Introduction[ edit ] The word "fractal" often has different connotations for laypeople than for mathematicians, where the layperson is more likely to be familiar with fractal art 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.
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. The difference for fractals is that the pattern reproduced must be detailed.
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. In contrast, consider the Koch snowflake.
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.
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.
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.
Bytwo French mathematicians, Pierre Fatou and Gaston Juliathough 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.
In  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 setcaptured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".
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. Koch snowflake Quasi self-similarity: A consequence of this structure is fractals may have emergent properties  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".
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 dimensionand is fully defined without a need for recursion.
Because of the butterfly effecta small change in a single variable can have an unpredictable outcome. Iterated function systems IFS — use fixed geometric replacement rules; may be stochastic or deterministic;  e.
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.
A fractal generated by a finite subdivision rule for an alternating link Finite subdivision rules — use a recursive topological algorithm for refining tilings  and they are similar to the process of cell division.
A fractal flame 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,  digital images, electrochemical patterns, circadian rhythms etc.
Fractal patterns have been reconstructed in physical 3-dimensional space : 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[ edit ] Further information: Patterns in nature 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.Benoit Mandelbrot’s style is deliciously eccentric and idiosyncratic, and this finely produced volume, a lovingly reworked edition of “Fractals”, his masterpiece, is a fitting tribute to the fruits of a career of near-obsessive inquiry.
The origin of such fractal patterns, named after the pioneering work by Benoit Mandelbrot, is a fundamental problem in many areas of science. the dynamics of short-term operational processes that occur in smaller spaces can match the dynamics of long-term developmental processes that occur in much larger spaces.
A fractal is defined to. A fractal is “a geometrical or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous mediu.
Fractal basin boundary in a geometrical optical system Fractals, A Very Short Introduction.
Oxford University Press. Benoit Mandelbrot: Fractals and the Art of Roughness, TED, February ; Technical Library on Fractals for controlling fluid;. International Business Machines, or IBM, nicknamed "Big Blue", is a multinational computer technology and IT consulting corporation headquartered in Armonk, New York, United caninariojana.com originated from the bringing together of several companies that worked to automate routine business transactions.
Subject: What is a fractal? Q2: What is a fractal?
What are some examples of fractals? A2: A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole.