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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, th ... [...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] |
Smith–Volterra–Cantor Set
In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line, and Volterra introduced a similar example in 1881. The Cantor set as we know it today followed in 1883. The Smith–Volterra–Cantor set is topologically equivalent to the middle-thirds Cantor set. Construction Similar to the construction of the Cantor set, the Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval , 1 The process begins by removing the middle 1/4 from the interval , 1(the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is :\left , \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Word Fractal
The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word. Definition This curve is built iteratively by applying, to the Fibonacci word 0100101001001...etc., the Odd–Even Drawing rule: For each digit at position ''k'' : # Draw a segment forward # If the digit is 0: #* Turn 90° to the left if ''k'' is even #* Turn 90° to the right if ''k'' is odd To a Fibonacci word of length F_n (the ''n''th Fibonacci number) is associated a curve \mathcal_n made of F_n segments. The curve displays three different aspects whether ''n'' is in the form 3''k'', 3''k'' + 1, or 3''k'' + 2. Properties Some of the Fibonacci word fractal's properties include: * The curve \mathcal, contains F_n segments, F_ right angles and F_ flat angles. * The curve never self-intersects and does not contain double points. At the limit, it contains an infinity of points asymptotically close. * The curve presents self-similarities at all scales. The reducti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dendrite Julia
Dendrites (from Greek δένδρον ''déndron'', "tree"), also dendrons, are branched protoplasmic extensions of a nerve cell that propagate the electrochemical stimulation received from other neural cells to the cell body, or soma, of the neuron from which the dendrites project. Electrical stimulation is transmitted onto dendrites by upstream neurons (usually via their axons) via synapses which are located at various points throughout the dendritic tree. Dendrites play a critical role in integrating these synaptic inputs and in determining the extent to which action potentials are produced by the neuron. Dendritic arborization, also known as dendritic branching, is a multi-step biological process by which neurons form new dendritic trees and branches to create new synapses. The morphology of dendrites such as branch density and grouping patterns are highly correlated to the function of the neuron. Malformation of dendrites is also tightly correlated to impaired nervous syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosper Curve
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. Algorithm 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. Logo A Logo program to draw the Gosper curve using turtle graphics: to rg :st :ln make "st :st - 1 make "ln :ln / sqrt 7 if :st > 0 g :st :ln rt 60 gl :st :ln rt 120 gl :st :ln lt 60 rg :st :ln lt 120 rg :st :ln rg :st :ln lt 60 gl :st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosper Island 4
Gosper may refer to: *Gosper County, Nebraska *Gosper curve *Bill Gosper, American mathematician *Kevan Gosper Richard Kevan Gosper, AO (born 19 December 1933) is an Australian former athlete who mainly competed in the 400 metres. He was formerly a Vice President of the International Olympic Committee, and combined Chairman and CEO of Shell Australi ..., 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. Algorithm 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. Logo A Logo program to draw the Gosper curve using turtle graphics: to rg :st :ln make "st :st - 1 make "ln :ln / sqrt 7 if :st > 0 g :st :ln rt 60 gl :st :ln rt 120 gl :st :ln lt 60 rg :st :ln lt 120 rg :st :ln rg :st :ln lt 60 gl :st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes 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. ... [...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 is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g. Julia of Corsica) but became rare during the Middle Ages, and was revived only with the Italian Renaissance. It became common in the English-speaking world only in the 18th century. Today, it is frequently used throughout the world. Statistics Julia was the 10th most popular name for girls born in the United States in 2007 and the 88th most popular name for women in the 1990 census there. It has been among the top 150 names given to girls in the United States for the past 100 years. It was the 89th most popular name for girls born in England and Wales in 2007; the 94th most popular name for girls born in Scotland in 2007; the 13th most popular name for girls born in Spain in 2006; the 5th most popular name for girls born in Swede ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia 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 o ... [...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 + \right \rfloor \right) \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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
