Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Fractal |
FractalNoun1. A geometric pattern that is repeated at every scale and so cannot be represented by classical geometry. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "fractal" was first used: 1975. (references) |
| Domain | Definition |
Computing | Fractal |
Mathematics | Benoit Mandelbrot coined the word in 1975 from the latin fractus which describes a broken stone-broken up and irregular. Fractals are geometrical shapes that, contrary to those of Euclid, are not regular at all. First, they are irregular all over. Secondly, they have the same degree of irregularity on all scales. A fractal object looks the same when examined from far away or nearby-it is self-similar. Source: European Union. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
In mathematics, a fractal is an object which is self-similar; they are repetitive in shape, but not in size. In other words, no matter how much you magnify a fractal, it will always look the same (or at least similar). Perhaps the simplest example of a fractal would be the spiral. Fractals are generally irregular; and thus, not definable by traditional geometry -- furthermore, fractals tend to have repetitive details, visible at any arbitrary scale. A fractal may have fractional Hausdorff (or box-counting) dimensions; they may also be defined recursively.
Mandelbrot set These characteristics of fractals, while intuitively appealing, are (aside from dimensionality) remarkably hard to condense into a mathematically precise definition. The problem with most definitions of fractal is that there are objects that one would like to call fractals but which do not satisfy the definition. To name a few problems: there is no precise meaning of "too irregular"; there are many ways that an object can be self-similar; not every fractal is defined recursively; and the many definitions of dimension admitting fractional values don't, in general, agree numerically (so an acceptable definition of fractal cannot be based on a single fractal dimension).
Approximate fractals (objects displaying complex structure over a vary broad, but finite, scale range) are easily found in nature. These naturally occurring fractals (like clouds, mountains, river networks, and systems of blood vessels) have both lower and upper cut-offs, but they are separated by several orders of magnitude. It is noteworthy that, despite being ubiquitous, fractals were not considered a legitimate object of study (or even defined!) until well into the 20th century.
Examples of fractals are the Mandelbrot set, Lyapunov fractal, Cantor set, Sierpinski carpet and triangle, Peano curve and the Koch snowflake. Fractals can be deterministic or stochastic. Chaotic dynamical systems are often (if not always) associated with fractals.
There are three broad categories of fractals that are commonly studied at this time:
Of all of these, only Iterated function systems usually display the well known "self-similarity" property--meaning that their complexity is invariant under scaling transforms. Fractals such as the Mandelbrot set are more loosely self-similar: they contain small copies of the entire fractal in distorted and degenerate forms.
- Iterated function systems. These have a fixed geometric replacement rule (Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake).
- Fractals defined by a recurrence relation at each point in a space (such as the complex plane). An example of this type are the Mandelbrot set and the Lyapunov fractal. These are also called escape-time fractals.
- Random fractals, generated by stochastic rather than deterministic processes, for example Fractal landscapes and Lévy flights.
Harrison extended Newtonian calculus to fractal domains, including the theorems of Gauss, Green, and Stokes.
Fractals are usually calculated by computers with fractal software. See External Links.
Random fractals have the greatest practical use, and can be used to describe many highly irregular real-world objects. Examples include clouds, mountains, turbulence, coastlines and trees. Fractal techniques have also been employed in image compression, as well as a variety of scientific disciplines.
See also: Fractal art, Graftals, Hausdorff dimension, Constructal theory
References, further reading
- 1 Fractal Geometry, by Kenneth Falconer; John Wiley & Son Ltd; ISBN 0471922870 (March 1990)
- The Fractal Geometry of Nature, by Benoit Mandelbrot; W H Freeman & Co; ISBN 0716711869 (hardcover, September 1982).
- The Science of Fractal Images, by Heinz-Otto Peitgen, Dietmar Saupe (Editor); Springer Verlag; ISBN 0387966080 (hardcover, August 1988)
- Fractals Everywhere, by Michael F. Barnsley; Morgan Kaufmann; ISBN 0120790610
External Links
- Fractal Properties
- more information on fractals from FAQS.org
- Many good Fractal examples
- Fractal Landscapes
- Ultra Fractal - fractal software for Windows
- Fractal Dimensions
- Fractal Calculus
- Mitchell-Green Gravity Set
- Fractint software for various operating systems
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Fractal."
Crosswords: Fractal |
| Specialty definitions using "fractal": fractal compression, fractal dimension, Fractals ♦ Landscape indexes ♦ Mandelbrot set, MPEG-4 ♦ National Fish and Wildlife Forensics Laboratory, Nonlinear Dynamics ♦ Sierpinski triangle. (references) |
| Non-English Usage: "Fractal" is also a word in the following language with the English translation in parentheses. Portuguese (fractal). |
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Theater & Movies | |||
Music |
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High Tech |
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Source: compiled by the editor from various references; see credits. | |||
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| "FL 11" by Lucien Aréstegüi Commentary: "FRACTAL LIFE 11." |
Source: photographs selected by the editor, with permission from the photographers. |
| "Fractal" is generally used as an adjective (general or positive) -- approximately 95.45% of the time. "Fractal" is used about 44 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Adjective (general or positive) | 95.45% | 42 | 52,864 |
| Noun (proper) | 4.55% | 2 | 245,945 |
| Total | 100.00% | 44 | N/A |
Source: compiled by the editor from several corpora; see credits.
Expressions using "fractal": fractal compression ♦ fractal dimension. Additional references. | |
| Hypenated Usage | |
Ending with "fractal": non-fractal. | |
| Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Expression | Frequency per Day | Expression | Frequency per Day |
fractal | 4,139 | 3d fractal | 24 |
genuine fractal | 183 | best fractal | 22 |
fractal art | 160 | chaos and fractal | 22 |
fractal gallery | 95 | background fractal | 21 |
fractal geometry | 79 | fractal graphic | 21 |
fractal software | 74 | flower fractal | 20 |
fractal image | 72 | fractal program | 18 |
fractal picture | 47 | 800x600 fractal | 18 |
fractal design | 47 | fractal design painter | 18 |
fractal wallpaper | 45 | ultra fractal | 18 |
fractal music | 41 | dimension fractal | 18 |
butterfly fractal | 39 | fractal studio underground | 18 |
fractal free ware | 34 | art fantasy fractal | 15 |
fractal art gallery | 31 | arquitectura fractal | 15 |
fractal screensaver | 30 | fractal free wallpaper | 15 |
fractal generator | 26 | fractal terrain | 14 |
fractal print | 26 | animated fractal | 14 |
fractal free | 25 | fractal screen saver | 14 |
black fractal white | 25 | fractal antenna | 13 |
boggling fractal mind | 25 | fractal infinities | 13 |
| Source: compiled by the editor from various references; see credits. | |||
| Language | Translations for "fractal"; alternative meanings/domain in parentheses. | |
Danish | fraktal. (various references) | |
Dutch | fractalen. (various references) | |
Finnish | fraktaali. (various references) | |
French | fractale. (various references) | |
German | Fraktal. (various references) | |
Greek | κλασματική συμπίεση. (various references) | |
Italian | frattale. (various references) | |
Japanese Kanji | フラクション活動 (flag, flannel, flap, flap pocket, flapper, flash, flashback, flashlight, flask, flat, flat collar, flat race, flush, fraction activity, fragment, fragmentation, frappe, fresco, frustration, hula dance, news flash). (various references) | |
Japanese Katakana | フラクタル . (various references) | |
Pig Latin | actalfray.(various references) | |
Portuguese | fractal. (various references) | |
Russian | рекурсивный (recursive), фракталь, дробная размерность. (various references) | |
Spanish | fractal. (various references) | |
Swedish | fraktal. (various references) | |
| Source: compiled by the editor from various translation references. | ||
| Language | Period | Translations |
| Latin | 500 BCE-Modern | fractus. (various references) |
| Source: compiled by the editor from various references. | ||
Derivations | |
Words beginning with "fractal": fractals. (additional references) | |
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"Fractal" is suggested in spellcheckers for the following: Factel, farakat, farctal, fracht, fractail, fractally, fractl, fractle, fractual, frata. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "fractal" (pronounced fra"ktul) |
| 5 | -a" k t u l | artiodactyl, tactile. |
| 4 | -k t u l | ductile, projectile, rectal. |
| 3 | -t u l | infantile, infertile, accidental, acquittal, anecdotal, battle, beetle, belittle, betel, bicoastal, bottle, brattle, Bristol, brittle, brutal, butyl, Cantle, capital, Capitol, cattle, chattel, chortle, coastal, coincidental, committal, compartmental, congenital, consonantal, continental, crustal, crystal, dental, detrimental, developmental, digital, disgruntle, dismantle, distal, elemental, embattle, entitle, environmental, experimental, extramarital, fatal, fertile, fetal, frontal, fundamental, futile, genital, gentle, glottal, governmental, horizontal, hospital, hostel, hostile, hurtle, immortal, immotile, incidental, incremental, spittle, startle, subtitle, subtle, supplemental, tattle, temperamental, instrumental, intercontinental, intergovernmental, judgmental, kettle, Kittel, Kittle, lentil, lintel, little, mantel, mantle, marital, mental, metal, mettle, monumental, mortal, motile, Myrtle, Natal, neonatal, nettle, noncommittal, nonfatal, nongovernmental, nonvolatile, occidental, occipital, orbital, oriental, ornamental, parental, parietal, pedestal, periodontal, petal, Pistil, pistol, pivotal, portal, postal, postnatal, Pottle, prattle, prefrontal, premarital, prenatal, quintal, rattle, rebuttal, recital, regimental, rental, resettle, scuttle, sentimental, settle, shuttle, skeletal, skittle, societal, throttle, title, tittle, tootle, total, transcendental, transcontinental, transmittal, turtle, unsentimental, unsettle, unsubtle, varietal, vegetal, versatile, vestal, vital, vittle, volatile, Whittle, Wintle. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
Direct Anagrams: flatcar. | |
| Words within the letters "a-a-c-f-l-r-t" | |
-2 letters: altar, artal, carat, craal, craft, fatal, ratal, talar. | |
-3 letters: acta, afar, alar, alfa, calf, carl, cart, fact, farl, flat, frat, raft, tala, talc. | |
-4 letters: aal, act, aft, ala, alt, arc, arf, art, car, cat, far, fat, lac, lar, lat, rat, tar. | |
-5 letters: aa, al, ar, at, fa, la, ta. | |
| Words containing the letters "a-a-c-f-l-r-t" | |
+1 letter: flatcars, fractals. | |
+2 letters: afterclap, factorial. | |
+3 letters: afterclaps, artificial, defalcator, factorable, factorials, fractional, malefactor, ultramafic. | |
+4 letters: artifactual, calefactory, conflagrant, craftsmanly, defalcators, facilitator, farcicality, frantically, fratricidal, interfacial, malefactors, parfocality, trafficable. | |
+5 letters: artificially, characterful, confabulator, facilitators, facilitatory, forecastable, fractionally, ultracareful. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Definition 2. Crosswords 3. Usage: Commercial 4. Images: Digital Art | 5. Usage Frequency 6. Expressions 7. Expressions: Internet 8. Translations: Modern | 9. Translations: Ancient 10. Derivations 11. Rhymes 12. Anagrams | 13. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
