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The Mandelbrot set is a famous example of a fractal
Frost crystals formed naturally on cold glass illustrate fractal process development in a purely physical system
.
A fractal is "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,"[1] a property called self-similarity.^ It is self-similar (at least approximately or stochastically ).
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Fractals can also be classified according to their self-similarity.
  • Fractal - encyclopedia article - Citizendium 11 January 2010 5:12 UTC locke.citizendium.org:8080 [Source type: Reference]
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Above all, fractal meant self-similar."
  • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

 Roots of mathematical interest in fractals can be traced back to the late 19th Century; however, the term "fractal" was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.[2]
A fractal often has the following features:[3]
.
.Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms).^ Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms).
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Above all, fractal meant self-similar."
  • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

 ^ Learn how fractal geometry can identify valuable reference...points can be identified through fractal geometry because they reflect mileposts...open or the close, but with fractal geometry they become visible.
  • fractal geometry Facts, information, pictures | Encyclopedia.com articles about fractal geometry 11 January 2010 5:12 UTC www.encyclopedia.com [Source type: Academic]

 .Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns.^ Similarity of fractals to natural objects .
  • Fractal@Everything2.com 11 January 2010 5:12 UTC everything2.com [Source type: FILTERED WITH BAYES]

 ^ Examples include clouds, snow flakes , crystals , mountain ranges , lightning , river networks , cauliflower or broccoli , and systems of blood vessels and pulmonary vessels .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Fractal objects in nature include shells, cauliflowers, mountains and clouds.
  • fractal geometry Facts, information, pictures | Encyclopedia.com articles about fractal geometry 11 January 2010 5:12 UTC www.encyclopedia.com [Source type: Academic]

 .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; for instance, it is regular enough to be described in Euclidean terms.^ 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.
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Fractals can also be classified according to their self-similarity.
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Above all, fractal meant self-similar."
  • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]
.Images of fractals can be created using fractal-generating software.^ It has a delicate framework of generating fractal images.
  • Introducing FerryMan Fractal on Renderosity.com 11 January 2010 5:12 UTC www.renderosity.com [Source type: General]

 ^ Images of fractals can be created using fractal generating software .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ History of fractals Fractal images Simple fractals Fractals used Fractal generator Fractal - 2 reference results Fractal 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 .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 .Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, such as when it is possible to zoom into a region of the fractal that does not exhibit any fractal properties.^ Fractal algorithms have made it possible to generate lifelike images of complicated, highly irregular natural objects, such as the rugged terrains of mountains and the intricate branch systems of trees.
  • fractal (mathematics) -- Britannica Online Encyclopedia 11 January 2010 5:12 UTC www.britannica.com [Source type: FILTERED WITH BAYES]

 ^ Fractal Explorer is a freeware fractal generator that can produce mysterious and beautiful mathematically-based images.

 ^ These are deep zoom levels (the first being over 1000x), but fractal details remain abundant in all three dimensions!
  • Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal 11 January 2010 5:12 UTC www.skytopia.com [Source type: FILTERED WITH BAYES]

 .Also, these may include calculation or display artifacts which are not characteristics of true fractals.^ I may be wrong, but I think he merely is saying that they leave these forces out of their calculations.
  • Galaxy map hints at fractal universe - space - 25 June 2008 - New Scientist 11 January 2010 5:12 UTC www.newscientist.com [Source type: FILTERED WITH BAYES]

 ^ For 68K Macs Mandella 8.7 * generation of many different types of fractals, allow editing of the color map, and other display & calculation options.
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 ^ Despite misleading claims by its proponents, the intentional disorganization characteristic of the deconstructivist architectural style is the opposite of the internal organization of a true fractal.
  • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

Contents

History

Animated construction of a Sierpiński Triangle, only going nine generations of infinite—click for larger image.
To create a 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.^ After one iterations, the length is increased by four-thirds.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 ^ As iteration count increases, the length and time of the orbit(s) increase, thus creating more overlaps and more incremented pixels.

 ^ With each iteration, the middle third from each lines segment of the previous set is simply removed.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 .The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains finite.^ When the length of the path x becomes zero, then the number p of paths of zero length is infinite.
  • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

 ^ Here M is a prescribed maximum number of iterations that rescues the computer from getting trapped in an infinite loop, and M = 500 is used in this example.
  • Sekino's Fractal Gallery 11 January 2010 5:12 UTC www.willamette.edu [Source type: FILTERED WITH BAYES]

 ^ Assuming this could be iterated an infinite number of times, the result would be a figure which is infinitely wiggly, having no straight lines whatsoever.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 For this reason, the Koch snowflake and similar constructions were sometimes called "monster curves."
.The mathematics behind fractals began to take shape in the 17th century when mathematician and philosopher Gottfried Leibniz considered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).^ The mathematics behind 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).
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Fractals can also be classified according to their self-similarity.
  • Fractal - encyclopedia article - Citizendium 11 January 2010 5:12 UTC locke.citizendium.org:8080 [Source type: Reference]
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Above all, fractal meant self-similar."
  • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]
.It was not until 1872 that 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.^ 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 .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ 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 .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Fractal dimension can be illustrated by considering a specific example: the snowflake curv e defined by Helge von Koch in 1904.
  • fractal (mathematics) -- Britannica Online Encyclopedia 11 January 2010 5:12 UTC www.britannica.com [Source type: FILTERED WITH BAYES]

 .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 curve.^ 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 .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ The Koch curve was named after Helge von Koch in 1904.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 ^ Fractal dimension can be illustrated by considering a specific example: the snowflake curv e defined by Helge von Koch in 1904.
  • fractal (mathematics) -- Britannica Online Encyclopedia 11 January 2010 5:12 UTC www.britannica.com [Source type: FILTERED WITH BAYES]

 .(The image at right is three Koch curves put together to form what is commonly called the Koch snowflake.^ 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 .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Fractal dimension can be illustrated by considering a specific example: the snowflake curv e defined by Helge von Koch in 1904.
  • fractal (mathematics) -- Britannica Online Encyclopedia 11 January 2010 5:12 UTC www.britannica.com [Source type: FILTERED WITH BAYES]

 ^ The three contractive mappings from the full triangle onto the subtriangles forms an IFS. These mappings will be of the form "shrink by half and move to the top, left, or right".
  • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 ) .Waclaw Sierpinski constructed his triangle in 1915 and, one year later, his carpet.^ In 1915, Waclaw Sierpinski constructed his triangle and, one year later, his carpet .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ One year later, FerryMan Fractal 1.0 was released to the Internet as freeware.
  • Introducing FerryMan Fractal on Renderosity.com 11 January 2010 5:12 UTC www.renderosity.com [Source type: General]

 ^ I don't have a whole lot else to say about it; this is one of the few flames that I'm genuinely happy with when I look back at it years later.

 .Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions.^ Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions.
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ These dimensions are known as topological dimensions, and have been used for many years to describe the shape and position of objects.
  • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

 ^ For some reason people often become more entranced by the fractal shapes than by the mechanisms underlying them.
  • The Oil Drum | Fractal Adaptive Cycles in Natural and Human Systems 11 January 2010 5:12 UTC www.theoildrum.com [Source type: FILTERED WITH BAYES]

 .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.^ Fractals can also be classified according to their self-similarity.
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ Above all, fractal meant self-similar."
  • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

 ^ A11a_: If a fractal is self-similar, you can specify mappings that map the whole onto the parts.
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 .Georg Cantor also gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals.^ Georg Cantor also gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals.
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ The Cantor set The Cantor set is a good example of an elementary fractal.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 ^ 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; for instance, it is regular enough to be described in Euclidean terms.
  • 3dfiction.com - Fractal Animations, Video Screensaver, Animated Backgrounds 11 January 2010 5:12 UTC 3dfiction.com [Source type: General]
.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.^ 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 .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 ^ It was named after Gaston Julia, who studied the iteration of polynomials and rational functions during the early twentieth century, making the Julia set much older than the Mandelbrot set.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 ^ They consist of very simple formulas, such as the classic Mandelbrot set, X ← X 2 + c, iterated on the complex number plane.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 .Without the aid of modern computer graphics, however, they lacked the means to visualize the beauty of many of the objects that they had discovered.^ They serve strictly as visual decoration for the car city, without relating in any way to the pedestrian city (which may in fact be nonexistent).
  • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

 ^ This object can then be displayed using computer graphics techniques such as ray tracing.
  • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 ^ Software for computing fractal dimension: Fractal Dimension Calculator is a Macintosh program which uses the box- counting method to compute the fractal dimension of planar graphical objects.
  • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
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.^ Most often, especially in natural fractals, self-similarity is approximate or statistical.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 ^ Self-Similarity, Code-Repetition and Fractal Dimension .
  • Pellionisz (1989) Neural Geometry: Towards a Fractal Model of Neurons 11 January 2010 5:12 UTC usa-siliconvalley.com [Source type: Academic]

 .Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension.^ The boundary of the Mandelbrot set and the Julia set of a generic c in M have Hausdorff dimension 2 and have topological dimension 1.
  • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 ^ Each of the following objects has a topological dimension of 1.
  • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

 ^ Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension .
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 .He illustrated this mathematical definition with striking computer-constructed visualizations.^ He illustrated this mathematical definition with striking computer-constructed visualizations.
  • Fractal encyclopedia topics | Reference.com 11 January 2010 5:12 UTC www.reference.com [Source type: Reference]

 These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".

Examples

A Julia set, a fractal related to the Mandelbrot set
.A class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, and Koch curve.^ The Sierpinski triangle below is a good example of this.
  • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

 ^ Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor.
  • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 ^ Julia set Koch Curve .
  • Fractal@Everything2.com 11 January 2010 5:12 UTC everything2.com [Source type: FILTERED WITH BAYES]

 .Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups.^ Other traditional rendering methods for 256-colour fractals include continuous potential, external decomposition and level-set methods like Fractint's Bof60 and Bof61.
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 ^ The Flash Fractal Map Viewer only supports a limited set of options on the context menu and the DHTML map may not include a context menu at all.
  • Online Fractal Map User Guide 11 January 2010 5:12 UTC www.fractaledge.com [Source type: FILTERED WITH BAYES]

 ^ When a group is designed okay with its children blended rightly and some ideas, for example, how about a lake effect on the group, come to you, then you can get it by applying a Lake fractal component to the group.
  • Introducing FerryMan Fractal on Renderosity.com 11 January 2010 5:12 UTC www.renderosity.com [Source type: General]

 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.^ Computer modeling reveals hidden, fractal order beneath apparently random, surface behavior typical of many chaotic systems.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 ^ Alien 5 by Fractals By-Vicky Digital, Fractal Time-discrete dynamical system fractal with Gaussian integer coloring algorithm, four layers.
  • Fractal Paintings and Art at Artist Rising 11 January 2010 5:12 UTC www.artistrising.com [Source type: General]

 ^ A scientific visualization software program to be used as a Dynamic Matrix Systems fractal generator.
  • Fractal Generating Software Programs & Links on Paul N. Lee's website 11 January 2010 5:12 UTC home.att.net [Source type: Reference]

 .Objects in the phase space of a dynamical system can be fractals (see attractor).^ Alien 5 by Fractals By-Vicky Digital, Fractal Time-discrete dynamical system fractal with Gaussian integer coloring algorithm, four layers.
  • Fractal Paintings and Art at Artist Rising 11 January 2010 5:12 UTC www.artistrising.com [Source type: General]

 ^ Fractals are a means by which time, or system dynamics, gets etched into form via self-similar, recursive loops that exist on multiple size scales.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 ^ A huge (>100 pages double-spaced) essay on chaos, fractals, and non-linear dynamics.
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 .Objects in the parameter space for a family of systems may be fractal as well.^ Consider a volume in phase space defined by all the initial conditions a system may have.
  • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 ^ Another property of a fractal object is the lack of a well-defined scale .
  • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

 ^ The amount of space filled by one of these objects is represented by the fractal dimension or index (D), which can be thought of as a "filling factor".
  • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

 .An interesting example is the Mandelbrot set.^ The Mandelbrot set merges with the "Tricorn" fractal (or "Mandelbar" set) to form this interesting 3D hybrid created by Eric Baird .
  • Skytopia - Mystery of the Real 3D Mandelbrot Fractal 11 January 2010 5:12 UTC www.skytopia.com [Source type: General]

 .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.^ The boundary of the Mandelbrot set and the Julia set of a generic c in M have Hausdorff dimension 2 and have topological dimension 1.
  • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 ^ (Note that these images are in fact based on the Julia set, which is closely related to the Mandelbrot).
  • Skytopia - Mystery of the Real 3D Mandelbrot Fractal 11 January 2010 5:12 UTC www.skytopia.com [Source type: General]

 ^ If c is not in the Mandelbrot set, the Julia set will be a Cantor dust.
  • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

Generating fractals

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.
Four common techniques for generating fractals are:
.
  • Escape-time fractals – (also known as "orbits" fractals) These are defined by a formula or recurrence relation at each point in a space (such as the complex plane).^ Nathan Cohen first looked at fractal capacitors in 1988 and included commonly known structures such as Koch islands.
    • Fractal Antenna Systems, Inc. 11 January 2010 5:12 UTC www.fractenna.com [Source type: FILTERED WITH BAYES]

     ^ The site life.csu.edu.au has a collection of fractal programs, papers, information related to complex systems, and gopher and World Wide Web connections.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ However, there are many complex biologic structures that cannot be easily modelled by simple shapes such as these.
    • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

     .Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal, the Nova fractal and the Lyapunov fractal.^ Draws a variety of fractals of Mandelbrot or Julia types.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ If c is not in the Mandelbrot set, the Julia set will be a Cantor dust.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ Mandelbrot/Julia sets.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .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.^ Another way of generating 3-D fractals is to use 3-D iterated function systems (IFS).
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ Fractals Everywhere_ by Barnsley is mostly about iterated function systems.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ An iterated function system is By taking a point and repeatedly applying these mappings you end up with a collection of points on the fractal.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Highway 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 landscapes and the Brownian tree.^ What are some examples of fractals?
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ Random fractals are quite familiar and many look like random walks (Brownian motion); dendrites; or lightning bolts.
    • Fractal Antenna Systems, Inc. 11 January 2010 5:12 UTC www.fractenna.com [Source type: FILTERED WITH BAYES]

     The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.
  • Strange attractors – Generated by iteration of a map or the solution of a system of initial-value differential equations that exhibit chaos.

Classification

.Fractals can also be classified according to their self-similarity.^ Above all, fractal meant self-similar."
  • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

 ^ The World-Wide Web itself has grown and has self-organized according to a self-similar, small-world structure (Barabási, 2002).
  • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

 ^ Fractals are a means by which time, or system dynamics, gets etched into form via self-similar, recursive loops that exist on multiple size scales.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 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.^ "Self-similarity is symmetry across scale.
    • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

     ^ Fractals are _generally_ self-similar and independent of scale.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ Above all, fractal meant self-similar."
    • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

     .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.^ "Self-similarity is symmetry across scale.
    • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

     ^ Most often, especially in natural fractals, self-similarity is approximate or statistical.
    • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

     ^ Fractals are _generally_ self-similar and independent of scale.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms.^ Above all, fractal meant self-similar."
    • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

     ^ The World-Wide Web itself has grown and has self-organized according to a self-similar, small-world structure (Barabási, 2002).
    • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

     ^ Fractals are a means by which time, or system dynamics, gets etched into form via self-similar, recursive loops that exist on multiple size scales.
    • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

     .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.^ "Self-similarity is symmetry across scale.
    • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

     ^ This is what they mean when they say a scaling fractal is self-similar.
    • The Fractal Dances of Nature 11 January 2010 5:12 UTC www.rps.psu.edu [Source type: FILTERED WITH BAYES]

     ^ Roughly, fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .Most reasonable definitions of "fractal" trivially imply some form of statistical self-similarity.^ Most often, especially in natural fractals, self-similarity is approximate or statistical.
    • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

     ^ Above all, fractal meant self-similar."
    • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

     ^ One of the most interesting is self-similarity .
    • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

     .(Fractal dimension itself is a numerical measure which is preserved across scales.^ Roughly, fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ The spaces between buildings are a fractal structure on the scale of the city itself.
    • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

     ^ Roughly, fractal dimension can be calculated by taking the limit of the quo- tient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.

In nature

.Approximate fractals are easily found in nature.^ Most often, especially in natural fractals, self-similarity is approximate or statistical.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 ^ He found that near/at DC, and only near/at DC, a fractal structure can approximate a capacitor and thus there can be bona fide fractal capacitors.
  • Fractal Antenna Systems, Inc. 11 January 2010 5:12 UTC www.fractenna.com [Source type: FILTERED WITH BAYES]

 .These objects display self-similar structure over an extended, but finite, scale range.^ "Self-similarity is symmetry across scale.
  • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

 ^ The World-Wide Web itself has grown and has self-organized according to a self-similar, small-world structure (Barabási, 2002).
  • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

 ^ A structure defined at an overall size x implies something similar at a size rx , where r is a scaling factor like 1/3 .
  • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

 .Examples include clouds, snow flakes, crystals, mountain ranges, lightning, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels.^ Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line (Mandelbrot, 1977, p.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 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.^ Mathematical formulas are used to model self similiar natural forms.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 ^ A: The Fractal Genetic Optimizer is a computer-based optimizing tool which we use to help identify the best fractal designs for a given antenna or electronics problem.
  • Fractal Antenna Systems, Inc. 11 January 2010 5:12 UTC www.fractenna.com [Source type: FILTERED WITH BAYES]

 ^ It introduces chaos fractals and dynamics using a combination of hands-on computer experimentation and precalculus math.
  • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 .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.^ Fractals exhibit self-similarity, meaning their inner structure has the same pattern as their outer structure - like a pine cone or a fern tree.
  • Fractal Field Breakthru-Technologies SCHEDULE and Online Links 11 January 2010 5:12 UTC www.goldenmean.info [Source type: FILTERED WITH BAYES]

 ^ Self-similarity of (top left and top right) branches of the dendritic tree is qualitatively apparent in Fig.4 (compare B with either A or C).
  • Pellionisz (1989) Neural Geometry: Towards a Fractal Model of Neurons 11 January 2010 5:12 UTC usa-siliconvalley.com [Source type: Academic]

 ^ Self-similarity is a newly discovered symmetry in nature by which parts of fractal objects relate to their wholes.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 .The connection between fractals and leaves are currently being used to determine how much carbon is contained in trees.^ SPD: contains generators for fractal mountain, tree, recursive tetrahedron.
  • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

 ^ But what separates this from the rest is the judicious use of contrast between larger and smaller objects, along with the nicely shaped contours of the fractal 'shell'.
  • Skytopia - Mystery of the Real 3D Mandelbrot Fractal 11 January 2010 5:12 UTC www.skytopia.com [Source type: General]

 ^ This section shows how a 3.3 level, Java Fractal platform can be used, in order to illustrate how the APIs defined in this specification can be used to create, assemble and reconfigure component configurations.
  • Fractal - Specification 11 January 2010 5:12 UTC fractal.ow2.org [Source type: Reference]

 [5]
.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).^ Above all, fractal meant self-similar."
  • Fractal Evolution 11 January 2010 5:12 UTC www.fractal.org [Source type: FILTERED WITH BAYES]

 ^ One can also see that it exhibits self-similarity.
  • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

 ^ Fractals have another key property -- that of coherence and self-similarity.
  • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

 [6]
A fractal that models the surface of a mountain (animation)
Photograph of a romanesco broccoli, showing a naturally occurring fractal
Fractal pentagram drawn with a vector iteration program

In creative works

Further information: Fractal art
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.^ What underlying mechanisms are at work here to cause overt patterns of fractal behavior.
  • The Oil Drum | Fractal Adaptive Cycles in Natural and Human Systems 11 January 2010 5:12 UTC www.theoildrum.com [Source type: FILTERED WITH BAYES]

 ^ Morigiwa K, Tauchi M, Fukuda Y. Fractal analysis of ganglion cell dendritic branching patterns of the rat and cat retinae.
  • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

 ^ Computer modeling reveals hidden, fractal order beneath apparently random, surface behavior typical of many chaotic systems.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 [7]
.Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns.^ Again, you can see striations in parts of the image which are an artifact of the technique used to produce the smooth, blurred effect found in both flames.

 ^ Trying to make fractals that looked like something is what motivated me to learn to use the software better, something I'm still working on even now.

 ^ Fractal images are used as an alternative to costly elaborate sets to produce fantasy landscapes.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 [8] It involves pressing paint between two surfaces and pulling them apart.
.Fractals are also prevalent in African art and architecture.^ Fractal geometry in traditional african architecture, Dynamics Newsletter , June.
  • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

 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.^ Often Priore found that every cell of the body—even the hair— must be irradiated and treated ("charged up") with the signal, for the disease pattern was in every cell.
  • Fractal Field Breakthru-Technologies SCHEDULE and Online Links 11 January 2010 5:12 UTC www.goldenmean.info [Source type: FILTERED WITH BAYES]

 [9]
.In a 1996 interview David Foster Wallace admitted that the structure of his novel Infinite Jest was inspired by fractals, specifically the Sierpinski triangle.^ "Inside the Sierpinski Temple" David Makin (aka MakinMagic) has also created variations of the fractal temple - this one's on the inside.
  • Skytopia - Mystery of the Real 3D Mandelbrot Fractal 11 January 2010 5:12 UTC www.skytopia.com [Source type: General]

 ^ The Sierpinski Triangle Unlinke the Koch Snowflake, which is generated with infinite additions, the Sierpinski triangle is created by infinite removals.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 ^ The Sierpinski Triangle is one of the easiest fractals to generate yourself.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

 [10]
.A fractal is formed when pulling apart two glue-covered acrylic sheets.^ Internally, a Fractal component is formed out of two parts: a controller (also called membrane), and a content (see Figure 5 ).
  • Fractal - Specification 11 January 2010 5:12 UTC fractal.ow2.org [Source type: Reference]

 ^ In the same manner of thinking, a wavy fractal scene will cover a dimension somewhere between two and three.
  • http://library.thinkquest.org/3493/frames/fractal.html 11 January 2010 5:12 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]
.High voltage breakdown within a 4″ block of acrylic creates a fractal Lichtenberg figure.^ Double Fractal is a program that creates fast and easily high complex images .
  • Fractal downloads at VicMan 11 January 2010 5:12 UTC www.vicman.net [Source type: General]
.Fractal branching occurs in a fractured surface such as a microwave-irradiated DVD.^ You see, now we have the fractal dimension of a contour line of the fracture surface — everything that’s at a certain elevation — and it’s 1.12 or 1.31 or something.
  • The Fractal Dances of Nature 11 January 2010 5:12 UTC www.rps.psu.edu [Source type: FILTERED WITH BAYES]

 ^ Everybody should be extremely skeptical — until we computer-generate a fractal model that looks like a real fracture surface.” .
  • The Fractal Dances of Nature 11 January 2010 5:12 UTC www.rps.psu.edu [Source type: FILTERED WITH BAYES]

 ^ Naturally-occurring fractals (such as cauliflowers, fern leaves, and the human lung) exhibit a nested structure with r not very different from 1/3 (Salingaros, 1995; Salingaros & West, 1999).
  • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

 [11]
A DLA cluster grown from a copper(II) sulfate solution in an electrodeposition cell
A "woodburn" fractal
A magnification of the phoenix set
A fractal flame created with the program Apophysis
Fractal made by the program Sterling
A fractal created using the program Apophysis and a julian transform

Applications

Main article: Fractal analysis
.As described above, random fractals can be used to describe many highly irregular real-world objects.^ Fractals in Nature Seen a fractal in the real world?
  • fractalforums.com - Welcome to Fractal Forums - Index 11 January 2010 5:12 UTC www.fractalforums.com [Source type: General]

 ^ These dimensions are known as topological dimensions, and have been used for many years to describe the shape and position of objects.
  • Fractal Analysis of Trabecular Bone — Musculoskeletal Radiology — UW Radiology 11 January 2010 5:12 UTC www.rad.washington.edu [Source type: Academic]

 ^ But what separates this from the rest is the judicious use of contrast between larger and smaller objects, along with the nicely shaped contours of the fractal 'shell'.
  • Skytopia - Mystery of the Real 3D Mandelbrot Fractal 11 January 2010 5:12 UTC www.skytopia.com [Source type: General]

 Other applications of fractals include:[12]

See also

References

  1. ^ Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company.. ISBN 0-7167-1186-9. 
  2. ^ Briggs, John (1992). Fractals:The Patterns of Chaos. London : Thames and Hudson, 1992.. pp. 148. ISBN 0500276935, 0500276935. 
  3. ^ Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd.. xxv. ISBN 0-470-84862-6. 
  4. ^ The Hilbert curve map is not a homeomorhpism, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert map in R2 are both 2. Note, however, that the topological dimension of the graph of the Hilbert map (a set in R3) is 1.
  5. ^ "Hunting the Hidden Dimension." Nova. PBS. WPMB-Maryland. 28 October 2008.
  6. ^ Hohlfeld R, Cohen N (1999). "Self-similarity and the geometric requirements for frequency independence in Antennae". Fractals 7 (1): 79–84. doi:10.1142/S0218348X99000098. 
  7. ^ Richard Taylor, Adam P. Micolich and David Jonas. Fractal Expressionism : Can Science Be Used To Further Our Understanding Of Art?
  8. ^ A Panorama of Fractals and Their Uses by Michael Frame and Benoît B. Mandelbrot
  9. ^ Ron Eglash. African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press 1999.
  10. ^ http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace
  11. ^ Peng, Gongwen; Decheng Tian (21 July 1990). "The fractal nature of a fracture surface". Journal of Physics A 23 (14): 3257–3261. doi:10.1088/0305-4470/23/14/022. http://www.iop.org/EJ/abstract/0305-4470/23/14/022. Retrieved 2007-06-02. 
  12. ^ "Applications". http://library.thinkquest.org/26242/full/ap/ap.html. Retrieved 2007-10-21. 

Further reading

  • Barnsley, Michael F., and Hawley Rising. Fractals Everywhere. .Boston: Academic Press Professional, 1993. ISBN 0-12-079061-0
  • Falconer, Kenneth.^ A29a_: Some references are: M. Barnsley, _Fractals Everywhere_, Academic Press Inc., 1988, 1993.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     Techniques in Fractal Geometry. .John Wiley and Sons, 1997. ISBN 0-471-92287-0
  • Jürgens, Hartmut, Heins-Otto Peitgen, and Dietmar Saupe.^ E. Peters, _Fractal Market Analysis - Applying Chaos Theory to Investment & Economics_, John Wiley & Sons, 1994, ISBN 0-471-58524-6.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ C. Pickover, _Keys to Infinity_, (1995) John Wiley: NY. ISBN 0-471-11857-5.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .Chaos and Fractals: New Frontiers of Science.^ H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_, Springer-Verlag, New York, 1988.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ This FAQ is posted monthly to the Usenet newsgroups: sci.fractals ("Objects of non-integral dimension and other chaos"), sci.answers , and news.answers .
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ Chaos and Fractals: New Frontiers of Science_.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .New York: Springer-Verlag, 1992. ISBN 0-387-97903-4
  • Benoît B. Mandelbrot The Fractal Geometry of Nature.^ How long is a coastline?_, see The Fractal Geometry of Nature 4.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ Indeed, in his epoch-making book on nature's geometries, Mandelbrot ( 1977) surmised that "it would be nice" if neurons, specifically Purkinje cells of the cerebellum, turned out to be fractals.
    • Pellionisz (1989) Neural Geometry: Towards a Fractal Model of Neurons 11 January 2010 5:12 UTC usa-siliconvalley.com [Source type: Academic]

     ^ C. Pickover, (1995) _Chaos in Wonderland: Visual Adventures in a Fractal World._ St. Martin's Press: New York.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .New York: W. H. Freeman and Co., 1982. ISBN 0-7167-1186-9
  • Peitgen, Heinz-Otto, and Dietmar Saupe, eds.^ New York: Freeman.
    • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

     ^ M. Schroeder, _Fractals, Chaos, and Power Laws: Minutes from an Infinite Paradise_, W. H. Freeman, New York, 1991.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ New York: W.H. Freeman.
    • The Fractal Psyche 11 January 2010 5:12 UTC www.goertzel.org [Source type: FILTERED WITH BAYES]

     .The Science of Fractal Images.^ B. Bani-Eqbal, Speeding up fractal image compression, Proceedings from IS&TSPIE 1995 Symposium on Electronic Imaging: Science & Technology Vol.
    • ����������� ������ ����������� 11 January 2010 5:12 UTC www.compression.ru [Source type: Academic]

     ^ D. Saupe, Lean domain pools for fractal image compression , in: Proceedings from IS&SPIE 1996 Symposium on Electronic Imaging: Science & Technology -- Still Image Compression II,Vol.
    • ����������� ������ ����������� 11 January 2010 5:12 UTC www.compression.ru [Source type: Academic]

     .New York: Springer-Verlag, 1988. ISBN 0-387-96608-0
  • Clifford A. Pickover, ed.^ C. Pickover, (1995) _Chaos in Wonderland: Visual Adventures in a Fractal World._ St. Martin's Press: New York.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ H. O. Peitgen and P. H. Richter, _The Beauty of Fractals_, Springer- Verlag, New York, 1986.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_, Springer-Verlag, New York, 1988.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .Chaos and Fractals: A Computer Graphical Journey - A 10 Year Compilation of Advanced Research.^ Another collection of fractal images is archived at ftp.maths.tcd.ie/pub/images/Computer [134.226.81.10].
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ Chaos, Fractals, Dimension: mathematics in the age of the computer by Glenn Elert.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ It introduces chaos fractals and dynamics using a combination of hands-on computer experimentation and precalculus math.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .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.^ J.L. McCauley, _Chaos, dynamics, and fractals : an algorithmic approach to deterministic chaos_, Cambridge University Press, 1993.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ E. Ott, _Chaos in dynamical systems_, Cambridge University Press, 1993.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ The Waite Group, 1993.
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ISBN 0-691-02445-6 paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.
  • Sprott, Julien Clinton (2003). .Chaos and Time-Series Analysis.^ It introduces fractals from geometrical ideas, covers a wide variety of topics, and covers things such as time series and R/S analysis that aren't usually considered.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     .Oxford University Press.^ Kauffman, S. (1995) At Home in the Universe , Oxford University Press, New York.
    • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

     ^ Alexander, C., Silverstein, M., Angel, S., Ishikawa, S. & Abrams, D. (1975) The Oregon Experiment , Oxford University Press, New York.
    • "Connecting the Fractal City", by Nikos A. Salingaros. 11 January 2010 5:12 UTC zeta.math.utsa.edu [Source type: FILTERED WITH BAYES]

     ^ Oxford University Press, 1997.
    • ����������� ������ ����������� 11 January 2010 5:12 UTC www.compression.ru [Source type: Academic]

     .ISBN 0-19-850839-5 and ISBN 978-0-19-850839-7. 
  • 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.^ NOTICE (from Michael Peters): HOP - Fractals in Motion opens the door to a completely new world of fractals!
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ R. L. Devaney, _Chaos, Fractals, and Dynamics_, Addison-Wesley, 1990.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     "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.^ This book contains detailed mathematical descriptions of chaos, the Mandelbrot set, etc.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]
    • sci.fractals FAQ 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ Fractal Studio: Mandelbrot set program; handles distributed computing.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ What's so amazing about this fractal is first how it contains detail in all 3 dimensions, but also how close it comes to the style and character of the 2D Mandelbrot set.
    • Skytopia - Mystery of the Real 3D Mandelbrot Fractal 11 January 2010 5:12 UTC www.skytopia.com [Source type: General]

     .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.^ H. O. Peitgen and P. H. Richter, _The Beauty of Fractals_, Springer- Verlag, New York, 1986.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_, Springer-Verlag, New York, 1988.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     ^ G. A. Edgar, _Measure Topology and Fractal Geometry_, Springer- Verlag Inc., 1990.
    • Fractal Frequently Asked Questions and Answers 11 January 2010 5:12 UTC www.faqs.org [Source type: Academic]

     Available in PDF version at [1].

External links

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[[File:|thumb|right|200px|A Sierpinski triangle, after 7 iterations.]]

File:Mandel zoom 00 mandelbrot
The Mandelbrot set is a famous example of a fractal.
File:Frost Water crystal on Mercury 20Feb2010
Frost crystals on a cold window show how fractals can work in nature

A fractal is any equation or pattern, that when seen as an image, produces a picture, which can be zoomed into infinity and will still produce the same picture. The word fractal was made by Benoît Mandelbrot in 1975 from the Latin word fractus, which means "broken" or "fractured". A simple example is a tree that branches infinitely into smaller branches, and those branches into smaller branches and so on. Fractals are not only beautiful, but also have many practical applications.

Examples

There are many types of fractals, made in a large variety of ways. One example is the Sierpinski triangle, where there are an infinite number of small triangles inside the large one. Another example is the Mandelbrot set, named for Benoît Mandelbrot. The Sierpinksi triangle is constructed using patterns, but the Mandelbrot set is based on an equation.

There are also many natural examples of fractals in nature including trees, snowflakes, some vegetables and coastlines.

Uses

Some fractals exist only for artistic reasons, but others are very useful. Fractals are very efficient shapes for radio antennas and are used in computer chips to efficiently connect all the components.

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