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Chaotic maps and iterated functions often generate fractals. Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them. This is often because there are several different iterative procedures that generate the same fractal. See also Universality (dynamical systems).
Escape-time fractals – use a formula or recurrence relation at each point in a space (such as the complex plane); usually quasi-self-similar; also known as "orbit" fractals; e.g., the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape ...
Julia set (in white) for the rational function associated to Newton's method for f : z → z 3 −1. Coloring of Fatou set in red, green and blue tones according to the three attractors (the three roots of f). For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following ...
The filled-in Julia set of a polynomial is defined as the set of all points of the dynamical plane that have bounded orbit with respect to () = {: () } where: C {\displaystyle \mathbb {C} } is the set of complex numbers
The complementary set to the union of all these, is the Julia set. The Fatou sets have common boundary, namely the Julia set. Therefore, each point of the Julia set is a point of accumulation for each of the Fatou sets. It is this property that causes the fractal structure of the Julia set (when the degree of the polynomial is larger than 2).
About properties and symmetries of the Burning Ship fractal, featured by Theory.org; Burning Ship Fractal, Description and C source code. Burning Ship with its Mset of higher powers and Julia Sets; Burningship, Video, Fractal webpage includes the first representations and the original paper cited above on the Burning Ship fractal.
The Mandelbrot set is widely considered the most popular fractal, [45] [46] and has been referenced several times in popular culture. The Jonathan Coulton song "Mandelbrot Set" is a tribute to both the fractal itself and to the man it is named after, Benoit Mandelbrot. [47]
Mandelbrot set rendered using a combination of cross and point shaped orbit traps. In mathematics, an orbit trap is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap".