each detail.

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Image:Microwaved-DVDjpgFractal Branching Occurs On A Microwave-irradiated

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