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Fractal

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A closeup of the [[Mandelbrot setEnlarge

A closeup of the [[Mandelbrot set

, the most famous fractal.]]

Fractal is a scientific concept used to describe geometric objects which satisfy most, if not all, of the following properties.

Fractals were originally studied as mathematical objects, and the term "fractal" has been given various precise definitions by mathematicians. Fractal geometry, the branch of mathematics which studies fractals, has recently found numerous applications to science, technology, and computer-generated art.

The word fractal was coined in 1975 by Benoît Mandelbrot.

Table of contents
1 History
2 Categories of fractals
3 Definitions
4 Examples
5 See also
6 References, further reading
7 Fractal generators
8 External links

History

[[Koch snowflakeEnlarge

[[Koch snowflake

]]

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 everywhere 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 abstract 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 Lévy who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described two fractal curves, the Lévy C curve and the Lévy 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 Carathéodory 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 Poincaré, 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 Benoît 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 describe self-similar objects which had no clear dimension. He derived the word fractal from the Latin fractus, meaning broken or irregular, (and not from the word fractional, as is commonly believed.)

How Long is the Coast of Britain?

Lewis Fry Richardson was a pacifist and a mathematician, studying the cause of war between two countries. He decided to search for a relation between the size of its mutual border and the probability of two countries entering a war. As part of this research, he investigated how the measured length of a border changes as the unit of measurement is changed. Richardson published empirical statistics which led to a conjectured relationship. This research was quoted by Mandelbrot in his 1967 paper How Long Is the Coast of Britain?

Suppose you try to measure Britain's coast using a 200 km ruler, specifying that both ends of the ruler must touch the coast. Then you cut the ruler in half, and again:

150px
150px
150px

The interesting part here is that the smaller the ruler, the bigger result you get. One might suppose that these values would tend to a finite number representing the "true" length of the coastline, but Richardson proved that the measurements of the coastline actually tended to infinity.

Image:How-long.jpg

At the time Richardson's research was ignored by the scientific community. Today, many see it as one element of the birth of the modern study of fractals.

Categories of fractals

1
2
3
4
Self-similarity in the Mandelbrot set. The top image is the full set, and each subsequent image is a closeup of the red square.

Fractals can be groups into three broad categories. These categories are determined from how the fractal is defined or generated:

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

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 few would argue that a definition of "fractal" should include the real line. (Ironically, Mandelbrot himself originally thought otherwise and wanted a definition of "fractal" that would include not only "true" fractals, but also traditional Euclidean objects.)

Because a fractal is defined mathematically, no natural object can be a fractal. However, natural objects can display fractal-like properties across a limited range of scales.

Definitions

The defining characteristics of fractals, while intuitively appealing, are remarkably hard to condense into a mathematically precise definition.

Problems with defining fractals include:

The following definitions of fractal have all been proposed, but each one has shortcomings :-

Examples

A natural fractal formed by pulling two glue-covered acrylic sheets apartEnlarge

A natural fractal formed by pulling two glue-covered acrylic sheets apart

Trees and ferns are fractal in nature and can be modelled on a computer using a recursive algorithm. This recursive nature is clear in these examples — take a branch from a tree or a frond from a fern and you will see it is a miniature replica of the whole. Not identical, but similar in nature.

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 (this value is the same, no matter what base is chosen), 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 would be impossible to tell the difference between the 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, with a fractal, increasing the magnification reveals more detail that was previously invisible.

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 Mandelbrot set contains whole discs, so has dimension 2. This is not surprising. What is truly surprising is that the boundary of the Mandelbrot set also has Hausdorff dimension 2 and topological dimension 1.

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.

One of the [[Julia setEnlarge

One of the [[Julia set

.]]

Harrison [1] 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 for a list.

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 fractal image compression, as well as a variety of scientific disciplines.

See also

References, further reading

Fractal generators

External links

Topics in mathematics related to spaces Edit
Topology | Geometry | Trigonometry | Algebraic geometry | Differential geometry and topology | Algebraic topology | Linear algebra | Fractal geometry | Compact space