List of fractal topics
Encyclopedia
This is a list of fractal
topics, by Wikipedia page, See also list of dynamical systems and differential equations topics.
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
topics, by Wikipedia page, See also list of dynamical systems and differential equations topics.
- 1/f noise
- Apollonian gasketApollonian gasketIn mathematics, an Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where each circle is tangent to the other two. It is named after Greek mathematician Apollonius of Perga.-Construction:...
- AttractorAttractorAn attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
- Box-counting dimension
- Cantor distribution
- Cantor dust
- Cantor functionCantor functionIn mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Devil's staircase.-Definition:See figure...
- Cantor setCantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
- Cantor spaceCantor spaceIn mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space...
- Chaos theoryChaos theoryChaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
- Coastline
- Constructal theoryConstructal theoryThe constructal law puts forth the idea that the generation of design in nature is a physics phenomenon that unites all animate and inanimate systems, and that this phenomenon is covered by the Constructal Law...
- DimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
- Dimension theoryDimension theoryIn mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.-See also:*Lebesgue covering dimension*Inductive dimensions *Dimension...
- Dragon curveDragon curveA dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems.-Heighway dragon:...
- Fatou set
- Fractal antennaFractal antennaA fractal antenna is an antenna that uses a fractal, self-similar design to maximize the length, or increase the perimeter , of material that can receive or transmit electromagnetic radiation within a given total surface area or volume.Such fractal antennas are also referred to as multilevel and...
- Fractal artFractal artFractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still 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...
- Fractal compressionFractal compressionFractal compression is a lossy compression method for digital images, based on fractals. The method is best suited for textures and natural images, relying on the fact that parts of an image often resemble other parts of the same image...
- Fractal flameFractal flameFractal flames are a member of the iterated function system class of fractals created by Scott Draves in 1992. Draves' open-source code was later ported into Adobe After Effects graphics software and translated into the Apophysis fractal flame editor....
- Fractal landscapeFractal landscapeA fractal landscape is a surface generated using a stochastic algorithm designed to produce fractal behaviour that mimics the appearance of natural terrain...
- Fractal transformFractal transformThe fractal transform is a technique invented by Michael Barnsley et al. to perform lossy image compression.This first practical fractal compression system for digital images resembles a vector quantization system using the image itself as the codebook....
- FractintFractintFractInt is a freeware program that can render and display many kinds of fractals.-Name:Its name comes from the words fractal and integer, since the first versions of it computed fractals by using only integer arithmetic , which led to much faster rendering on x86 computers without math coprocessors...
- Graftal
- Gravity set
- Iterated function systemIterated function systemIn mathematics, iterated function systems or IFSs are a method of constructing fractals; the resulting constructions are always self-similar....
- Horseshoe mapHorseshoe mapIn the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator...
- How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional DimensionHow Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2...
- Julia setJulia setIn the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
- Koch snowflakeKoch snowflakeThe Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described...
- L-systemL-systemAn L-system or Lindenmayer system is a parallel rewriting system, namely a variant of a formal grammar, most famously used to model the growth processes of plant development, but also able to model the morphology of a variety of organisms...
- Lebesgue covering dimensionLebesgue covering dimensionLebesgue covering dimension or topological dimension is one of several inequivalent notions of assigning a topological invariant dimension to a given topological space.-Definition:...
- Lévy C curveLévy C curveIn mathematics, the Lévy C curve is a self-similar fractal that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and G...
- Lévy flightLévy flightA Lévy flight is a random walk in which the step-lengths have a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions...
- List of fractals by Hausdorff dimension
- Lorenz attractorLorenz attractorThe Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape...
- Lyapunov fractalLyapunov fractalIn mathematics, Lyapunov fractals are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B.A Lyapunov fractal is constructed by mapping the regions of stability and chaotic...
- Mandelbrot setMandelbrot setThe Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
- Menger spongeMenger spongeIn mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge...
- Minkowski-Bouligand dimensionMinkowski-Bouligand dimensionthumb|450px|Estimating the box-counting dimension of the coast of Great BritainIn fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rn, or more generally in...
- Multifractal analysis
- Olbers' paradoxOlbers' paradoxIn astrophysics and physical cosmology, Olbers' paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. It is one of the pieces of evidence for a non-static universe such as the current Big Bang model. The argument is also...
- Perlin noisePerlin noisePerlin noise is a computer-generated visual effect developed by Ken Perlin, who won an Academy Award for its use in the motion picture Tron...
- Power lawPower lawA power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
- Rectifiable curve
- Scale-free networkScale-free networkA scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P of nodes in the network having k connections to other nodes goes for large values of k as...
- Self-similaritySelf-similarityIn 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...
- Sierpinski carpetSierpinski carpetThe Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...
- Sierpiński curveSierpinski curveSierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n \rightarrow \infty completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling...
- Sierpinski triangleSierpinski triangleThe Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral...
- Space-filling curveSpace-filling curveIn mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square...
- T-Square (fractal)T-Square (fractal)In mathematics, the T-square is a two-dimensional fractal. As all two-dimensional fractals, it has a boundary of infinite length bounding a finite area...
- Topological dimension