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List Of Fractal Topics
This is a list of fractal topics, by Wikipedia page, See also list of dynamical systems and differential equations topics. *1/f noise *Apollonian gasket *Attractor * Box-counting dimension *Cantor distribution * Cantor dust *Cantor function *Cantor set *Cantor space *Chaos theory *Coastline *Constructal theory *Dimension *Dimension theory *Dragon curve *Fatou set *Fractal * Fractal antenna *Fractal art * Fractal compression * Fractal flame *Fractal landscape * Fractal transform *Fractint *Graftal *Iterated function system *Horseshoe map * How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension *Julia set *Koch snowflake *L-system *Lebesgue covering dimension *Lévy C curve * Lévy flight *List of fractals by Hausdorff dimension *Lorenz attractor * Lyapunov fractal *Mandelbrot set *Menger sponge *Minkowski–Bouligand dimension *Multifractal analysis *Olbers' paradox *Perlin noise *Power law *Rectifiable curve *Scale-free network *Self-similarity * Sierp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
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. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fatou Set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") 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". The Julia set of a function is commonly denoted \operatorname(f), and the Fatou set is denoted \operatorname(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. Formal definition Let f(z) be a non-constant holomorphic function from the Riemann sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
How 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 Benoit Mandelbrot, first published in ''Science'' on 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of ''fractals'', although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals. Overview The paper examines the coastline paradox: the property that the measured length of a stretch of coastline depends on the scale of measurement. Empirical evidence suggests that the smaller the increment of measurement, the longer the measured length becomes. If one were to measure a stretch of coastline with a yardstick, one would get a shorter result than if the same stretch were measured with a ruler. This is because one would be laying the ruler along a more curvilinear route th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horseshoe Map
In 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. The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe. Most points eventually leave the square under the action of the map. They go to the side caps where they will, under iteration, converge to a fixed point in one of the caps. The points that remain in the square under repeated iteration form a fractal set and are part of the invariant set of the map. The squishing, stretching and folding of the horseshoe map are typical of chaotic systems, but not necessary or even sufficient. In the horseshoe map, the squeezing and stretching are uniform. They compensate each oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Function System
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. They were introduced in 1981. IFS fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and drawn in 2D. The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). The canonical example is the Sierpiński triangle. The functions are normally contractive, which means they bring points closer together and make shapes smaller. Hence, the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of which is also made up of copies of itself, ad infinitum. This is the source of its self-similar fractal nature. Definition Formally, an iterated function system is a finite set of contraction mappings on a complete metric space. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graftal
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial " axiom" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. L-systems were introduced and developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist at the University of Utrecht. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes of plant development. L-systems have also been used to model the morphology of a variety of organisms and can be used to generate self-similar fractals. Origins As a biologist, Lindenmayer worked with yeast and filamentous fungi and studied the growth patterns of various types of bacteria, such as the cyanobacteria '' Anabae ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractint
Fractint is a freeware computer program to render and display many kinds of fractals. The program originated on MS-DOS, then ported to the Atari ST, Linux, and Macintosh. During the early 1990s, Fractint was the definitive fractal generating program for personal computers. The name is a portmanteau of ''fractal'' and ''integer'', since the first versions of Fractint used only integer arithmetic (also known as fixed-point arithmetic), for faster rendering on computers without math coprocessors. Since then, floating-point arithmetic and arbitrary-precision arithmetic modes have been added. Features FractInt can draw most kinds of fractals that have appeared in the literature. It also has a few "fractal types" that are not strictly speaking fractals, but may be more accurately described as display hacks. These include cellular automata. History Fractint originally appeared in 1988 as FRACT386, a computer program for rendering fractals very quickly on the Intel 80386 pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Transform
The 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. Fractal transform compression Start with a digital image A1. Downsample it by a factor of 2 to produce image A2. Now, for each block B1 of 4x4 pixels in A1, find the corresponding block B2 in A2 most similar to B1, and then find the grayscale or RGB offset and gain from A2 to B2. For each destination block, output the positions of the source blocks and the color offsets and gains. Fractal transform decompression Starting with an empty destination image A1, repeat the following algorithm several times: Downsample A1 down by a factor of 2 to produce image A2. Then copy blocks from A2 to A1 as directed by the compressed data, multiplying by the respective gains and adding the respective color offsets. This algorithm is guar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Landscape
A fractal landscape is a surface that is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the result of the procedure is not a deterministic fractal surface, but rather a random surface that exhibits fractal behavior. Many natural phenomena exhibit some form of statistical self-similarity that can be modeled by fractal surfaces.''Advances in multimedia modeling: 13th International Multimedia Modeling'' by Tat-Jen Cham 2007 pag/ref> Moreover, variations in surface texture provide important visual cues to the orientation and slopes of surfaces, and the use of almost self-similar fractal patterns can help create natural looking visual effects. The modeling of the Earth's rough surfaces via fractional Brownian motion was first proposed by Benoit Mandelbrot. Because the intended result of the process is to produce a landscape, rather than a mathematical function, processes are frequently appl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Flame
Fractal 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 softwareChris Gehman and Steve Reinke (2005). ''The Sharpest Point: Animation at the End of Cinema''. YYZ Books. pp 269. and translated into the Apophysis fractal flame editor. Fractal flames differ from ordinary iterated function systems in three ways: * Nonlinear functions are iterated in addition to affine transforms. * Log-density display instead of linear or binary (a form of tone mapping) * Color by structure (i.e. by the recursive path taken) instead of monochrome or by density. The tone mapping and coloring are designed to display as much of the detail of the fractal as possible, which generally results in a more aesthetically pleasing image. Algorithm The algorithm consists of two steps: creating a histogram and then rendering the histogram. Creating the histogram First, one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Compression
Fractal 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 algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image. Iterated function systems Fractal image representation may be described mathematically as an iterated function system (IFS). For binary images We begin with the representation of a binary image, where the image may be thought of as a subset of \mathbb^2. An IFS is a set of contraction mappings ''ƒ''1,...,''ƒN'', :f_i:\mathbb^2\to \mathbb^2. According to these mapping functions, the IFS describes a two-dimensional set ''S'' as the fixed point of the Hutchinson operator :H(A)=\bigcup_^N f_i(A), \quad A \subset \mathbb^2. That is, ''H'' is an operator mapping sets to sets, and ''S'' is the unique set satisfy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
