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In mathematics, a fractal is a detailed, recursive, and infinitely self-similar mathematical set whose strictly exceeds its and which is encountered ubiquitously in nature. Fractals exhibit similar patterns at increasingly small scales, also known as expanding symmetry or unfolding symmetry. If this replication is exactly the same at every scale, as in the, it is called a pattern. Fractals can also be nearly the same at different levels, as illustrated here in small magnifications of the.

One way that fractals are different from finite is the way in which they. Doubling the edge lengths of a multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an. This power is called the of the fractal, and it usually exceeds the fractal's.

As mathematical equations, fractals are usually nowhere. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line - although it is still its fractal dimension indicates that it also resembles a surface.

The mathematical of fractals have been traced throughout the years as a formal path of published works, starting in the 17th century with notions of, then moving through increasingly rigorous mathematical treatment of the concept to the study of but not functions in the 19th century by the seminal work of,, and, and on to the coining of the word in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. The term "fractal" was first used by mathematician in 1975. Mandelbrot based it on the Latin meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional to geometric.:405

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." More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the strictly exceeds the." Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use as a generic term applicable to all the variants."

The consensus is that theoretical fractals are infinitely self-similar,, and detailed mathematical constructs having fractal dimensions, of which many have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in,,, architecture and. Fractals are of particular relevance in the field of, since the graphs of most chaotic processes are fractals.



The word "fractal" often has different connotations for laypeople than for mathematicians, where the layperson is more likely to be familiar with than a mathematical conception. The mathematical concept is difficult to define formally 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 fractals, 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 in parallel mirrors or the, 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.:166; 18

This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractional or greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric 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 Koch snowflake. It is also 1-dimensional for the same 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.

This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere ". In a concrete sense, this means fractals cannot be measured in traditional ways. 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 to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of with a tape measure. But in measuring a wavy fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the wavy pattern 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.


A is 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

The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the way. According to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher pondered (although he made the mistake of thinking that only the was self-similar in this sense). In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.: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". Thus, it was not until two centuries had passed that on July 18, 1872 presented the first definition of a with a that would today be considered a fractal, having the non- property of being everywhere but at the Royal Prussian Academy of Sciences.:7 In addition, the quotient difference becomes arbitrarily large as the summation index increases. Not long after that, in 1883,, who attended lectures by Weierstrass, published examples of of the real line known as, which had unusual properties and are now recognized as fractals.:11–24 Also in the last part of that century, and introduced a category of fractal that has come to be called "self-inverse" fractals.:166

A, a fractal related to the Mandelbrot set

One of the next milestones came in 1904, when, extending ideas of Poincaré 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.:25 Another milestone came a decade later in 1915, when constructed his famous then, one year later, his. By 1918, two French mathematicians, and, though working independently, arrived essentially simultaneously at results describing what are now seen as fractal behaviour associated with mapping and iterative functions and leading to further ideas about (i.e., points that attract or repel other points), which have become very important in the study of fractals. Very shortly after that work was submitted, by March 1918, expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have noninteger dimensions. The idea of self-similar curves was taken further by, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the.

Uniform mass center triangle fractal 2x 120 degrees recursive IFS

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).:179 That changed, however, in the 1960s, when started writing about self-similarity in papers such as, which built on earlier work by. In 1975 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, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".

In 1980, gave a presentation at the where he introduced his software for generating and rendering fractally generated landscapes.


One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"; this is generally helpful but limited. Authors disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of self-similarity and an unusual relationship with the space a fractal is embedded in. One point agreed on is that fractal patterns are characterized by, but whereas these numbers quantify (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns. In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose is greater than its. It has been noted that this dimensional requirement is not met by fractal such as the.

According to Falconer, rather than being strictly defined, fractals should, in addition to being nowhere differentiable and able to have a, be generally characterized by a of the following features;

  • Self-similarity, which may be manifested as:
  • Exact self-similarity: identical at all scales; e.g.
  • Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the 's satellites are approximations of the entire set, but not exact copies.
  • Statistical self-similarity: repeats a pattern so numerical or statistical measures are preserved across scales; e.g., ; the well-known example of the, 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
  • Qualitative self-similarity: as in a time series
  • 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 (related to the next criterion in this list).
  • Irregularity locally and globally that is not easily described in traditional language. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls".
  • Simple and "perhaps " definitions see

As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without 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 as, and is fully defined without a need for recursion.

Common techniques for generating fractals[]

Images of fractals can be created by. Because of the a small change in a single variable can have a outcome.

  • – use fixed geometric replacement rules; may be stochastic or deterministic; e.g.,,, Haferman carpet,,,,,,
  • – use iterations of a map or solutions of a system of initial-value differential or difference equations that exhibit chaos (e.g., see image, or the )
  • – use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells), blood vessels, pulmonary structure, etc. or patterns such as and tilings
  • Escape-time fractals – use a or at each point in a space (such as the ); usually quasi-self-similar; also known as "orbit" fractals; e.g., the,,, and. 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.,,,,, trajectories of and the (i.e., dendritic fractals generated by modeling or reaction-limited aggregation clusters).

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. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for. Some specific applications of fractals to technology are listed. 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 which are not characteristics of true fractals.

Modeled fractals may be sounds, digital images, electrochemical patterns,, etc. Fractal patterns have been reconstructed in physical 3-dimensional space:10 and virtually, often called "" modeling. Models of fractals are generally created using that implements techniques such as those outlined above. As one illustration, trees, ferns, cells of the nervous system, blood and lung vasculature, and other branching can be modeled on a computer by using recursive and techniques. The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a from a 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 algorithms.

Natural phenomena with fractal features[]

Further information:

Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees. Phenomena known to have fractal features include:

  • Frost crystals occurring naturally on cold glass form fractal patterns

  • Fractal basin boundary in a geometrical optical system

  • A fractal is formed when pulling apart two glue-covered sheets

  • High voltage breakdown within a 4 in (100 mm) block of acrylic creates a fractal

  • Fractal defrosting patterns, polar Mars. The patterns are formed by sublimation of frozen CO2. Width of image is about a kilometer.

In creative works[]

Further information: and

Since 1999, more than 10 scientific groups have performed fractal analysis on over 50 of 's (1912–1956) paintings which were created by pouring paint directly onto his horizontal canvases Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks. Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.

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

Cyberneticist has suggested that fractal geometry and mathematics are prevalent in, games,, 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. also suggested the similar properties in Indonesian traditional art,, and found in traditional houses.

In a 1996 interview with, admitted that the structure of the first draft of he gave to his editor Michael Pietsch was inspired by fractals, specifically the (a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket".

  • A fractal that models the surface of a mountain (animation)

  • 3D recursive image

  • Recursive fractal butterfly image

Physiological responses[]

Humans appear to be especially well-adapted to processing fractal patterns with D values between 1.3 and 1.5. When humans view fractal patterns with D values between 1.3 and 1.5, this tends to reduce physiological stress.

Ion production capabilities[]

If a circle boundary is drawn around the two-dimensional view of a fractal, the fractal will never cross the boundary, this is due to the scaling of each successive iteration of the fractal being smaller. When fractals are iterated many times, the perimeter of the fractal increases, while the area will never exceed a certain value. A fractal in three-dimensional space is similar, however, a difference between fractals in two dimensions and three dimensions, is that a three dimensional fractal will increase in surface area, but never exceed a certain volume. This can be utilized to maximize the efficiency of, when choosing electron emitter construction and material. If done correctly, the efficiency of the emission process can be maximized.

Applications in technology[]

Main article:

See also[]

  1. The original paper, Lévy, Paul (1938). "Les Courbes planes ou gauches et les surfaces composées de parties semblables au tout". Journal de l'École Polytechnique: 227–247, 249–291., is translated in, pages 181–239.
  2. The Hilbert curve map is not a, 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.


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Further reading[]

  • Barnsley, Michael F.; and Rising, Hawley; Fractals Everywhere. Boston: Academic Press Professional, 1993.  
  • Duarte, German A.; Fractal Narrative. About the Relationship Between Geometries and Technology and Its Impact on Narrative Spaces. Bielefeld: Transcript, 2014.  
  • Falconer, Kenneth; Techniques in Fractal Geometry. John Wiley and Sons, 1997.  
  • Jürgens, Hartmut; ; and Saupe, Dietmar; Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992.  
  • ;. New York: W. H. Freeman and Co., 1982.  
  • Peitgen, Heinz-Otto; and Saupe, Dietmar; eds.; The Science of Fractal Images. New York: Springer-Verlag, 1988.  
  • ; ed.; Chaos and Fractals: A Computer Graphical Journey – A 10 Year Compilation of Advanced Research. Elsevier, 1998.  
  • Jones, Jesse; Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993.  .
  • Lauwerier, Hans; Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991.  , cloth.   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. Oxford University Press.  .
  • Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric;, Addison Wesley, 1995.  
  • Lesmoir-Gordon, Nigel; The Colours of Infinity: The Beauty, The Power and the Sense of Fractals. 2004.   (The book comes with a related DVD of the documentary introduction to the fractal concept and the.)
  • Liu, Huajie; Fractal Art, Changsha: Hunan Science and Technology Press, 1997,  .
  • Gouyet, Jean-François; Physics and Fractal Structures (Foreword by B. Mandelbrot); Masson, 1996.  , and New York: Springer-Verlag, 1996.  . Out-of-print. Available in PDF version at. (in French). Jfgouyet.fr. Retrieved 2010-10-17.
  • Bunde, Armin; Havlin, Shlomo (1996).. Springer.
  • Bunde, Armin; Havlin, Shlomo (1995).. Springer.
  • ben-Avraham, Daniel; Havlin, Shlomo (2000).. Cambridge University Press.
  • Falconer, Kenneth (2013). Fractals, A Very Short Introduction. Oxford University Press.

External links[]

  • at the Web Archives (archived 2001-11-16)
  • presented by,
  • ,, first aired August 24, 2011
  • ,, February 2010
  • (2007)

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