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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 car ... [...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 car ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector (mathematics And Physics)
In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces. Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers. The term ''vector'' is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length. Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors. A vector space formed by geometric vectors is called a ... [...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] |
List Of Fractals
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, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pisot Number
Charles Pisot (2 March 1910 – 7 March 1984) was a French mathematician. He is chiefly recognized as one of the primary investigators of the numerical set associated with his name, the Pisot–Vijayaraghavan numbers. He followed the classical path of great French mathematicians by studying at the École Normale Supérieure on Ulm street, where he was received first at the agrégation in 1932. He then began his academic career at the Bordeaux University before being offered a chair at the Science Faculty of Paris and at the École Polytechnique. He was a member of Bourbaki. Also of interest is the recently solved Pisot conjecture on rational functions. (For a technical account and bibliography see Umberto Zannier's paper in the ''Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the correspondin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is '' locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include ''regular tilings'' with regular polygonal tiles all of the same shape, and ''semiregular tilings'' with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An ''aperiodic tiling'' uses a small set of tile shapes that cannot form a repeating pattern. A ''tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. ... [...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 sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tribonacci Number
In mathematics, the Fibonacci numbers form a sequence defined recursively by: :F_n = \begin 0 & n = 0 \\ 1 & n = 1 \\ F_ + F_ & n > 1 \end That is, after two starting values, each number is the sum of the two preceding numbers. The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers. Extension to negative integers Using F_ = F_n - F_, one can extend the Fibonacci numbers to negative integers. So we get: :... −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8, ... and F_ = (-1)^ F_n. See also NegaFibonacci coding. Extension to all real or complex numbers There are a number of possible generalizations of the Fibonacci numbers which include the real numbers (and sometimes the complex numbers) in their domain. These each involve the golden ratio , and are based on Binet's formula :F_n = \frac. The analy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
