A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," [cite book
last = Mandelbrot
first = B.B.
title = The Fractal Geometry of Nature
publisher = W.H. Freeman and Company.
date = 1982
id = ISBN 0-7167-1186-9] a property called
self-similarity. The term was coined by Benoît Mandelbrotin 1975 and was derived from the Latin"" meaning "broken" or "fractured."
A fractal often has the following features: [cite book
last = Falconer
first = Kenneth
title = Fractal Geometry: Mathematical Foundations and Applications
publisher = John Wiley & Sons, Ltd.
date = 2003
pages = xxv
id = ISBN 0-470-84862-6]
* It has a fine structure at arbitrarily small scales.
* It is too irregular to be easily described in traditional Euclidean geometric language.
* It is self-similar (at least approximately or
* It has a
Hausdorff dimensionwhich is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
* It has a simple and
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the
real line(a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics.
Images of fractals can be created using
fractal generating software.
Sierpiński Triangle, only going nine generations of infinite—click for larger image.]
Koch snowflake, one begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral "bump." One then performs the same replacement on every line segment of the resulting shape, ad infinitum. With every iteration, the perimeter of this shape increases by one third of the previous length. The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains . For this reason, the Koch snowflake and similar constructions were sometimes called "monster curves." ]
mathematicsbehind fractals began to take shape in the 17th century when mathematician and philosopher Leibniz considered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).
It took until 1872 before a function appeared whose graph would today be considered fractal, when
Karl Weierstrassgave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable. 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. In 1915, Waclaw Sierpinskiconstructed his triangle and, one year later, his carpet. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. 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 a new fractal curve, the Lévy C curve. Georg Cantoralso gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals.
Iterated functions in the
complex planewere investigated in the late 19th and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatouand Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualize the beauty of many of the objects that they had discovered.
In the 1960s,
Benoît Mandelbrotstarted investigating self-similarity in papers such as " How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", which built on earlier work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".
A relatively simple class of examples is given by the
Cantor sets, Sierpinski triangleand carpet, Menger sponge, dragon curve, space-filling curve, and Koch curve. Additional examples of fractals include the Lyapunov fractaland the limit sets of Kleinian groups. Fractals can be deterministic(all the above) or stochastic(that is, non-deterministic). For example, the trajectories of the Brownian motionin the plane have a Hausdorff dimension of 2.
Chaotic dynamical systems are sometimes associated with fractals. Objects in the
phase spaceof a dynamical systemcan be fractals (see attractor). Objects in the parameter spacefor a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension of 2—but what is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by Mitsuhiro Shishikurain 1991. A closely related fractal is the Julia set.
Even simple smooth curves can exhibit the fractal property of self-similarity. For example the
power-lawcurve (also known as a Pareto distribution) produces similar shapes at various magnifications.
Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set.
Four common techniques for generating fractals are:
:* Escape-time fractals — (also known as "orbits" fractals) These are defined by a
recurrence relationat each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal, the Nova fractaland the Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.:* Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.:* Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapesand the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregationor reaction-limited aggregationclusters.:* Strange attractors — Generated by iteration of a map or the solution of a system of initial-value differential equations that exhibit chaos.
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 functionsystems 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 definitions of "fractal" trivially imply some form of statistical self-similarity. ( Fractal dimensionitself 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.
Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes,
crystals, mountain ranges, lightning, river networks, caulifloweror broccoli, and systems of blood vessels and pulmonary vessels. Coastlines may be loosely considered fractal in nature.
Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive
algorithm. This recursive nature is obvious in these examples — a branch from a tree or a frondfrom a fern is a miniature replica of the whole: not identical, but similar in nature.
In 1999, certain self similar fractal shapes were shown to have a property of "frequency invariance" — the same electromagnetic properties no matter what the frequency — from
Maxwell's equations(see fractal antenna). [Hohlfeld,R., and Cohen, N.,"SELF-SIMILARITY AND THE GEOMETRIC REQUIREMENTS FOR FREQUENCY INDEPENDENCE IN ANTENNAE ", Fractals, Vol. 7, No. 1 (1999) 79-84] Iterated function system pentagramdrawn with a vector iterationprogram
In creative works
Fractal patterns have been found in the paintings of American artist
Jackson Pollock. While Pollock's paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work. [ [http://www.phys.unsw.edu.au/PHYSICS_!/FRACTAL_EXPRESSIONISM/fractal_taylor.html Richard Taylor, Adam P. Micolich and David Jonas. "Fractal Expressionism : Can Science Be Used To Further Our Understanding Of Art?"] ] Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns. [ [http://classes.yale.edu/Fractals/Panorama/ A Panorama of Fractals and Their Uses] by Michael Frame and Benoît B. Mandelbrot] It involves pressing paint between two surfaces and pulling them apart.
Fractals are also prevalent in
African artand architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles. [ [http://www.rpi.edu/~eglash/eglash.dir/afractal/afractal.htm Ron Eglash. "African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press 1999."] ] acrylic sheets.
Lichtenberg figure. DVDcite journal|title=The fractal nature of a fracture surface|journal= Journal of Physics A|date=21 July 1990|first=Gongwen|last=Peng|coauthors=Decheng Tian|volume=23|issue=14|pages=3257–3261|doi=10.1088/0305-4470/23/14/022|url=http://www.iop.org/EJ/abstract/0305-4470/23/14/022|format=|accessdate=2007-06-02] Romanesco broccolishowing very fine natural fractals
DLA cluster grown from a
copper(II) sulfatesolution in an electrodepositioncell fractal flamecreated with the program Apophysis
As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications of fractals include: [cite web|url=http://library.thinkquest.org/26242/full/ap/ap.html|title=Applications|accessdate=2007-10-21]
* Classification of
histopathologyslides in medicine
Fractal landscapeor Coastline complexity
* Enzyme/enzymology (
* Generation of new music
* Generation of various
* Signal and image compression
Fractal in soil mechanics
* Computer and video game design, especially
computer graphicsfor organic environments and as part of procedural generation
* Fractography and
Fractal antennas — Small size antennas using fractal shapes
* Small angle scattering theory of fractally rough systems
hippies' t-shirts and other fashion
* Generation of patterns for camouflage, such as
Technical analysisof price series (see Elliott wave principle)
* Contraction mapping theorem
List of fractals by Hausdorff dimension
* Publications in fractal geometry
* Barnsley, Michael F., and Hawley Rising. "Fractals Everywhere". Boston: Academic Press Professional, 1993. ISBN 0-12-079061-0
* Falconer, Kenneth. " Techniques in Fractal Geometry". John Willey and Sons, 1997. ISBN 0-471-92287-0
* Jürgens, Hartmut, Heins-Otto Peitgen, and Dietmar Saupe. "Chaos and Fractals: New Frontiers of Science". New York: Springer-Verlag, 1992. ISBN 0-387-97903-4
Benoît B. Mandelbrot"The Fractal Geometry of Nature". New York: W. H. Freeman and Co., 1982. ISBN 0-7167-1186-9
* Peitgen, Heinz-Otto, and Dietmar Saupe, eds. "The Science of Fractal Images". New York: Springer-Verlag, 1988. ISBN 0-387-96608-0
Clifford A. Pickover, ed. "Chaos and Fractals: A Computer Graphical Journey - A 10 Year Compilation of Advanced Research". Elsevier, 1998. ISBN 0-444-50002-2
Jesse Jones, "Fractals for the Macintosh", Waite Group Press, Corte Madera, CA, 1993. ISBN 1-878739-46-8.
Hans Lauwerier, "Fractals: Endlessly Repeated Geometrical Figures", Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. ISBN 0-691-08551-X, cloth. ISBN 0-691-02445-6 paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.
* Bernt Wahl, Peter Van Roy, Michael Larsen, and Eric Kampman [http://www.fractalexplorer.com "Exploring Fractals on the Macintosh"] , Addison Wesley, 1995. ISBN 0-201-62630-6
*Nigel Lesmoir-Gordon. "The Colours of Infinity: The Beauty, The Power and the Sense of Fractals." ISBN 1-904555-05-5 (The book comes with a related DVD of the
Arthur C. Clarkedocumentary introduction to the fractal concept and the Mandelbrot set.
* Gouyet, Jean-François. " Physics and Fractal Structures" (Foreword by B. Mandelbrot); Masson, 1996. ISBN 2-225-85130-1, and New York: Springer-Verlag, 1996. ISBN 0-387-94153-1. Out-of-print. Available in PDF version at [http://www.jfgouyet.fr/fractal/fractauk.html] .
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