Fractal Wiki

8/12/2019 Fractal Wiki

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12/4/13 Fractal - Wikipedia, the free encyclopedia

en.wikipedia.org/wiki/Fractal

Figure 1a. The Mandelbrot set

illustrates self-similarity. As you

zoom in on the image at finer and

finer scales, the same pattern re-

appears so that it is virtually

impossible to know at which level

you are looking.

Figure 1b. Mandelbrot zoomed 6x

Figure 1c. Mandelbrot zoomed

100x

Figure 1d. Even 2000 times

magnification of the Mandelbrot

set uncovers fine detail resembling

the full set.

FractalFrom Wikipedia, the free encyclopedia

Afractalis a mathematical set that has a fractal dimension that usually

exceeds its topological dimension[1]and may fall between the integers.[2]

Fractals are typically self-similarpatterns, where self-similarmeans they are

"the same from near as from far".[3]Fractals may be exactly the same at

every scale, or,as illustrated in Figure 1, they may be nearly the same at

different scales.[2][4][5][6]The definition of fractal goes beyond self-similarity

er seto exclude trivial self-similarity and include the ideaof a detailedatternrepeating itself.[2]:166; 18[4][7]

As mathematical equations, fractals are usually nowhere

differentiable.[2][6][8]An infinite fractal curve can be perceived of as winding

through space differently from an ordinary line, still being a 1-dimensional

line yet having a fractal dimension indicating it also resembles a

surface.

[1]:48[2]:15

The mathematical roots of the idea of fractals have been traced through a

formal path of published works, starting in the17th century with notions of

recursion, then moving through increasingly rigorous mathematical treatment

of the concept to the study of continuous but not differentiable functions in

the 19th century, and on to the coining of the wordfractalin the 20thcentury with a subsequent burgeoning of interest in fractals and computer-

based modelling in the 21st century.[9][10]The term "fractal" was first used

by mathematician Benot Mandelbrot in 1975. Mandelbrot based it on the

Latinfrctusmeaning "broken" or "fractured", and used it to extend theconcept of theoretical fractional dimensions to geometric patterns innature.[2]:405[7]

There is some disagreement amongst authorities about how the concept of a

fractal should be formally defined. Mandelbrot himself summarized it as

"beautiful, damn hard, increasingly useful. That's fractals."[11]The general

consensus is that theoretical fractals are infinitely self-similar, iterated, and

detailed mathematical constructs having fractal dimensions, ofwhich many

examples have been formulated and studied in great depth.[2][4][5]Fractals

are not limited to geometric patterns, but can also describe processes intime.[3][6][12]Fractal patterns with various degrees of self-similarity have

been rendered or studied in images, structures and sounds[13]and found in

nature,[14][15][16][17][18]technology,[19][20][21][22]art,[23][24][25]and law.[26]

Contents

1 Introduction
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en.wikipedia.org/wiki/Fractal 2

2 History

3 Characteristics

4 Common techniques for generating fractals

5 Simulated fractals

6 Natural phenomena with fractal features

7 In creative works

8 Applications in technology

9 In law10 See also

10.1 Fractal-generating programs

11 Notes

12 References

13 Further reading

14 External links

IntroductionThe word "fractal" often has different connotations for laypeople than mathematicians, where the layperson is more

likely to be familiar with fractal art than a mathematical conception. The mathematical concept is difficult to formally

define even for mathematicians, but key features can be understood with little mathematical background.

The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other

device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractal

however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals

nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g.,

people have pondered self-similarity informally such as in the infinite regress in parallel mirrors or the homunculus,the little man inside the head of the little man inside the head...). The difference for fractals is that the pattern

reproduced must be detailed.[2]:166; 18[4][7]

This idea of being detailed relates to another feature that can be understood without mathematical background:

Having a fractional or fractal dimension greater than its topological dimension, for instance, refers to how a fractal

scales compared to how geometric shapes are usually perceived. A regular line, for instance, is conventionally

understood to be 1-dimensional; if such a curve is divided into pieces each 1/3 the length of the original, there are

always 3 equal pieces. In contrast, consider the curve in Figure 2. It is also 1-dimensional for the same reason as

the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be

measured. The fractal curve divided into parts 1/3 the length of the original line becomes 4 pieces rearranged to

repeat the original detail, and this unusual relationship is the basis of its fractal dimension.

This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable"

In a concrete sense, this means fractals cannot be measured in traditional ways. [2][6][8]To elaborate, in trying to

find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough

lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve

the normal manner of measuring with a tape measure. But in measuring a wavy fractal curve such as the one in

Figure 2, one would never find a small enough straight segment to conform to the curve, because the wavy pattern
http://en.wikipedia.org/wiki/Rectifiable_curvehttp://en.wikipedia.org/wiki/Differentiablehttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Reasonhttp://-/?-http://en.wikipedia.org/wiki/Shapeshttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Homunculushttp://en.wikipedia.org/wiki/Infinite_regresshttp://en.wikipedia.org/wiki/Fractal_art


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Figure 2. Koch snowflake, a

fractal that 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"

Figure 3. A Julia set, a

fractal related to the

Mandelbrot set

would always re-appear, albeit at a smaller size, essentially pulling a little more of the tape measure into the total

length measured each time one attempted to fit it tighter and tighter to the curve. This is perhaps counter-intuitive,

but it is how fractals behave.[2]

History

The history of fractals traces a path from chiefly theoretical studies to modern

applications in computer graphics, with several notable people contributingcanonical fractal forms along the way.[9][10]According to Pickover, the

mathematics behind fractals began to take shape in the 17th century when the

mathematician and philosopher Gottfried Leibniz pondered recursive self-

similarity (although he made the mistake of thinking that only the straight line was

self-similar in this sense).[27]In his writings, Leibniz used the term "fractional

exponents", but lamented that "Geometry" did not yet know of them.[2]:405

Indeed, according to various historical accounts, after that point few

mathematicians tackled the issues and the work of those who did remained

obscured largely because of resistance to such unfamiliar emerging concepts,

which were sometimes referred to as mathematical "monsters".[8][9][10]Thus, it

was not until two centuries had passed that in 1872 Karl Weierstrass presented

the first definition of a function with a graph that would today be considered

fractal, having the non-intuitive property of being everywhere continuous but

nowhere differentiable.[9]:7[10]Not long after that, in 1883, Georg Cantor, who

attended lectures by Weierstrass,[10]published examples of subsets of the real

line known as Cantor sets, which had unusual properties and are now recognized

as fractals.[9]:1124Also in the last part of that century, Felix Klein and Henri Poincar introduced a category of

fractal that has come to be called "self-inverse" fractals.[2]:166

One of the next milestones came in 1904, when Helge von Koch, extending ideas

of Poincar and dissatisfied with Weierstrass's abstract and analytic definition,

gave a more geometric definition including hand drawn images of a similar

function, which is now called the Koch curve (see Figure 2)[9]:25.[10]Another

milestone came a decade later in 1915, when Wacaw Sierpiski constructed his

famous triangle then, one year later, his carpet. By 1918, two French

mathematicians, Pierre Fatou and Gaston Julia, though working independently,

arrived essentially simultaneously at results describing what are now seen as

fractal behaviour associated with mapping complex numbers and iterative

functions and leading to further ideas about attractors and repellors (i.e., pointsthat attract or repel other points), which have become very important in the study

of fractals (see Figure 3 and Figure 4).[6][9][10]Very shortly after that work was

submitted, by March 1918, Felix Hausdorff expanded the definition of "dimension", significantly for the evolution o

the definition of fractals, to allow for sets to have noninteger dimensions.[10]The idea of self-similar curves was

taken further by Paul Lvy, who, in his 1938 paperPlane or Space Curves and Surfaces Consisting of PartsSimilar to the Wholedescribed a new fractal curve, the Lvy C curve.[notes 1]
http://en.wikipedia.org/wiki/L%C3%A9vy_C_curvehttp://en.wikipedia.org/wiki/Paul_L%C3%A9vy_(mathematician)http://en.wikipedia.org/wiki/Felix_Hausdorffhttp://-/?-http://-/?-http://en.wikipedia.org/wiki/Strange_attractorshttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Gaston_Juliahttp://en.wikipedia.org/wiki/Pierre_Fatouhttp://en.wikipedia.org/wiki/Sierpinski_carpethttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Wac%C5%82aw_Sierpi%C5%84skihttp://-/?-http://en.wikipedia.org/wiki/Koch_curvehttp://en.wikipedia.org/wiki/Helge_von_Kochhttp://en.wikipedia.org/wiki/Henri_Poincar%C3%A9http://en.wikipedia.org/wiki/Felix_Kleinhttp://en.wikipedia.org/wiki/Cantor_sethttp://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Georg_Cantorhttp://en.wikipedia.org/wiki/Nowhere_differentiablehttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Intuition_(knowledge)http://en.wikipedia.org/wiki/Graph_of_a_functionhttp://en.wikipedia.org/wiki/Weierstrass_functionhttp://en.wikipedia.org/wiki/Karl_Weierstrasshttp://en.wikipedia.org/wiki/Straight_linehttp://en.wikipedia.org/wiki/Recursionhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Julia_sethttp://en.wikipedia.org/wiki/File:Julia_set_(indigo).pnghttp://en.wikipedia.org/wiki/Koch_snowflakehttp://en.wikipedia.org/wiki/File:Von_Koch_curve.gif


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Figure 4. A strange attractor

that exhibits multifractal

scaling

uniform mass center triangle

fractal

Different researchers have postulated that without the aid of modern computer graphics, early investigators were

limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate

some of the implications of many of the patterns they had discovered (the Julia

set, for instance, could only be visualized through a few iterations as very simple

drawings hardly resembling the image in Figure 3).[2]:179[8][10]That changed,

however, in the 1960s, when Benot Mandelbrot started writing about self-

similarity in papers such asHow Long Is the Coast of Britain? Statistical Self-

Similarity and Fractional Dimension,[28]

which built on earlier work by LewisFry Richardson. In 1975[7]Mandelbrot solidified hundreds of years of thought

and mathematical development in coining the word "fractal" and illustrated his

mathematical definition with striking computer-constructed visualizations. These

images, such as of his canonical Mandelbrot set pictured in Figure 1, captured the

popular imagination; many of them were based on recursion, leading to the

popular meaning of the term "fractal".[29]Currently, fractal studies are essentially

exclusively computer-based.[8][9][27]

Characteristics

One often cited description that Mandelbrot published to describe geometric

fractals 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";[2]

this is generally helpful but limited. Authorities disagree on the exact definition of

ractal, but most usually elaborate on the basic ideas of self-similarity and anunusual relationship with the space a fractal is embedded in.[2][3][4][6][30]One

point agreed on is that fractal patterns are characterized by fractal dimensions, but

whereas these numbers quantify complexity (i.e., changing detail with changing

scale), they neither uniquely describe nor specify details of how to constructparticular fractal patterns.[31]In 1975 when Mandelbrot coined the word

"fractal", he did so to denote an object whose HausdorffBesicovitch dimension is greater than its topological

dimension.[7]It has been noted that this dimensional requirement is not met by fractal space-filling curves such as

the Hilbert curve.[notes 2]

According to Falconer, rather than being strictly defined, fractals should, in addition to being nowhere differentiabl

and able to have a fractal dimension, be generally characterized by a gestalt of the following features;[4]

Self-similarity, which may be manifested as:

Exact self-similarity: identical at all scales; e.g. Koch snowflakeQuasi self-similarity: approximates the same pattern at different scales; may contain smacopies of the entire fractal in distorted and degenerate forms; e.g., the Mandelbrot set's

satellites are approximations of the entire set, but not exact copies, as shown in Figure Statistical self-similarity: repeats a pattern stochastically so numerical or statisticalmeasures are preserved across scales; e.g., randomly generated fractals; the well-knowexample of the coastline of Britain, for which one would not expect to find a segment

scaled and repeated as neatly as the repeated unit that defines, for example, the Koch

snowflake[6]
http://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Self-Similarity_and_Fractional_Dimensionhttp://-/?-http://en.wikipedia.org/wiki/Stochastichttp://-/?-http://en.wikipedia.org/wiki/Mandelbrot_sethttp://-/?-http://en.wiktionary.org/wiki/gestalthttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Hilbert_curvehttp://en.wikipedia.org/wiki/Space-filling_curvehttp://en.wikipedia.org/wiki/Lebesgue_covering_dimensionhttp://en.wikipedia.org/wiki/Hausdorff%E2%80%93Besicovitch_dimensionhttp://en.wikipedia.org/wiki/Complexityhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Shapehttp://-/?-http://en.wikipedia.org/wiki/Mandelbrot_sethttp://en.wikipedia.org/wiki/Lewis_Fry_Richardsonhttp://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Self-Similarity_and_Fractional_Dimensionhttp://en.wikipedia.org/wiki/Beno%C3%AEt_Mandelbrothttp://-/?-http://en.wikipedia.org/wiki/File:Uniform_Triangle_Mass_Center_grade_5_fractal.gifhttp://en.wikipedia.org/wiki/Multifractalhttp://en.wikipedia.org/wiki/Strange_attractorhttp://en.wikipedia.org/wiki/File:Karperien_Strange_Attractor_200.gif


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Letter as a fractal

Figure 5. Self-similar

branching pattern modeled in

silico using L-systems

principles[18]

Qualitative self-similarity:as in a time series[12]

Multifractal scaling: characterized by more than one fractal dimension or scaling rule

Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may

have emergent properties[32](related to the next criterion in this list).

Irregularity locally and globally that is not easily described in traditional Euclidean geometric language

For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up

surfaces" and "swirls upon swirls".[1]Simple and "perhaps recursive" definitionssee Common techniques for generating fractals

As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar withou

having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks

detail, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimension, and i

fully defined without a need for recursion.[2][6]

Common techniques for generating

fractalsImages of fractals can be created by fractal generating programs.

Iterated function systems use fixed geometric

replacement rules; may be stochastic or deterministic;[33]

e.g., Koch snowflake, Cantor set, Haferman carpet,[34]

Sierpinski carpet, Sierpinski gasket, Peano curve, Harter-

Heighway dragon curve, T-Square, Menger sponge

Strange attractors use iterations of a map or solutions of

a system of initial-value differential equations that exhibitchaos (e.g., see multifractal image)

L-systems- use string rewriting; may resemble branchingpatterns, such as in plants, biological cells (e.g., neurons and

immune system cells[18]), blood vessels, pulmonary

structure,[35]etc. (e.g., see Figure 5) or turtle graphics patterns such

as space-filling curves and tilings

Escape-time fractals use a formula or recurrence relation at eachpoint in a space (such as the complex plane); usually quasi-self-

similar; also known as "orbit" fractals; e.g., the Mandelbrot set, Juliaset, Burning Ship fractal, Nova fractal and Lyapunov fractal. 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.

Random fractals use stochastic rules; e.g., Lvy flight,percolation clusters, self avoiding walks, fractal landscapes,

trajectories of Brownian motion and the Brownian tree (i.e.,

dendritic fractals generated by modeling diffusion-limited

aggregation or reaction-limited aggregation clusters).[6]
http://en.wikipedia.org/w/index.php?title=Reaction-limited_aggregation&action=edit&redlink=1http://en.wikipedia.org/wiki/Diffusion-limited_aggregationhttp://en.wikipedia.org/wiki/Brownian_treehttp://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Fractal_landscapeshttp://en.wikipedia.org/wiki/Self-avoiding_walkhttp://en.wikipedia.org/wiki/Percolation_theoryhttp://en.wikipedia.org/wiki/L%C3%A9vy_flighthttp://en.wikipedia.org/wiki/Lyapunov_fractalhttp://en.wikipedia.org/wiki/Nova_fractalhttp://en.wikipedia.org/wiki/Burning_Ship_fractalhttp://en.wikipedia.org/wiki/Julia_sethttp://en.wikipedia.org/wiki/Mandelbrot_sethttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Recurrence_relationhttp://en.wikipedia.org/wiki/Formulahttp://en.wikipedia.org/w/index.php?title=Tilings&action=edit&redlink=1http://en.wikipedia.org/wiki/Space-filling_curveshttp://en.wikipedia.org/wiki/Turtle_graphicshttp://-/?-http://en.wikipedia.org/wiki/L-systemhttp://-/?-http://en.wikipedia.org/wiki/Strange_attractorhttp://en.wikipedia.org/wiki/Menger_spongehttp://en.wikipedia.org/wiki/T-Square_(fractal)http://en.wikipedia.org/wiki/Dragon_curvehttp://en.wikipedia.org/wiki/Peano_curvehttp://en.wikipedia.org/wiki/Sierpinski_gaskethttp://en.wikipedia.org/wiki/Sierpinski_carpethttp://en.wikipedia.org/w/index.php?title=Haferman_carpet&action=edit&redlink=1http://en.wikipedia.org/wiki/Cantor_sethttp://en.wikipedia.org/wiki/Koch_snowflakehttp://en.wikipedia.org/wiki/Iterated_function_systemshttp://en.wikipedia.org/wiki/Fractal-generating_softwarehttp://en.wikipedia.org/wiki/Topological_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://-/?-http://en.wikipedia.org/wiki/Recursionhttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Emergent_propertieshttp://en.wikipedia.org/wiki/Multifractalhttp://en.wikipedia.org/wiki/L-systemshttp://en.wikipedia.org/wiki/In_silicohttp://en.wikipedia.org/wiki/File:KarperienFractalBranch.jpghttp://en.wikipedia.org/wiki/File:Letter_Oe_fractal.png


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A fractal generated by a

finite subdivision rule for an

alternating link

A fractal flame

Finite subdivision rulesuse a recursive topological algorithm for

refining tilings[36]and they are similar to the process of cell

division.[37]The iterative processes used in creating the Cantor set

and the Sierpinski carpet are examples of finite subdivision rules, as

is barycentric subdivision.

Simulated fractals

Fractal patterns have been modeled extensively, albeit within a range of scales

rather than infinitely, owing to the practical limits of physical time and space.

Models may simulate theoretical fractals or natural phenomena with fractal

features. The outputs of the modelling process may be highly artistic renderings,

outputs for investigation, or benchmarks for fractal analysis. Some specific

applications of fractals to technology are listed elsewhere. Images and other

outputs of modelling are normally referred to as being "fractals" even if they do

not have strictly fractal characteristics, such as when it is possible to zoom into a

region of the fractal image that does not exhibit any fractal properties. Also, these

may include calculation or display artifacts which are not characteristics of true

fractals.

Modeled fractals may be sounds,[13]digital images, electrochemical patterns,

circadian rhythms,[38]etc. Fractal patterns have been reconstructed in physical 3-

dimensional space[21]:10and virtually, often called "in silico" modeling.[35]Models

of fractals are generally created using fractal-generating software that implements

techniques such as those outlined above.[6][12][21]As one illustration, trees, ferns, cells of the nervous system,[18]

blood and lung vasculature,[35]and other branching patterns in nature can be modeled on a computer by using

recursive algorithms and L-systems techniques.[18]

The recursive nature of some patterns is obvious in certainexamplesa branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar

in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects. A

limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that

the phenomenon being modeled is formed by a process similar to the modeling algorithm.

Natural phenomena with fractal features

Further information: Patterns in nature

Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connectionbetween fractals and leaves, for instance, is currently being used to determine how much carbon is contained in

trees.[39]

Examples of phenomena known or anticipated to have fractal features are listed below:

clouds

river networks

fault lines

animal coloration patterns

Romanesco broccoli

crystals[42]

blood vessels and pulmonary vessels[35]

ocean waves[43]
http://en.wikipedia.org/wiki/Wind_wavehttp://en.wikipedia.org/wiki/Pulmonary_vesselshttp://en.wikipedia.org/wiki/Blood_vesselhttp://en.wikipedia.org/wiki/Crystalhttp://en.wikipedia.org/wiki/Romanesco_broccolihttp://en.wikipedia.org/wiki/Fault_linehttp://en.wikipedia.org/wiki/Riverhttp://en.wikipedia.org/wiki/Patterns_in_naturehttp://en.wikipedia.org/w/index.php?title=Random_fractal&action=edit&redlink=1http://en.wikipedia.org/wiki/Fernhttp://en.wikipedia.org/wiki/Frondhttp://en.wikipedia.org/wiki/L-systemshttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Patterns_in_naturehttp://en.wikipedia.org/wiki/Fractal-generating_softwarehttp://en.wikipedia.org/wiki/In_silicohttp://en.wikipedia.org/wiki/Circadian_rhythmhttp://en.wikipedia.org/wiki/Artifact_(error)http://-/?-http://en.wikipedia.org/wiki/Fractal_analysishttp://-/?-http://en.wikipedia.org/wiki/Barycentric_subdivisionhttp://en.wikipedia.org/wiki/Sierpinski_carpethttp://en.wikipedia.org/wiki/Cantor_sethttp://en.wikipedia.org/wiki/Cell_divisionhttp://en.wikipedia.org/wiki/Topologicalhttp://en.wikipedia.org/wiki/Finite_subdivision_rulehttp://en.wikipedia.org/wiki/Fractal_flamehttp://en.wikipedia.org/wiki/File:Apophysis-100303-104.jpghttp://en.wikipedia.org/wiki/Alternating_linkhttp://en.wikipedia.org/wiki/Finite_subdivision_rulehttp://en.wikipedia.org/wiki/File:Finite_subdivision_of_a_radial_link.png


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Frost crystals formed naturally

on cold glass illustrate fractal

process development in apurely physical system

A fractal is formed when

pulling apart two glue-covered

acrylic sheets

High voltage breakdown within

a 4 block of acrylic creates a

fractal Lichtenberg figure

Romanesco broccoli, showing

self-similar form approximating

a natural fractal

mountain ranges

craters

lightning bolts

coastlines

Mountain Goat horns

heart rates[14]

heartbeat[15]

earthquakes[22][40]

snow flakes[41]

DNA

various vegetables (cauliflower &

broccoli)

soil pores [44]

Psychological subjective perception[45]

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.[25]

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns. [46]It involves

pressing paint between two surfaces and pulling them apart.
http://en.wikipedia.org/wiki/Max_Ernsthttp://en.wikipedia.org/wiki/Decalcomaniahttp://en.wikipedia.org/wiki/Jackson_Pollockhttp://en.wikipedia.org/wiki/Fractal_arthttp://en.wikipedia.org/wiki/Psychological_subjective_perceptionhttp://en.wikipedia.org/wiki/DNAhttp://en.wikipedia.org/wiki/Snowflakehttp://en.wikipedia.org/wiki/Heart_soundshttp://en.wikipedia.org/wiki/Crater_(disambiguation)http://en.wikipedia.org/wiki/Mountainhttp://en.wikipedia.org/wiki/Self-similarhttp://en.wikipedia.org/wiki/Romanesco_broccolihttp://en.wikipedia.org/wiki/File:Fractal_Broccoli.jpghttp://en.wikipedia.org/wiki/Lichtenberg_figurehttp://en.wikipedia.org/wiki/File:Lichtenberg_figure_in_block_of_Plexiglas.jpghttp://en.wikipedia.org/wiki/Acrylhttp://en.wikipedia.org/wiki/File:Glue1_800x600.jpghttp://en.wikipedia.org/wiki/File:Frost_patterns_2.jpg


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A fractal that models the surface of a

mountain (animation)

Cyberneticist Ron Eglash has suggested that fractal geometry and

mathematics are prevalent in African art, games, divination, trade, 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.[24][47]

In a 1996 interview with Michael Silverblatt, David Foster Wallace

admitted that the structure of the first draft ofInfinite Jesthe gave to hiseditor Michael Pietsch was inspired by fractals, specifically the Sierpinski

triangle (aka Sierpinski gasket) but that the edited novel is "more like a

lopsided Sierpinsky Gasket".[23]

Applications in technology

Main article: Fractal analysis

fractal antennas[48]

Fractal transistor[49]

Fractal heat exchangers

Digital imaging

Urban growth[50][51]

Classification of histopathology

slides

Fractal landscape or Coastline

complexity

Enzyme/enzymology (Michaelis-Menten kinetics)

Generation of new music

Signal and image compression

Creation of digital photographic

enlargements

Fractal in soil mechanics

Computer and video game design

computer graphics

organic environments

procedural generation

Fractography and fracture mechanics

Small angle scattering theory of fractally

rough systems

T-shirts and other fashion

Generation of patterns for camouflage,

such as MARPATDigital sundial

Technical analysis of price series

Fractals in networks

medicine[21]

neuroscience[16][1

diagnostic

imaging[20]

pathology[52][53]

geology[54]

geography[55]

archaeology[56][57

soil mechanics[19]

seismology[22]

search and

rescue[58]

technical

analysis[59]

In lawIf a rule or principle of law is conceptualized as defining a two-dimensional "area" of conduct, conduct within whic

should be legal and conduct outside of which should be illegal, it has been observed that the border of that area

must be a fractal, because of the infinite and recursive potential exceptions and extensions necessary to account

appropriately for all variations in fact pattern that may arise.[26]

See also
http://en.wikipedia.org/wiki/Technical_analysishttp://en.wikipedia.org/wiki/Search_and_rescuehttp://en.wikipedia.org/wiki/Seismologyhttp://en.wikipedia.org/wiki/Soil_mechanicshttp://en.wikipedia.org/wiki/Archaeologyhttp://en.wikipedia.org/wiki/Geographyhttp://en.wikipedia.org/wiki/Geologyhttp://en.wikipedia.org/wiki/Pathologyhttp://en.wikipedia.org/wiki/Diagnostic_imaginghttp://en.wikipedia.org/wiki/Neurosciencehttp://en.wikipedia.org/wiki/Medicinehttp://en.wikipedia.org/wiki/Fractal_dimension_on_networkshttp://en.wikipedia.org/wiki/Digital_sundialhttp://en.wikipedia.org/wiki/MARPAThttp://en.wikipedia.org/wiki/Fashionhttp://en.wikipedia.org/wiki/T-shirthttp://en.wikipedia.org/wiki/SAXShttp://en.wikipedia.org/wiki/Fracture_mechanicshttp://en.wikipedia.org/wiki/Procedural_generationhttp://en.wikipedia.org/wiki/Lifehttp://en.wikipedia.org/wiki/Computer_graphicshttp://en.wikipedia.org/wiki/Game_designhttp://en.wikipedia.org/wiki/Fractal_in_soil_mechanicshttp://en.wikipedia.org/wiki/Fractal_compressionhttp://en.wikipedia.org/wiki/Signal_(information_theory)http://en.wikipedia.org/wiki/Algorithmic_compositionhttp://en.wikipedia.org/wiki/Michaelis-Menten_kineticshttp://en.wikipedia.org/wiki/Complexityhttp://en.wikipedia.org/wiki/Coasthttp://en.wikipedia.org/wiki/Fractal_landscapehttp://en.wikipedia.org/wiki/Histopathologyhttp://en.wikipedia.org/wiki/Categorisationhttp://en.wikipedia.org/wiki/Fractal_antennahttp://en.wikipedia.org/wiki/Fractal_analysishttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Infinite_Jesthttp://en.wikipedia.org/wiki/David_Foster_Wallacehttp://en.wikipedia.org/wiki/Michael_Silverblatthttp://en.wikipedia.org/wiki/Architecturehttp://en.wikipedia.org/wiki/Tradehttp://en.wikipedia.org/wiki/Divinationhttp://en.wikipedia.org/wiki/Gamehttp://en.wikipedia.org/wiki/African_arthttp://en.wikipedia.org/wiki/Ron_Eglashhttp://en.wikipedia.org/wiki/File:Animated_fractal_mountain.gif


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Banach fixed point theorem

Bifurcation theory

Box counting

Butterfly effect

Complexity

Constructal theory

CymaticsDiamond-square algorithm

Droste effect

Feigenbaum function

Fractal compression

Fractal cosmology

Fractal derivative

Fractal-generating software

Fracton

Golden ratio

Graftal

Greeble

LacunarityList of fractals by Hausdorff dimension

Mandelbulb

Mandelbox

Multifractal system

Newton fractal

Patterns in nature

Percolation

Power law

Publications in fractal

geometry

Random walkSacred geometry

Self-reference

Strange loop

Symmetry

Turbulence

Fractal-generating programs

There are many fractal generating programs available, both free and commercial. Some of the fractal generatingprograms include:

Apophysis - open source software for Microsoft Windows based systems

Electric Sheep - open source distributed computing software

Fractint - freeware with available source code

Sterling - Freeware software for Microsoft Windows based systems

SpangFract - For Mac OS

Ultra Fractal - A proprietary fractal generator for Microsoft Windows and Mac OS X based systems

XaoS - A cross platform open source fractal zooming program

Terragen - a fractal terrain generator.

Most of the above programs make two-dimensional fractals, with a few creating three-dimensional fractal objects,

such as quaternions, mandelbulbs and mandelboxes.

Notes

1. ^The original paper, Lvy, Paul (1938). "Les Courbes planes ou gauches et les surfaces composes de parties

semblables au tout".Journal de l'cole Polytechnique: 227247, 249291., is translated in Edgar, pages 181-239.2. ^The Hilbert curve map is not a homeomorphism, so it does not preserve topological dimension. The topological

dimension and Hausdorff dimension of the image of the Hilbert map in R2are both 2. Note, however, that the

topological dimension of thegraphof the Hilbert map (a set in R3) is 1.

References

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dimension"
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vascular constructive optimization" (http://books.google.com/books?id=t9l9GdAt95gC). In Losa, Gabriele A.;

Nonnenmacher, Theo F. Fractals in biology and medicine. Springer. pp. 5566. ISBN 978-3-7643-7172-2.>36. ^J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5

(2001), pp. 153196.

37. ^J. W. Cannon, W. Floyd and W. Parry. Crystal growth, biological cell growth and geometry.(http://books.google.co.uk/books?

id=qZHyqUli9y8C&pg=PA65&lpg=PA65&dq=%22james+w.+cannon%22+maths&source=web&ots=RP1svsBqg

&sig=kxyEXFBqOG5NnJncng9HHniHyrc&hl=en&sa=X&oi=book_result&resnum=1&ct=result#PPA65,M1)Pattern Formation in Biology, Vision and Dynamics, pp. 6582. World Scientific, 2000. ISBN 981-02-3792-

8,ISBN 978-981-02-3792-9.

38. ^Fathallah-Shaykh, Hassan M. (2011). "Fractal Dimension of the Drosophila Circadian Clock".Fractals19(4):423430. doi:10.1142/S0218348X11005476 (http://dx.doi.org/10.1142%2FS0218348X11005476).

39. ^"Hunting the Hidden Dimension."Nova. PBS. WPMB-Maryland. 28 October 2008.40. ^Sornette, Didier (2004). Critical phenomena in natural sciences: chaos, fractals, selforganization, and disorder

concepts and tools. Springer. pp. 128140. ISBN 978-3-540-40754-6.41. ^Meyer, Yves; Roques, Sylvie (1993).Progress in wavelet analysis and applications: proceedings of the

International Conference "Wavelets and Applications," Toulouse, France - June 1992(http://books.google.com/?id=aHux78oQbbkC&pg=PA25&dq=snowflake+fractals+book#v=onepage&q=snowflake%20fractals%20book&f=

alse). Atlantica Sguier Frontires. p. 25. ISBN 978-2-86332-130-0. Retrieved 2011-02-05.42. ^Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000).Pattern formation in biology,

vision and dynamics(http://books.google.com/?id=qZHyqUli9y8C&pg=PA78&dq=crystal+fractals+book#v=onepage&q=crystal%20fractals%20book&f=false).

World Scientific. p. 78. ISBN 978-981-02-3792-9.

43. ^Addison, Paul S. (1997).Fractals and chaos: an illustrated course(http://books.google.com/?id=l2E4ciBQ9qEC&pg=PA45&dq=lightning+fractals+book#v=onepage&q=lightning%20fractals%20book&f=fals

. CRC Press. pp. 4446. ISBN 978-0-7503-0400-9. Retrieved 2011-02-05.

44. ^Ozhovan M.I., Dmitriev I.E., Batyukhnova O.G. Fractal structure of pores of clay soil. Atomic Energy, 74, 241

243 (1993)

45. ^Pincus, David (September 2009). "The Chaotic Life: Fractal Brains Fractal Thoughts"

(http://www.psychologytoday.com/blog/the-chaotic-life/200909/fractal-brains-fractal-thoughts).psychologytoday.com.46. ^Frame, Michael; and Mandelbrot, Benot B.;A Panorama of Fractals and Their Uses

(http://classes.yale.edu/Fractals/Panorama/)

47. ^Nelson, Bryn; Sophisticated Mathematics Behind African Village Designs Fractal patterns use repetition onlarge, small scale(http://www.sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/2000/02/23/MN36684.DTL)San Francisco Chronicle, Wednesday, February 23, 2009

48. ^Hohlfeld, Robert G.; Cohen, Nathan (1999). "Self-similarity and the geometric requirements for frequency

independence in Antennae".Fractals7(1): 7984. doi:10.1142/S0218348X99000098(http://dx.doi.org/10.1142%2FS0218348X99000098).

49. ^Reiner, Richard; Waltereit, Patrick; Benkhelifa, Fouad; Mller, Stefan; Walcher, Herbert; Wagner, Sandrine;

Quay, Rdiger; Schlechtweg, Michael et al. (2012). "Fractal structures for low-resistance large area AlGaN/GaNpower transistors" (http://ieeexplore.ieee.org/xpl/login.jsp?

tp=&arnumber=6229091&url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F6222389%2F6228999%2F062290

91.pdf%3Farnumber%3D6229091).Proceedings of ISPSD: 341. doi:10.1109/ISPSD.2012.6229091(http://dx.doi.org/10.1109%2FISPSD.2012.6229091). ISBN 978-1-4577-1596-9.

50. ^Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions".

PLoS ONE6 (9): e24791. doi:10.1371/journal.pone.0024791 (http://dx.doi.org/10.1371%2Fjournal.pone.0024791PMC 3176775 (//www.ncbi.nlm.nih.gov/pmc/articles/PMC3176775). PMID 21949753


51. ^"Applications" (http://library.thinkquest.org/26242/full/ap/ap.html). Retrieved 2007-10-21.
http://library.thinkquest.org/26242/full/ap/ap.htmlhttp://www.ncbi.nlm.nih.gov/pubmed/21949753http://en.wikipedia.org/wiki/PubMed_Identifierhttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC3176775http://en.wikipedia.org/wiki/PubMed_Centralhttp://dx.doi.org/10.1371%2Fjournal.pone.0024791http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Special:BookSources/978-1-4577-1596-9http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://dx.doi.org/10.1109%2FISPSD.2012.6229091http://en.wikipedia.org/wiki/Digital_object_identifierhttp://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6229091&url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F6222389%2F6228999%2F06229091.pdf%3Farnumber%3D6229091http://dx.doi.org/10.1142%2FS0218348X99000098http://en.wikipedia.org/wiki/Digital_object_identifierhttp://www.sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/2000/02/23/MN36684.DTLhttp://classes.yale.edu/Fractals/Panorama/http://www.psychologytoday.com/blog/the-chaotic-life/200909/fractal-brains-fractal-thoughtshttp://en.wikipedia.org/wiki/Special:BookSources/978-0-7503-0400-9http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/?id=l2E4ciBQ9qEC&pg=PA45&dq=lightning+fractals+book#v=onepage&q=lightning%20fractals%20book&f=falsehttp://en.wikipedia.org/wiki/Special:BookSources/978-981-02-3792-9http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/?id=qZHyqUli9y8C&pg=PA78&dq=crystal+fractals+book#v=onepage&q=crystal%20fractals%20book&f=falsehttp://en.wikipedia.org/wiki/Special:BookSources/978-2-86332-130-0http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/?id=aHux78oQbbkC&pg=PA25&dq=snowflake+fractals+book#v=onepage&q=snowflake%20fractals%20book&f=falsehttp://en.wikipedia.org/wiki/Special:BookSources/978-3-540-40754-6http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Hunting_the_Hidden_Dimension&action=edit&redlink=1http://dx.doi.org/10.1142%2FS0218348X11005476http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Special:BookSources/9789810237929http://en.wikipedia.org/wiki/Special:BookSources/9810237928http://books.google.co.uk/books?id=qZHyqUli9y8C&pg=PA65&lpg=PA65&dq=%22james+w.+cannon%22+maths&source=web&ots=RP1svsBqga&sig=kxyEXFBqOG5NnJncng9HHniHyrc&hl=en&sa=X&oi=book_result&resnum=1&ct=result#PPA65,M1http://en.wikipedia.org/wiki/Special:BookSources/978-3-7643-7172-2http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/books?id=t9l9GdAt95gC


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52. ^Smith, Robert F.; Mohr, David N.; Torres, Vicente E.; Offord, Kenneth P.; Melton III, L. Joseph (1989). "Ren

insufficiency in community patients with mild asymptomatic microhematuria".Mayo Clinic proceedings. MayoClinic64(4): 409414. PMID 2716356 (//www.ncbi.nlm.nih.gov/pubmed/2716356).

53. ^Landini, Gabriel (2011). "Fractals in microscopy". Journal of Microscopy241(1): 18. doi:10.1111/j.1365-2818.2010.03454.x (http://dx.doi.org/10.1111%2Fj.1365-2818.2010.03454.x). PMID 21118245


54. ^Cheng, Qiuming (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology29(7): 919932. doi:10.1023/A:1022355723781 (http://dx.doi.org/10.1023%2FA%3A1022355723781).

55. ^Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions".PLoS ONE6 (9): e24791. doi:10.1371/journal.pone.0024791 (http://dx.doi.org/10.1371%2Fjournal.pone.0024791PMC 3176775 (//www.ncbi.nlm.nih.gov/pmc/articles/PMC3176775). PMID 21949753


56. ^Burkle-Elizondo, Gerardo; Valdz-Cepeda, Ricardo David (2006). "Fractal analysis of Mesoamerican pyramids"

Nonlinear dynamics, psychology, and lif e sciences10(1): 105122. PMID 16393505(//www.ncbi.nlm.nih.gov/pubmed/16393505).

57. ^Brown, Clifford T.; Witschey, Walter R. T.; Liebovitch, Larry S. (2005). "The Broken Past: Fractals in

Archaeology".Journal of Archaeological Method and Theory12: 37. doi:10.1007/s10816-005-2396-6(http://dx.doi.org/10.1007%2Fs10816-005-2396-6).

58. ^Saeedi, Panteha; Sorensen, Soren A. "An Algorithmic Approach to Generate After-disaster Test Fields for Searc

and Rescue Agents" (http://www.iaeng.org/publication/WCE2009/WCE2009_pp93-98.pdf).Proceedings of theWorld Congress on Engineering 2009: 9398. ISBN 978-988-17-0125-1.59. ^Bunde, A.; Havlin, S. (2009). "Fractal Geometry, A Brief Introduction to". Encyclopedia of Complexity and

Systems Science. p. 3700. doi:10.1007/978-0-387-30440-3_218 (http://dx.doi.org/10.1007%2F978-0-387-30440-3_218). ISBN 978-0-387-75888-6.

Further reading

Barnsley, Michael F.; and Rising, Hawley;Fractals Everywhere. Boston: Academic Press Professional,1993. ISBN 0-12-079061-0

Falconer, Kenneth; Techniques in Fractal Geometry. John Wiley and Sons, 1997. ISBN 0-471-92287-Jrgens, Hartmut; Peitgen, Heins-Otto; and Saupe, Dietmar; Chaos and Fractals: New Frontiers ofScience. New York: Springer-Verlag, 1992. ISBN 0-387-97903-4Mandelbrot, Benoit B.; The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982.ISBN 0-7167-1186-9

Peitgen, Heinz-Otto; and Saupe, Dietmar; eds.; The Science of Fractal Images. New York: Springer-Verlag, 1988. ISBN 0-387-96608-0

Pickover, Clifford A.; ed.; Chaos and Fractals: A Computer Graphical Journey - A 10 YearCompilation of Advanced Research. Elsevier, 1998. ISBN 0-444-50002-2

Jones, Jesse;Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993. ISBN 1-878739-46-8.

Lauwerier, Hans;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 program

in an appendix.

Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 978-0-19850839-7. ISBN 0-19-850839-5.

Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric;Exploring Fractals on theMacintosh(http://www.fractalexplorer.com), Addison Wesley, 1995. ISBN 0-201-62630-6
http://en.wikipedia.org/wiki/Special:BookSources/0201626306http://www.fractalexplorer.com/http://en.wikipedia.org/wiki/Special:BookSources/0198508395http://en.wikipedia.org/wiki/Special:BookSources/978-0-19-850839-7http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0691024456http://en.wikipedia.org/wiki/Special:BookSources/069108551Xhttp://en.wikipedia.org/wiki/Special:BookSources/1878739468http://en.wikipedia.org/wiki/Jesse_Joneshttp://en.wikipedia.org/wiki/Special:BookSources/0444500022http://en.wikipedia.org/wiki/Clifford_A._Pickoverhttp://en.wikipedia.org/wiki/Special:BookSources/0387966080http://en.wikipedia.org/wiki/Special:BookSources/0716711869http://en.wikipedia.org/wiki/The_Fractal_Geometry_of_Naturehttp://en.wikipedia.org/wiki/Benoit_Mandelbrothttp://en.wikipedia.org/wiki/Special:BookSources/0387979034http://en.wikipedia.org/wiki/Special:BookSources/0471922870http://en.wikipedia.org/wiki/Special:BookSources/0120790610http://en.wikipedia.org/wiki/Special:BookSources/978-0-387-75888-6http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://dx.doi.org/10.1007%2F978-0-387-30440-3_218http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Special:BookSources/978-988-17-0125-1http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://www.iaeng.org/publication/WCE2009/WCE2009_pp93-98.pdfhttp://dx.doi.org/10.1007%2Fs10816-005-2396-6http://en.wikipedia.org/wiki/Digital_object_identifierhttp://www.ncbi.nlm.nih.gov/pubmed/16393505http://en.wikipedia.org/wiki/PubMed_Identifierhttp://www.ncbi.nlm.nih.gov/pubmed/21949753http://en.wikipedia.org/wiki/PubMed_Identifierhttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC3176775http://en.wikipedia.org/wiki/PubMed_Centralhttp://dx.doi.org/10.1371%2Fjournal.pone.0024791http://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1023%2FA%3A1022355723781http://en.wikipedia.org/wiki/Digital_object_identifierhttp://www.ncbi.nlm.nih.gov/pubmed/21118245http://en.wikipedia.org/wiki/PubMed_Identifierhttp://dx.doi.org/10.1111%2Fj.1365-2818.2010.03454.xhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://www.ncbi.nlm.nih.gov/pubmed/2716356http://en.wikipedia.org/wiki/PubMed_Identifier


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Lesmoir-Gordon, Nigel; "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 t

the fractal concept and the Mandelbrot set).

Liu, Huajie;Fractal Art, Changsha: Hunan Science and Technology Press, 1997, ISBN 9787535722348Gouyet, Jean-Franois;Physics and Fractal Structures(Foreword by B. Mandelbrot); Masson, 1996.ISBN 2-225-85130-1, and New York: Springer-Verlag, 1996. ISBN 978-0-387-94153-0. Out-of-print

Available in PDF version at."Physics and Fractal Structures" (http://www.jfgouyet.fr/fractal/fractauk.html) (i

(French)). Jfgouyet.fr. Retrieved 2010-10-17.Bunde, Armin; Havlin, Shlomo (1996).Fractals and Disordered Systems(http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php). Springer.

Bunde, Armin; Havlin, Shlomo (1995).Fractals in Science(http://havlin.biu.ac.il/Shlomo%20Havlin%20books_f_in_s.php). Springer.

ben-Avraham, Daniel; Havlin, Shlomo (2000).Diffusion and Reactions in Fractals and DisorderedSystems(http://havlin.biu.ac.il/Shlomo%20Havlin%20books_d_r.php). Cambridge University Press.

External links

Fractals (http://www.dmoz.org/Science/Math/Chaos_And_Fractals//) at the Open Directory Project

Scaling and Fractals (http://havlin.biu.ac.il/nas1/index.html) presented by Shlomo Havlin, Bar-Ilan University

Hunting the Hidden Dimension (http://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html),

PBSNOVA, first aired August 24, 2011Benoit Mandelbrot: Fractals and the Art of Roughness

(http://www.ted.com/talks/benoit_mandelbrot_fractals_the_art_of_roughness.html), TED (conference),

February 2010

Zoom Video in Mandelbox (http://www.youtube.com/watch?v=7Pf6jZWguCc)(Example of 3D fractal)

Video fly through of an animated Mandelbulb world (http://www.youtube.com/watch?v=VB-XUoDqYfs)

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