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Sierpiński Curve
Sierpiński curves are a recursively defined sequence of Geometric continuity, continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n \to \infty completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve. Because the Sierpiński curve is space-filling, its Hausdorff dimension (in the limit n \to \infty ) is 2 . The Euclidean distance, Euclidean length of the nth iteration curve S_n is : l_n = (1+\sqrt 2) 2^n - (2-\sqrt 2) , i.e., it grows ''exponentially'' with n beyond any limit, whereas the limit for n \to \infty of the area enclosed by S_n is 5/12 \, that of the square (in Euclidean metric). Uses of the curve The Sierpiński curve is useful in several practical applications because it is more symmetrical than other commonly studied space-filling curves. For example, it has been used as a basis for the rapid construction of an approximate soluti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's ances ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sierpiński Triangle
The Sierpiński triangle (sometimes spelled ''Sierpinski''), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński. Constructions There are many different ways of constructing the Sierpinski triangle. Removing triangles The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets: # Start with an equilateral triangle. # Subdivide it into four smaller congruent equilateral triangles and remove the central triangle. # Repeat step 2 with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Science And Technology In Poland
Education has been of prime interest to Poland's rulers since the early 12th century. The catalog of the library of the Cathedral Chapter in Kraków dating from 1110 shows that Polish scholars already then had access to western European literature. In 1364, King Kazimierz the Great founded the Cracow Academy, which would become one of the great universities of Europe. The Polish people have made considerable contributions in the fields of science, technology and mathematics. The list of famous scientists in Poland begins in earnest with the polymath, astronomer and mathematician Nicolaus Copernicus, who formulated the heliocentric theory and made an important contribution to the scientific revolution. In 1773 King Stanisław August Poniatowski established the Commission of National Education, the world's first ministry of education. After the third partition of Poland, in 1795, no Polish state existed. The 19th and 20th centuries saw many Polish scientists working abroad. One of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Curves
A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length. A famous example is the boundary of the Mandelbrot set. Fractal curves in nature Fractal curves and fractal patterns are widespread, in nature, found in such places as broccoli, snowflakes, feet of geckos, frost crystals, and lightning bolts. See also Romanesco broccoli, dendrite crystal, trees, fractals, Hofstadter's butterfly, Lichtenberg figure, and self-organized criticality. Dimensions of a fractal curve Most of us are used to mathematical curves having dimension one, but as a general rule, fractal curves have different dimensions, also see also fractal dimension and list of fractals by Hausdorf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion (computer Science)
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science. Most computer programming languages support recursion by allowing a function to call itself from within its own code. Some functional programming languages (for instance, Clojure) do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in computability theory that these recursive-only languages are Turing complete; this means that they are as powerful (they can be used to solve the same problems) as imperative languages based on control structures such as and . Repeatedly calling a function from within itself may cause the call stack to have a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Fractals By Hausdorff Dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff dimension, 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* [https://web.archive.org/web/20060923100014/http://library.thinkquest.org/26242/full/index.html Fractals unleashed] IFStile - software that computes the dimension of the boundary of self-affine tiles {{DEFAULTSORT:Fractals By Hausdorff Dimension Fractals, Hausdorff Dimension Fractal curves, Hausdorff Dimension Mathematics-related lists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Curve
In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve. Construction Peano's curve may be constructed by a sequence of steps, where the ''i''th step constructs a set ''Si'' of squares, and a sequence ''Pi'' of the centers of the squares, from the set and sequence constructed in the previous step. As a base case, ''S''0 consists of the single unit square, and ''P''0 is the one-element sequence consisting of its center point. In step ''i'', each square ''s'' of ''S''''i'' − 1 is partitioned into nine smaller equal squares, and its center point ''c'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Murray Polygon
In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve. Construction Peano's curve may be constructed by a sequence of steps, where the ''i''th step constructs a set ''Si'' of squares, and a sequence ''Pi'' of the centers of the squares, from the set and sequence constructed in the previous step. As a base case, ''S''0 consists of the single unit square, and ''P''0 is the one-element sequence consisting of its center point. In step ''i'', each square ''s'' of ''S''''i'' − 1 is partitioned into nine smaller equal squares, and its center point ''c'' is replac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moore Graph
In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). If the degree of such a graph is and its diameter is , its girth must equal . This is true, for a graph of degree and diameter , if and only if its number of vertices equals :1+d\sum_^(d-1)^i, an upper bound on the largest possible number of vertices in any graph with this degree and diameter. Therefore, these graphs solve the degree diameter problem for their parameters. Another equivalent definition of a Moore graph is that it has girth and precisely cycles of length , where and are, respectively, the numbers of vertices and edges of . They are in fact extremal with respect to the number of cycles whose length is the girth of the graph. Moore graphs were named by after Edward F. Moore, who posed the question of describing and classifying these graphs. As well as having the maximum possible numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koch Snowflake
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch. The Koch snowflake can be built up iteratively, in a sequence of stages. The first stage is an equilateral triangle, and each successive stage is formed by adding outward bends to each side of the previous stage, making smaller equilateral triangles. The areas enclosed by the successive stages in the construction of the snowflake converge to \tfrac times the area of the original triangle, while the perimeters of the successive stages increase without bound. Consequently, the snowflake encloses a finite area, but has an infinite perimeter. Construction The Koch snowflake can be constructed by starting with an equilateral triangle, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Curve
The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890. Because it is space-filling, its Hausdorff dimension is 2 (precisely, its image is the unit square, whose dimension is 2 in any definition of dimension; its graph is a compact set homeomorphic to the closed unit interval, with Hausdorff dimension 2). The Hilbert curve is constructed as a limit of piecewise linear curves. The length of the nth curve is \textstyle 2^n - , i.e., the length grows exponentially with n, even though each curve is contained in a square with area 1. Images File:Hilbert curve 1.svg, Hilbert curve, first order File:Hilbert curve 2.svg, Hilbert curves, first and second orders File:Hilbert curve 3.svg, Hilbert curves, first to third orders File:Hilbert curve production rules!.svg, Production rules F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
