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List Of Fractals By Hausdorff Dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension. Deterministic fractals Random and natural fractals See also * Fractal dimension * Hausdorff dimension * Scale invariance Notes and references Further reading * * * * External links The fractals on MathworldOther fractals on Paul Bourke's websiteFractals on mathcurve.com* ttps://web.archive.org/web/20060923100014/http://library.thinkquest.org/26242/full/index.html Fractals unleashedIFStile - software that computes the dimension of the boundary of self-affine tiles {{DEFAULTSORT:Fractals By Hausdorff Dimension Hausdorff Dimension Hausdorff Dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benoit Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) 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. In 1936, at the age of 11, Mandelbrot and his family emigrated from Warsaw, Poland, to France. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and in the United States and receiving a master's degree in aeronautics from the California Institute of Technology. He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosper Island 4
Gosper may refer to: *Gosper County, Nebraska *Gosper curve *Bill Gosper (born 1943), American mathematician *Kevan Gosper Richard Kevan Gosper, AO (19 December 1933 – 19 July 2024) was an Australian athlete who mainly competed in the 400 metres. He was a Vice President of the International Olympic Committee, and combined chairman and CEO of Shell Australia. ... (1933–2024), Australian athlete and 1956 Olympic medalist * John J. Gosper (1843–1913), Nebraska Secretary of State (1873–1875) and Secretary of Arizona Territory (1875–1882). {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosper Island
The Gosper curve, named after Bill Gosper, also known as the Peano-Gosper Curve and the flowsnake (a spoonerism of snowflake), is a space-filling curve whose limit set is rep-7. It is a fractal curve similar in its construction to the dragon curve and the Hilbert curve. The Gosper curve can also be used for efficient hierarchical hexagonal clustering and indexing. Lindenmayer system The Gosper curve can be represented using an L-system with rules as follows: * Angle: 60° * Axiom: A * Replacement rules: ** A \mapsto A-B--B+A++AA+B- ** B \mapsto +A-BB--B-A++A+B In this case both A and B mean to move forward, + means to turn left 60 degrees and - means to turn right 60 degrees - using a "turtle"-style program such as Logo. Properties The space filled by the curve is called the Gosper island. The first few iterations of it are shown below: The Gosper Island can tile the plane. In fact, seven copies of the Gosper island can be joined to form a shape that is similar, but sca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rauzy Fractal
In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution : s(1)=12,\ s(2)=13,\ s(3)=1 \,. It was studied in 1981 by Gérard Rauzy, with the idea of generalizing the dynamic properties of the Fibonacci morphism. That fractal set can be generalized to other maps over a 3-letter alphabet, generating other fractal sets with interesting properties, such as periodic tiling of the plane and self-similarity in three homothetic parts. Definitions Tribonacci word The infinite tribonacci word is a word constructed by iteratively applying the ''Tribonacci'' or ''Rauzy map'' : s(1)=12, s(2)=13, s(3)=1.Lothaire (2005) p.525Pytheas Fogg (2002) p.232 It is an example of a morphic word. Starting from 1, the Tribonacci words are:Lothaire (2005) p.546 * t_0 = 1 * t_1 = 12 * t_2 = 1213 * t_3 = 1213121 * t_4 = 1213121121312 We can show that, for n>2, t_n = t_t_t_; hence the name " Tribonacci". Fractal construction Consider, now, the space R^3 with carte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia Z2+0,25
Julia may refer to: People *Julia (given name), including a list of people with the name *Julia (surname), including a list of people with the name *Julia gens, a patrician family of Ancient Rome *Julia (clairvoyant) (fl. 1689), lady's maid of Queen Christina of Sweden in Rome, alleged clairvoyant and predictor Science and technology *Julia (programming language), a computer language with features suited for numerical analysis and computational science *Julia (unidentified sound), an underwater sound record by the NOAA *Julia (gastropod), a genus of minute bivalved gastropods in the family Juliidae *Julia butterfly, ''Dryas iulia'', misspelled as ''Dryas julia'' Television * ''Julia'' (1968 TV series), a 1968–1971 American series starring Diahann Carroll * ''Julia'' (2022 TV series), an American drama series * ''Julia'' (Mexican TV series), a 1979 Mexican telenovela * ''Julia'' (Polish TV series), a 2012 Polish soap opera * ''Julia'' (Venezuelan TV series), a 1983 Venezuelan TV ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia Set
In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under iterated function, repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small Perturbation theory, 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 "chaos theory, 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. Form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Wave
A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse). Definitions Definition A triangle wave of period ''p'' that spans the range , 1is defined as x(t) = 2 \left, \frac - \left\lfloor \frac + \frac \right\rfloor \, where \lfloor\ \rfloor is the floor function. This can be seen to be the absolute value of a shifted sawtooth wave. For a triangle wave spanning the range the expression becomes x(t)= 2 \left , 2 \left( \frac - \left\lfloor \frac + \frac \right\rfloor \right) \ - 1. A more general equation for a triangle wave with amplitude a and period p using the modulo operation and absolute value is y(x) = \frac \left, \left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation ' is most commonly reserved for the closed interval . Properties The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Takagi Curve
In mathematics, the blancmange curve is a self-affine fractal curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the curve named after Takagi and Georg Landsberg. The name ''blancmange'' comes from its resemblance to a Blancmange pudding. It is a special case of the more general de Rham curve. Definition The blancmange function is defined on the unit interval by : \operatorname(x) = \sum_^\infty , where s(x) is the triangle wave, defined by s(x)=\min_, x-n, , that is, s(x) is the distance from ''x'' to the nearest integer. The Takagi–Landsberg curve is a slight generalization, given by : T_w(x) = \sum_^\infty w^n s(2^x) for a parameter w; thus the blancmange curve is the case w=1/2. The value H=-\log_2 w is known as the Hurst parameter. The function can be extended to all of the real line: applying the definition given above shows that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
