Benoit B.Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.
Chaos theory is an interdisciplinary scientific theory and branch of mathematics focused on underlying patterns and deterministic laws highly sensitive to initial conditions in dynamical systems that were thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.
In mathematics, fractal is a term used to describe geometric shapes 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. 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 shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.
The Mandelbrot set is the set of complex numbers for which the function does not diverge to infinity when iterated from , i.e., for which the sequence , , etc., remains bounded in absolute value.
In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.
Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art.
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic".
Heinz-Otto Peitgen is a German mathematician and was President of Jacobs University from January 1, 2013 to December 31, 2013. Peitgen contributed to the study of fractals, chaos theory, and medical image computing, as well as helping to introduce fractals to the broader public.
Adrien Douady was a French mathematician.
The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function:
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray.
The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve-like properties of coastlines, i.e., the fact that a coastline typically has a fractal dimension. The first recorded observation of this phenomenon was by Lewis Fry Richardson and it was expanded upon by Benoit Mandelbrot.
John Hamal Hubbard is an American mathematician and professor at Cornell University and the Université de Provence. He is well known for the mathematical contributions he made with Adrien Douady in the field of complex dynamics, including a study of the Mandelbrot set. One of their most important results is that the Mandelbrot set is connected.
In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set and also in real quadratic maps of the interval for which the critical point is strictly preperiodic. By analogy, the term Misiurewicz point is also used for parameters in a multibrot set where the unique critical point is strictly preperiodic.. These points are named after mathematician Michał Misiurewicz who first studied them.
The filled-in Julia set of a polynomial is :
The Douady rabbit is any of various particular filled Julia sets associated with the parameter near the center period 3 buds of Mandelbrot set for complex quadratic map.
Fractal-generating software is any type of graphics software that generates images of fractals. There are many fractal generating programs available, both free and commercial. Mobile apps are available to play or tinker with fractals. Some programmers create fractal software for themselves because of the novelty and because of the challenge in understanding the related mathematics. The generation of fractals has led to some very large problems for pure mathematics.
Robert Luke Devaney is an American mathematician, the Feld Family Professor of Teaching Excellence at Boston University. His research involves dynamical systems and fractals.
Nessim Sibony is a French mathematician, specializing in the theory of several complex variables and complex dynamics in higher dimension. Since 1981, he has been a professor at the University of Paris-Sud in Orsay.
