Fractal
In mathematics, fractal is a term used to describe geometric shapes containing detailed cut at arbitrarily small scales, ordinarily having a fractal dimension strictly exceeding the topological dimension. many fractalssimilar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the rank is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.
One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two the ratio of the new to the old side length raised to the power to direct or creation of two the conventional dimension of the filled polygon. Likewise, if the radius of a filled sphere is doubled, its volume scales by eight, which is two the ratio of the new to the old radius to the power to direct or defining of three the conventional dimension of the filled sphere. However, if a fractal's one-dimensional lengths are any doubled, the spatial content of the fractal scales by a power that is non necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from the conventional dimension which is formally called the topological dimension.
Analytically, many fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is for still topologically 1-dimensional, its fractal dimension indicates that it locally fills space more efficiently than an ordinary line.
Starting in the 17th century with notions of fractal in the 20th century with a subsequent burgeoning of interest in fractals as well as computer-based modelling in the 20th century.
There is some disagreement among mathematicians approximately 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 defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified in addition to expanded the definition to this: "A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is at least about a reduced-size copy of the whole." Still later, Mandelbrot portrayed "to ownership fractal without a pedantic definition, to use fractal dimension as a generic term relevant to all the variants".
The consensus among mathematicians is that theoretical fractals are infinitely self-similar nature, technology, art, architecture and law. Fractals are of specific relevance in the field of chaos theory because they show up in the geometric depictions of near chaotic processes typically either as attractors or as boundaries between basins of attraction.