# Dictionary Definition

fractal n : (mathematics) a geometric pattern
that is repeated at every scale and so cannot be represented by
classical geometry

# User Contributed Dictionary

## English

### Pronunciation

- /ˈfræktəl/

### Noun

- A geometric figure that repeats itself under several levels of magnification, a shape that appears irregular at all scales of length, e.g. a fern.
- A geometric figure, built up from a simple shape, by generating the same or similar changes on successively smaller scales; it shows self-similarity on all scales

#### Derived terms

#### Translations

self-similar shape

### Adjective

- Having the form of a fractal

# Extensive Definition

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," a property called
self-similarity.
The term was coined by Benoît
Mandelbrot in 1975 and was derived from the Latin fractus
meaning "broken" or "fractured."

A fractal often has the following features:

- 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 stochastically).
- It has a Hausdorff dimension which 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 recursive definition.

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.

## History

The mathematics behind fractals
began to take shape in the 17th century when 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
Weierstrass gave 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
Sierpinski constructed 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 Cantor
also 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
plane were 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, 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".

## Examples

A relatively simple class of examples is given by
the Cantor
sets, Sierpinski
triangle and carpet,
Menger
sponge, dragon
curve, space-filling
curve, and Koch
curve. Additional examples of fractals include the Lyapunov
fractal and 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
motion in the plane have a Hausdorff dimension of 2.

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 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
Shishikura in 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-law curve
(also known as a Pareto
distribution) produces similar shapes at various
magnifications.

## Generating fractals

## 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.

Decalcomania,
a technique used by artists such as Max Ernst, can
produce fractal-like patterns. It involves pressing paint between
two surfaces and pulling them apart.

Fractals are also prevalent in African art
and 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.

″ block of acrylic creates a fractal
Lichtenberg
figure.

## Applications

As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications of fractals include:- Classification of histopathology slides in medicine
- Fractal landscape or Coastline complexity
- Enzyme/enzymology (Michaelis-Menten kinetics)
- Generation of new music
- Generation of various art forms
- Signal and image compression
- Seismology
- Fractal in Soil Mechanics
- Computer and video game design, especially computer graphics for organic environments and as part of procedural generation
- Fractography and fracture mechanics
- Fractal antennas — Small size antennas using fractal shapes
- Small angle scattering theory of fractally rough systems
- Neo-hippies' t-shirts and other fashion
- Generation of patterns for camouflage, such as MARPAT
- Digital sundial
- Technical analysis of price series (see Elliott wave principle)

## See also

- Bifurcation theory
- Butterfly effect
- Chaos theory
- Complexity
- Constructal theory
- Contraction mapping theorem
- Diamond-square algorithm
- Droste effect
- Feigenbaum function
- Fractal art
- Fractal compression
- Fractal flame
- Fractal landscape
- Fracton
- Graftal
- List of fractals by Hausdorff dimension
- Publications in fractal geometry
- Newton fractal
- Recursion
- Recursionism
- Reentrant
- Sacred geometry
- Self-reference
- Strange loop
- Turbulence

## References

## Further reading

- 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.
- Chaos and Time-Series Analysis
- Bernt Wahl, Peter Van Roy, Michael Larsen, and Eric Kampman 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. Clarke documentary 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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