Fractal geometry
The term fractal is now used as a scientific concept, as well as a strictly mathematical idea. In the first sense, it means a geometric shape that is self-similar on all scales. In other words, no matter how much you magnify a fractal, it always looks the same (or at least similar).
More specifically, in mathematics a fractal is a set with Hausdorff dimension ≠ an integer 0, 1, 2, ... . 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 < d < 1. A simple recipe, such as excluding the digit 7 from decimal expansions, is self-similar under 10-fold enlargement, and also has dimension log 9/log 10, showing the connection of the two concepts.
Fractals are generally irregular (not smooth) in shape, and thus are not objects definable by traditional geometry. That means that fractals tend to have significant detail, visible at any arbitrary scale; when there is self-similarity, this can occur because 'zooming in' simply shows similar pictures. Such sets are usually defined instead by recursion.
For example, a normal 'euclidean' shape, such as a circle, looks flatter and flatter as it is magnified. At infinite magnification it is impossible to tell the difference between a circle and a straight line. Fractals are not like this. The conventional idea of curvature, which represents the reciprocal of the radius of an approximating circle, cannot usefully apply because it scales away. Instead, in a fractal, increasing the magnification reveals detail that you simply couldn't see before.
The secondary characteristics of fractals, while intuitively appealing, are remarkably hard to condense into a mathematically precise definition. Strictly, a fractal should have fractional (that is, noninteger) Hausdorff (or box-counting) dimension. There are objects that have the appearance of fractals but which do not satisfy this definition.
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. The celebrated Mandelbrot set contains whole discs, so has dimension 2.
A few problems with defining fractals include
- 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
The many definitions of dimension giving fractional values don't always agree numerically (so an acceptable definition of fractal cannot be based on a single fractal dimension).
Approximate fractals are easily found in nature. These objects display complex structure over an extended, but finite, scale range. 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. Despite being ubiquitous, fractals were not much studied until well into the twentieth century, and general definitions came later.
Three broad categories of fractals are commonly studied at this time:
- Iterated function systems. These have a fixed geometric replacement rule (Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake).
- 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 and the Lyapunov fractal. These are also called escape-time fractals.
- 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 because they 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.
See also: Fractal art, fractal landscape, graftal, Hausdorff dimension, constructal theory, Gaston Julia, Benoit Mandelbrot
References, further reading
- 1 Fractal Geometry, by Kenneth Falconer; John Wiley & Son Ltd; ISBN 0471922870 (March 1990)
- The Fractal Geometry of Nature, by Benoit Mandelbrot; W H Freeman & Co; ISBN 0716711869 (hardcover, September 1982).
- The Science of Fractal Images, by Heinz-Otto Peitgen, Dietmar Saupe (Editor); Springer Verlag; ISBN 0387966080 (hardcover, August 1988)
- Fractals Everywhere, by Michael F. Barnsley; Morgan Kaufmann; ISBN 0120790610
Fractal generators
External links
Referenced By
Math | Mathematic | Mathematical | Mathematics | MathematicsAndStatistics | Maths
|
