Fractal
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2 Categories of fractals 3 Definitions 4 Examples 5 References, further reading 6 Fractal generators 7 External links |
Objects that we now call fractals were discovered and explored long before we had a word for them. In 1872 Karl Weierstrass found an example of a function with the non-intuitive property that it is everyhere continuous but nowhere differentiable - the graph of this function would now be called a fractal. In 1904 Helge von Koch, dissatisfied with Weierstrass's very abtract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake. The idea of self-similar curves was taken further by Paul Pierre LÃÂévy who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described two fractal curves, the LÃÂévy C curve and the LÃÂévy dragon curve
Georg Cantor gave examples of subsets of the real line with unusual properties - these Cantor sets are also now recognised as fractals. In an attempt to understand objects such as Cantor sets, mathematicians such as Constantin CarathÃÂéodory and Felix Hausdorff generalised the intuitive concept of dimension to include non-integer values. Iterated functions in the complex plane had been investigated in the late 19th and early 20th centuries by Henri PoincarÃÂé, Felix Klein, Pierre Fatou and Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualise the beauty of the objects that they had discovered.
In the 1960s Benoit Mandelbrot started investigating self-similarity in papers such as How Long is the Coast of Britain ? Statistical Self-Similarity and Fractional Dimension. Taking a highly visual approach, Mandelbrot recognised connections between these previously unrelated strands of mathematics. In 1975 Mandelbrot coined the word "fractal" to decribe self-similar objects.
Three broad categories of fractals are commonly studied at this time:
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.
A few problems with defining fractals include:
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.
History
Categories of fractals
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.Definitions
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).Examples
 Mandelbrot set |
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.
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.
 Natural fractal (1). Click image for description |
 Natural fractal (2). Click image for description |
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.
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 below.
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
Fractal generators
External links
| Topics in mathematics related to spaces |
| Topology | Geometry | Trigonometry | Algebraic geometry | Differential geometry and topology | Algebraic topology | Linear algebra | Fractal geometry | Compact space |