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 area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once 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, interconnection, 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 Texas can cause a tornado in Brazil.
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. 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 a two dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers for which the function does not diverge to infinity when iterated starting at , 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.
The Sierpiński triangle, also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.
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".
In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension.
Ralph William Gosper Jr., known as Bill Gosper, is an American mathematician and programmer. Along with Richard Greenblatt, he may be considered to have founded the hacker community, and he holds a place of pride in the Lisp community. The Gosper curve and the Gosper's algorithm are named after him.
Pickover stalks are certain kinds of details to be found empirically in the Mandelbrot set, in the study of fractal geometry. They are so named after the researcher Clifford Pickover, whose "epsilon cross" method was instrumental in their discovery. An "epsilon cross" is a cross-shaped orbit trap.
Algorithmic art or algorithm art is art, mostly visual art, in which the design is generated by an algorithm. Algorithmic artists are sometimes called algorists.
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. Although the "paradox of length" was previously noted by Hugo Steinhaus, the first systematic study of this phenomenon was by Lewis Fry Richardson, and it was expanded upon by Benoit Mandelbrot.
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.
Lateral computing is a lateral thinking approach to solving computing problems. Lateral thinking has been made popular by Edward de Bono. This thinking technique is applied to generate creative ideas and solve problems. Similarly, by applying lateral-computing techniques to a problem, it can become much easier to arrive at a computationally inexpensive, easy to implement, efficient, innovative or unconventional solution.
Michał Misiurewicz is a Polish mathematician. He is known for his contributions to chaotic dynamical systems and fractal geometry, notably the Misiurewicz point.
The seven states of randomness in probability theory, fractals and risk analysis are extensions of the concept of randomness as modeled by the normal distribution. These seven states were first introduced by Benoît Mandelbrot in his 1997 book Fractals and Scaling in Finance, which applied fractal analysis to the study of risk and randomness. This classification builds upon the three main states of randomness: mild, slow, and wild.
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.
Robert Luke Devaney is an American mathematician, the Feld Family Professor of Teaching Excellence at Boston University. His research involves dynamical systems and fractals.
Fractal expressionism is used to distinguish fractal art generated directly by artists from fractal art generated using mathematics and/or computers. Fractals are patterns that repeat at increasingly fine scales and are prevalent in natural scenery. Fractal expressionism implies a direct expression of nature's patterns in an art work.
