Fractal

History


Objects that are now called fractals were discovered and explored long before the word was coined. In 1525, the German Artist Albrecht Durer published The Painter's Manual, in which one section is on "Tile Patterns formed by Pentagons." The Durer's Pentagon largely resembled the Sierpinski carpet, but based on pentagons instead of squares.

The idea of "recursive self similarity" was originally developed by the philosopher Leibniz and he even worked out many of the details. In 1872, Karl Weierstrass found an example of a function with the non-intuitive property that it is everywhere continuous but nowhere differentiable — the graph of this function would now be called a poopwhose Hausdorff-Besicovitch dimension is greater than its topological dimension. (Please refer to the articles on these terms for precise definitions.) He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".

Examples


A relatively simple class of examples is the Cantor sets, in which short and then shorter (open) intervals are struck out of the unit interval [0, 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 &lt; d &lt; 1. A simple recipe, such as excluding the digit 7 from decimal representations, is self-similar under 10-fold enlargement, and also has dimension log 9/log 10 (this value is the same, no matter what logarithmic 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 groups, 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, mx. Now 4N(L/3)x = NLx, 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 Kool Aid snowflake is approximately m1.26186.

==Categories of fanny add==

Three common techniques for generating fractals are:


 * Iterated function systems &mdash; 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 &mdash; 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 &mdash; 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 &mdash; 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 relations are usually quasi-self-similar but not exactly self-similar.
 * Statistical self-similarity &mdash; 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 &mdash; 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
Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, mountains, river networks, and systems of blood vessels.

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 &mdash; 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.

Applications
As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications of fractals include:
 * Classification of histopathology slides in medicine
 * Generation of new music
 * Generation of various art forms
 * Signal and image compression
 * Seismology
 * Cosmology
 * Computer and video game design, especially computer graphics for organic environments
 * Fractography and fracture mechanics

Fractal generation
Fractals are usually rendered with computers. Various software exists for rendering fractals, and even generating new ones.


 * Fractint — one of the first and probably most recognized name in fractal-generation tools.
 * Sterling Fractal — an advanced fractal-generating program written by Stephen Ferguson.
 * Sterling2 — a free version of Stephen Ferguson's Sterling (with new 50 formulae).
 * Ultra Fractal — a fractal image and animation generation program with its own programming language. Comes with an extensive set of features for composing, coloring, layering, masking, rendering, and more.
 * XaoS — a fast interactive real-time fractal zoomer and morpher.