Fractal
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The word "fractal" has two related meanings. In colloquial usage, it denotes a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification and is therefore often referred to as "infinitely complex." In mathematics a fractal is a geometric object that satisfies a specific technical condition, namely having a Hausdorff dimension greater than its Lebesgue covering dimension. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin fractus, meaning "broken" or "fractured."
HistoryEdit
Objects that are now called fractals were discovered and explored long before the word was coined. In 1525, the German Artist Albrecht Durer published The Painter's Manual, in which one section is on "Tile Patterns formed by Pentagons." The Durer's Pentagon largely resembled the Sierpinski carpet, but based on pentagons instead of squares.
The idea of "recursive self similarity" was originally developed by the philosopher Leibniz and he even worked out many of the details. 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 a new fractal curve, the Lévy C curve.
Georg Cantor gave examples of subsets of the real line with unusual properties — these Cantor sets are also now recognised as fractals. 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 visualize the beauty of many 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. This built on earlier work by Lewis Fry Richardson. In 1975, Mandelbrot coined the word fractal to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. (Please refer to the articles on these terms for precise definitions.) 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".
ExamplesEdit
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 representations, 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.
Additional examples of fractals include the Lyapunov fractal, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, limit sets of Kleinian groups, and the Koch curve. Fractals can be deterministic or stochastic (i.e. non-deterministic).
Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (see Attractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so has dimension 2 and is not fractal--but what is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2. (M. Shishikura proved that in 1991.)
The fractional dimension of the boundary of the Koch snowflakeEdit
The following analysis of the Koch Snowflake suggests how self-similarity can be used to analyze fractal properties. This argument is only a sketch, but provides some of the flavor of the field.
The total length of a number, N, of small steps, L, is the product NL. Applied to the boundary of the Koch snowflake this gives a boundless length as L approaches zero. But this distinction is not satisfactory, as different Koch snowflakes do have different sizes. A solution is to measure, not in meter, m, nor in square meter, m^{2}, but in some other power of a meter, m^{x}. Now 4N(L/3)^{x} = NL^{x}, because a three times shorter steplength requires four times as many steps, as is seen from the figure. Solving that equation gives x = (log 4)/(log 3) ≈ 1.26186. So the unit of measurement of the boundary of the Kool Aid snowflake is approximately m^{1.26186}.
==Categories of fanny add==
The whole Mandelbrot set |
Mandelbrot zoomed 6x |
Mandelbrot Zoomed 100x |
Mandelbrot Zoomed 2000x Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set. |
Three common techniques for generating fractals are:
- 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.
- Escape-time fractals — 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, the Burning Ship fractal and the Lyapunov fractal.
- Random fractals, generated by stochastic rather than deterministic processes, for example, fractal landscapes, Lévy flight and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, Diffusion Limited Aggregation or Reaction Limited Aggregation clusters.
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 definitions 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 since its Hausdorff dimension and topological dimension are both equal to one, it is not a fractal.
Fractals in natureEdit
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, mountains, river networks, and systems of blood vessels.
Trees and ferns are fractal in nature and can be modeled on a computer using a recursive algorithm. This recursive nature is clear in these examples — a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature.
The surface of a mountain can be modeled on a computer using a fractal: Start with a triangle in 3D space and connect the central points of each side by line segments, resulting in 4 triangles. The central points are then randomly moved up or down, within a defined range. The procedure is repeated, decreasing at each iteration the range by half. The recursive nature of the algorithm guarantees that the whole is statistically similar to each detail.
ApplicationsEdit
As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications [1] of fractals include:
- Classification of histopathology slides in medicine
- Generation of new music
- Generation of various art forms
- Signal and image compression
- Seismology
- Cosmology
- Computer and video game design, especially computer graphics for organic environments
- Fractography and fracture mechanics
Fractal generationEdit
Fractals are usually rendered with computers. Various software exists for rendering fractals, and even generating new ones.
- Fractint — one of the first and probably most recognized name in fractal-generation tools.
- Sterling Fractal — an advanced fractal-generating program written by Stephen Ferguson.
- Sterling2 — a free version of Stephen Ferguson's Sterling (with new 50 formulae).
- Ultra Fractal — a fractal image and animation generation program with its own programming language. Comes with an extensive set of features for composing, coloring, layering, masking, rendering, and more.
- XaoS — a fast interactive real-time fractal zoomer and morpher.
See alsoEdit
- Bifurcation theory
- Butterfly effect
- Chaos theory
- Complexity
- Constructal theory
- Diamond-square algorithm
- Fractal art
- Fractal landscape
- Fractal metaphysics
- Fractal compression
- Graftal
- Publications in fractal geometry
- Newton fractal
- Recursion
- Turbulence
- Feigenbaum function
ReferencesEdit
- Barnsley, Michael F., and Hawley Rising. Fractals Everywhere. Boston: Academic Press Professional, 1993. ISBN 0120790610
- Falconer, Kenneth. Fractal Geometry: Mathematical Foundations and Applications. West Sussex: John Wiley & Sons, Ltd., 2003. ISBN 0470848618
- Jürgens, Hartmut, Heins-Otto Peitgen, and Dietmar Saupe. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992. ISBN 038797903
- Mandelbrot, Benoît B. The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982. ISBN 0716711869
- Peitgen, Heinz-Otto, and Dietmar Saupe, eds. The Science of Fractal Images. New York: Springer-Verlag, 1988. ISBN 0387966080
- 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. Probably the earliest good computer-generator for the masses; the book came with a floppy (unknown if it will still run on later Macintoshs). Good introduction geared toward students at junior-high and high school level. With brief history including Peano and Koch leading to Hausdorff dimension. Examples of imaginary-number math, how to generate a fractal. With formulas and brief explanations for the 69 generator functions supported by the floppy. References a 1985 Scientific American article in A.K. Dewdney's "Computer Recreations" that "...inspired countless programmers to write their own Mandelbrot programs" including, apparently, the author.
- 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.
External linksEdit
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- The Chaos Hypertextbook. An introductory primer on chaos and fractals.
- Fractals, in Layman's Terms
- Fractals, fractal dimension, chaos, plane filling curves at cut-the-knot
- Fractal properties
- Information on fractals from FAQS.org
- Fractal dimensions
- Fractal calculus
- Fractal Dimension
- Natural fractals in Grand Canyon
- One Dimensional Dynamical Systems. From UIUC a brief introduction
- Fractal Mountain - JAVA applet
- Multiplatform generator programs
- Xaos — GNU GPL licenced fractal generator for Windows, Mac, Linux; supporting zooming and animation in real time, featuring autopilot
- FLAM3 — Advanced iterated function system designer and renderer for all platforms.
- Fract — A Web-based fractal zoomer
- Linux generator programs
- Gnofract4d — Interactive editor which can use many fractint formulas
- Review of fractal software packages which run under X11 on Linux
- Windows generator programs
- Fractovia's listing of fractal generators is a fairly complete listing of free fractal generators.
- Ktaza: freeware by S. Ferguson
- Online Fractal Generator Java-Plugin required.
- Ultra Fractal — software for Microsoft Windows
- Apophysis — A free flame and IFS fractal generator. Used for creating fractal artwork.
- ChaosPro — for Microsoft Windows
- MSPlotter - a free Windows-based fractal generator, using fractals to create bitmap images and AVI video clips.
- Fractal Explorer — free Windows-based generator
- Yet Another Fractal Explorer — Lyapunov fractal renderer with zooming feature
- Mac generator programs
- Altivec Fractal Carbon Mac-based benchmarking utility, using fractals to determine performance.
- IFSLab A Iterated function system fractal generator for Mac OS X.
- MorphOS generator programs
- Zone Explorer with support for custom formulas
- Fractal Art Galleries
- Fractal Artwork, Tutorials, Information
- Soler's Fractal gallery
- Fractovia — authoritative source of fractal generators.
- Fractal Artwork
- Fractal landscapes
- Mitchell-Green gravity set
- Fractal art galleries
- Fractal Zoom movies
- WebFractales : Galleries and softwares
- Fractal art with papers and programs
- Fractal Art by Wolter Schraa
- Fractal Recursions — Images and Animations.