Fractal
In colloquial usage, a fractal is 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." Mathematicians avoid giving the strict definition and prefer to call fractal a
geometric object that usually
* has fine structure at each scale and can not be easily described in traditional Euclidean geometry language.
* is self-similar
* has
Hausdorff dimension greater than its topological dimension
* has a simple and recursive definition
Encyclopedia
In colloquial usage, a fractal is 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." Mathematicians avoid giving the strict definition and prefer to call fractal a
geometric object that usually
- has fine structure at each scale and can not be easily described in traditional Euclidean geometry language.
- is self-similar
- has Hausdorff dimension greater than its topological dimension
- has a simple and recursive definition
- has natural appearance.
Fractals have all or most of these features .
Not all self-similar objects are fractals — for example, the real line is formally self-similar and has natural appearance but fails to have other fractal characteristics. The term
fractal was coined in 1975 by
Benoît Mandelbrot, from the Latin
fractus, meaning "broken" or "fractured."
History
Objects that are now described as fractals were discovered and described centuries ago. Ethnomathematics like Ron Eglash's African Fractals describes pervasive fractal geometry in indigeneous African craft work. In 1525, the German Artist
Albrecht Dürer published
The Painter's Manual, in which one section is on "Tile Patterns formed by Pentagons." The Dürer'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 nonintuitive 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. In 1915 Waclaw Sierpinski constructed his
triangle and, one year later, his
carpet. Actually, these fractals were described as curves, which is hard to realize with the well known 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 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 is a paper by...
. 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. 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".
Examples
A relatively simple class of examples is given by the
Cantor sets,
Sierpinski triangle and
carpet,
Menger sponge,
dragon curve,
space-filling curve,
Koch curve. Additional examples of fractals include the
Lyapunov fractal and the limit sets of Kleinian groups. Fractals can be deterministic or stochastic . For example the trajectories of the Brownian motion in the plane have
Hausdorff dimension 2.
Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals . 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 it has the Hausdorff dimension equal to its topological dimension of 2 —but what is truly surprising is that the boundary of the Mandelbrot set also has the Hausdorff dimension of 2 , a result proved by M. Shishikura in 1991. A closely related fractal is the
Julia set.
The following analysis of the Koch Snowflake suggests how self-similarity can be used to analyze fractal properties.
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, m², but in some other power of a meter, m
x. Now 4
Nx =
NLx, because a three times shorter steplength requires four times as many steps, as is seen from the figure. Solving that equation gives
x = / ˜ 1.26186. So the unit of measurement of the boundary of the Koch snowflake is approximately m
1.26186.
More generally, suppose that a fractal consists of
N identical parts that are similar to the entire fractal with the scale factor of
L and that the intersection between part is of the Lebesgue measure 0. Then the Hausdorff dimension of the fractal is . For example, the Hausdorf dimension of
- the Cantor set is ,
- the Sierpinski gasket is ,
- the Sierpinski carpet is ,
and so on.
Even more generally one may assume that each of
N parts is similar to the fractal with a different scale factor , . Then the Hausdorff dimension can be calculated by solving the following equation in the variable
s:
-
Generating fractals
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| Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set. |
Three common techniques for generating fractals are:
...
s — 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.
Classification of fractals
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 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. Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.
Fractals in nature
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 by using a recursive algorithm. This recursive nature is obvious 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 by 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.
Applications
As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications of fractals include:
See also
References
- 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
- Falconer, Kenneth. Fractal Geometry: Mathematical Foundations and Applications. West Sussex: John Wiley & Sons, Ltd., 2003. ISBN 0-470-84861-8
- 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
- Mandelbrot, Benoît B. 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. Probably the earliest good computer-generator for the masses; the book came with a floppy . 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 links
- . An introductory primer on chaos and fractals.
- at cut-the-knot
- . From UIUC a brief introduction
- — Creation of simple fractals using arraying technique
- - JAVA applet
Multiplatform generator programs
- — free fractal real-time browser for Windows, Mac, Linux; supporting zooming and animation in real time, featuring autopilot. GNU GPL licensed.
- — free advanced iterated function system
...
designer and renderer for all platforms. Windows binaries available. GNU GPL licensed.
- — A web-based fractal zoomer, sending calculated images as bitmaps to the browser. Rather slow.
- — Java applet drawing Mandelbrot and Julia sets. Rather slow. Closed sourced.
- — A fast web-based mandelbrot explorer. Generated fractals can be saved, commented and rated in users gallery. Closed sourced.
Linux generator programs
...
grayscale renderer. GNU GPL-licensed. See also its .
Windows generator programs
- is a fairly complete listing of free fractal generators.
- — software for Microsoft Windows. Free trial version available.
- — A free flame and IFS fractal generator. Used for creating fractal artwork. GNU GPL licensed.
- — freeware for Microsoft Windows featuring real-time exploration, animation and more
- — a freeware Windows-based fractal generator, using fractals to create bitmap images and AVI video clips.
- — freeware Windows-based generator. Closed sourced, with source available for a fee.
- — free Lyapunov fractal renderer with zooming feature. GNU GPL licensed.
- — freeware 3D strange attractor rendering software for Windows
- — Freeware multi-agent simulator in Microsoft Excel. Samples include Mandelbrot Explorer and fractal tree projects.
- — freeware fractal generator for DOS and Windows, with a available
- — freeware for Microsoft Windows
- — a free fractal generator. Capable of animations, but with low quality. GNU GPL licensed.
- - a free fractal generator, open code; a strategy switch.
- - a free fractal generator with simple depth mapping.
Mac generator programs
...
fractal generator for Mac OS X.
MorphOS generator programs
- with support for custom formulas
