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A fractal is a set which is self-similar; fractals are repetitive in shape, but not in size. In other words, no matter how much you magnify a fractal, it will always look the same (or at least similar). More specifically, in mathematics a fractal is a set with Hausdorff dimension > topological dimension.

Fractals are generally irregular, and thus not definable by traditional geometry -- furthermore, fractals tend to have repetitive details, visible at any arbitrary scale. A fractal may have fractional Hausdorff (or box-counting) dimensions; they may also be defined recursively.

These characteristics of fractals, while intuitively appealing, are (aside from dimensionality) remarkably hard to condense into a mathematically precise definition. The problem with most definitions of fractal is that there are objects that one would like to call fractals but which do not satisfy the definition. To name a few problems: there is no precise meaning of "too irregular"; there are many ways that an object can be self-similar; not every fractal is defined recursively; and the many definitions of dimension admitting fractional values don't, in general, agree numerically (so an acceptable definition of fractal cannot be based on a single fractal dimension).

Approximate fractals (objects displaying complex structure over a very broad, but finite, scale range) are easily found in nature. These naturally occurring fractals (like clouds, mountains, river networks, and systems of blood vessels) have both lower and upper cut-offs, but they are separated by several orders of magnitude. It is noteworthy that, despite being ubiquitous, fractals were not considered a legitimate object of study (or even defined!) until well into the 20th century.

Some common examples of fractals include the Mandelbrot set, Lyapunov fractal, Cantor set, Sierpinski carpet and triangle, Peano curve and the Koch snowflake. Fractals can be deterministic or stochastic. Chaotic dynamical systems are often (if not always) associated with fractals.

There are three broad categories of fractals that are commonly studied at this time:

1. Iterated function systems. These have a fixed geometric replacement rule (Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake).
2. Fractals defined by a recurrence relation at each point in a space (such as the complex plane). An example of this type are the Mandelbrot set and the Lyapunov fractal. These are also called escape-time fractals.
3. Random fractals, generated by stochastic rather than deterministic processes, for example Fractal landscapes and Lévy flights.

Of all of these, only Iterated function systems usually display the well known "self-similarity" property--meaning that their complexity is invariant under scaling transforms. Fractals such as the Mandelbrot set are more loosely self-similar: they contain small copies of the entire fractal in distorted and degenerate forms.

Harrison extended Newtonian calculus to fractal domains, including the theorems of Gauss, Green, and Stokes.

Fractals are usually calculated by computers with fractal software. See External Links.

Random fractals have the greatest practical use, and can be used to describe many highly irregular real-world objects. Examples include clouds, mountains, turbulence, coastlines and trees. Fractal techniques have also been employed in image compression, as well as a variety of scientific disciplines.  