Add to Wishlist Install A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. It is also known as expanding symmetry or evolving symmetry.
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.
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 effect a small change in a single variable can have a unpredictable outcome.
Iterated function systems — use fixed geometric replacement rules; may be stochastic or deterministic;  e.1) You PC doesn't have a host application that can open and render MIME Application/Fractals content. Like file extensions, each MIME Type corresponds to a particular host application that can be used to read and write binary data sets according to the matching MIME Type specification.
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Fractal-generating software is any computer program that generates images of fractals. Linux has a good selection of fractal software to choose from. To provide an insight into the quality of software that is available, we have compiled a list of 6 absorbing fractal tools.
Engineers are using the ideas of fractal geometry in a variety of applications. Often we are faced with a task that is similar to something that nature has already found a solution for.
Aug 25, · Best Answer: compression of images "Fractal compression is a lossy compression method used to compress images using fractals. The method is best suited for photographs of natural scenes (trees, mountains, ferns, clouds).
The fractal compression technique relies on the fact that in certain images, parts Status: Resolved. Starting with numerical algorithms resulting in new kinds of amazing fractal patterns on the sphere, this book describes the theory underlying these phenomena and indicates possible future applications.