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To plot each point of the famous
Mandelbrot set, the computer repeatedly iterates an equation. If the result never strays far from the origin, the corresponding point is colored black. Other colors represent the number of iterations at which the result exceeds a given distance from the origin.
A fractal is a geometric object which is 'broken up' in a radical way. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin fractus or 'broken', in order to call attention to such objects. They are in a number of major aspects different from the more usual 'smooth' objects of traditional geometry. This is immediately apparent, visually.
In many cases a fractal can be generated (for example on a computer screen) by a repeating pattern, typically a recursive or iterative process. This may give it many interesting features, most notably self-similarity and infinite detail regardless of magnification. Fractals can combine structure and irregularity.
Fractals of many kinds were originally studied as mathematical objects, and the term "fractal" has been given various precise definitions by mathematicians. Fractal geometry is the branch of mathematics which studies fractals and the special way they behave. It has often been applied in science, technology, and computer-generated art.
Conceptual roots of the theory can be traced to attempts to measure a fractal's perimeter (or area, or volume) in cases where definitions based on calculus fail. Traditional mathematical methods zoom in, in order to simplify the local picture. In constrast, the existence of fractals points up the ways in which that approach may fail, if unlimited amounts of ever-finer detail becomes apparent.
History
A
Koch snowflake is the result of infinite additions of triangles to the perimeter of the starting triangle. With every addition of triangles (
iteration), the perimeter gets longer—it grows to infinity, although the enclosed surface does not and remains finite.
Objects that are now called fractals were discovered and explored long before the word was coined. 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.
Once computer visualization was applied to fractal geometry, it presented a powerful visual argument for fractal geometry connecting far larger domains of mathematics and science than had previously been considered, particularly in the realm of non-linear dynamics, chaos theory (though a few use the term xaos instead to differentiate between ordered non-linear behavior and the common meaning of the word), and complexity. One example is plotting Newton's method as a fractal, showing how the boundaries between different solutions are fractal, and that the solutions themselves are strange attractors. Fractal geometry was also used for data compression and for modelling complex organic and geological systems, for example the growth of trees or the development of river basins.
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 going to 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 the coast of Britain is measured using a 200 km ruler, specifying that both ends of the ruler must touch the coast. Now cut the ruler in half and repeat the measurement, then repeat again:
Notice that the smaller the ruler, the bigger the result. It might be supposed that these values would converge to a finite number representing the "true" length of the coastline, but Richardson proved that the measurements of the coastline actually tended to infinity. However, to say that the coastline of Britain is actually infinitely long must be incorrect, since it would require the ability to construct infinitely small rulers, something which particle physics says cannot be done, as there is a lower limit to the smallness of a measurement.
At the time, Richardson's research was ignored by the scientific community. Today, many see it as one element in the birth of the modern study of fractals.
Categories of fractals
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 350 times magnification of the Mandelbrot set uncovers fine detail resembling the full set. |
Fractals can be grouped 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:
- Exact self-similarity — 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 — 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 — This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definition 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 — e.g., the real line (a straight Euclidean line) is exactly self-similar, but the argument that euclidean objects are fractal is a distinct minority position. Mandelbrot argued that a definition of "fractal" should include not only "true" fractals, but also traditional Euclidean objects, because irrational numbers on the number line represent complex, non-repeating properties.
Because a fractal possesses infinite granularity, 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:
- There is no precise meaning of "too irregular".
- There is no single definition of "dimension".
- There are many ways that an object can be self-similar.
- Not every fractal is defined recursively.
The following definitions of fractal have all been proposed, but each one has shortcomings :-
- An object that is self-similar in some sense (including non-linear self similarity and statistical self-similarity) - this is a simple intuitive definition, but it is very difficult to make it mathematically precise. It also encompasses the objects of traditional Euclidean geometry, which are not generally considered to be fractals.
- An object with non-integer Hausdorff dimension - but this arbitrarily excludes some objects that are generally considered to be fractals, such as the Peano curve and the boundary of the Mandelbrot set.
- A set with Hausdorff dimension that strictly exceeds its topological dimension - this is the most widely accepted mathematical definition, but it requires a degree of mathematical sophistication to be understood.
Examples
Pulling two glue-covered acrylic sheets apart forms a natural fractal.
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 logarithmic 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.
Romanesco broccoli showing very fine natural fractals
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.
Harrison [1] (http://math.berkeley.edu/~harrison/research/publications/) 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
- 1 Fractal Geometry, by Kenneth Falconer; John Wiley & Son Ltd; ISBN 0471922870 (March 1990)
- The Fractal Geometry of Nature, by Benoît 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
- Ultra Fractal (http://www.ultrafractal.com/) - popular software for Microsoft Windows
- Fractint - available for most platforms
- Makin' Magic Fractals (http://skyscraper.fortunecity.com/terabyte/966)
- ChaosPro (http://www.chaospro.de) - for Microsoft Windows
- Xaos (http://xaos.sourceforge.net/) - Realtime generator - Windows, Mac, Linux, etc
- FLAM3 (http://flam3.com/) - Advanced iterated function system designer and renderer for all platforms.
- Sterling Fractal - Advanced fractal-generating program by Stephen Ferguson.
- Online Fractal Generator (http://www.wackerart.de/fractal.html) Java-Plugin required.
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