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In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set .
Apollonian gasket; Apollonian sphere packing; Blancmange curve; Cantor dust; Cantor set; Cantor tesseract [citation needed]; Circle inversion fractal; De Rham curve; Douady rabbit; Dragon curve
Starting in the 1950s Benoit Mandelbrot and others have studied self-similarity of fractal curves, and have applied theory of fractals to modelling natural phenomena. Self-similarity occurs, and analysis of these patterns has found fractal curves in such diverse fields as economics, fluid mechanics, geomorphology, human physiology and linguistics.
Each branch carries 3 branches (here 90° and 60°). The fractal dimension of the entire tree is the fractal dimension of the terminal branches. NB: the 2-branches tree has a fractal dimension of only 1. 1.5850: Sierpinski triangle: Also the limiting shape of Pascal's triangle modulo 2.
In Ian Stewart's 2001 book Flatterland, there is a character called the Mandelblot, who helps explain fractals to the characters and reader. [ 52 ] The unfinished Alan Moore 1990 comic book series Big Numbers used Mandelbrot's work on fractal geometry and chaos theory to underpin the structure of that work.
The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and further developed in 2009 by Daniel White and Paul Nylander using spherical coordinates. A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers .
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. [1] They were introduced in 1981. IFS fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and ...
Barnsley's 1988 book Fractals Everywhere is based on the course which he taught for undergraduate and graduate students in the School of Mathematics, Georgia Institute of Technology, called Fractal Geometry. After publishing the book, a second course was developed, called Fractal Measure Theory. [1]