Zoom in of the Mandelbrot setIn, a fractal is a subset of a for which the strictly exceeds the. Fractals appear the same at different levels, as illustrated in successive magnifications of the; because of this, fractals are encountered ubiquitously in nature. Fractals exhibit similar patterns at increasingly small scales called self similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the, it is called affine self-similar. Fractal geometry lies within the mathematical branch of.One way that fractals are different from finite is the way in which they.

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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.Analytically, 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. (to level 6), a fractal with a of 1 and a of 1.893Starting in the 17th century with notions of, fractals have moved 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.There is some disagreement among mathematicians 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. Fractals are of particular relevance in the field of, since the graphs of most chaotic processes are fractals.

A simple fractal tree created through javascriptThe word 'fractal' often has different connotations for the lay public as opposed to mathematicians, where the public are more likely to be familiar with than the mathematical concept. 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; 18This idea of being detailed relates to another feature that can be understood without mathematical background: Having a 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 one-dimensional; if such a curve is rep-tiled into pieces each 1/3 the length of the original, there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 3 2 = 9 pieces. We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/ r, there are a total of r n pieces. Now, consider the. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3.

So, strictly by analogy, we can consider the 'dimension' of the Koch curve as being the unique real number D that satisfies 3 D = 4, which by no means is an integer! This number is what mathematicians call the fractal dimension of the Koch curve. The fact that the Koch curve has a non-integer fractal dimension is what makes it a fractal.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 an infinitely 'wiggly' fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, 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. The result is that one must need infinite tape to perfectly cover the entire curve, i.e.

The snowflake has infinite perimeter.History. 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 bumpThe 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.

2x 120 degrees recursiveDifferent 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. 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.Definition and characteristics 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 the unusual relationship fractals have with the space they are 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. However, this requirement is not met by such as the.Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. Self-similar branching pattern modeled using principlesImages of fractals can be created. Because of the, a small change in a single variable can have an outcome. (IFS) – 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,.

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). A fractal generated by a for an. – use a recursive algorithm for refining tilings and they are similar to the process of. The iterative processes used in creating the and the are examples of finite subdivision rules, as is.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.

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: andSince 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.Ethnomathematician Ron Eglash has discussed the planned layout of using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses.

He commented that 'When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn’t even discovered yet.' 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'.

The model we have today is the ION+ 860P, which is a platinum-rated unit and is standard ATX size with a 140mm case length. The 150mm length is right in between the pretty standard 160mm and the 140mm units that Silverstone has been showing off.

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This, of course, means this will give more room for tucking excess cable in front of the supply.