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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. The Koch snowflake has been constructed as an example of a continuous curve where drawing a ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Of Revolution
In geometry, a solid of revolution is a Solid geometry, solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution''), which may not Intersection (geometry), intersect the generatrix (except at its boundary). The Surface (mathematics), surface created by this revolution and which bounds the solid is the ''surface of revolution''. Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's centroid theorem, Pappus's second centroid theorem). A representative disc is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length ) around some axis (located units away), so that a cylinder (geometry), cylindrical volume of units is enclosed. Finding the volume Two common methods for finding the volume of a solid of revolution are the Disc integration, disc met ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rewrite System
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects. Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting. Example cases Logic In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thue–Morse Sequence
In mathematics, the Thue–Morse or Prouhet–Thue–Morse sequence is the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. It is sometimes called the fair share sequence because of its applications to fair division or parity sequence. The first few steps of this procedure yield the strings 0, 01, 0110, 01101001, 0110100110010110, and so on, which are the prefixes of the Thue–Morse sequence. The full sequence begins: :01101001100101101001011001101001.... The sequence is named after Axel Thue, Marston Morse and (in its extended form) Eugène Prouhet. Definition There are several equivalent ways of defining the Thue–Morse sequence. Direct definition To compute the ''n''th element ''tn'', write the number ''n'' in binary. If the number of ones in this binary expansion is odd then ''tn'' = 1, if even then ''tn'' = 0. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turtle Graphics
In computer graphics, turtle graphics are vector graphics using a relative cursor (the "turtle") upon a Cartesian plane (x and y axis). Turtle graphics is a key feature of the Logo programming language. It is also a simple and didactic way of dealing with moving frames. Overview The turtle has three attributes: a location, an orientation (or direction), and a pen. The pen, too, has attributes: color, width, and on/off state (also called ''down'' and ''up''). The turtle moves with commands that are relative to its own position, such as "move forward 10 spaces" and "turn left 90 degrees". The pen carried by the turtle can also be controlled, by enabling it, setting its color, or setting its width. A student could understand (and predict and reason about) the turtle's motion by imagining what they would do if they were the turtle. Seymour Papert called this "body syntonic" reasoning. A full turtle graphics system requires control flow, procedures, and recursion: many turtl ... [...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 (an aperiodic set of prototiles). 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koch Similarity Tiling
Koch may refer to: People * Koch (surname), people with this surname * Koch dynasty, a dynasty in Assam and Bengal, north east India * Koch family * Koch people (or Koche), an ethnic group originally from the ancient Koch kingdom in north east India ** Koch languages, Sino-Tibetan language family *** Koch language, a language spoken in India and Bangladesh * Koch, an alternate name of the Rabha people, in northeast India and surrounding countries ** Rabha language, their language * Koch Kingdom, in and around Assam Places * Koch (crater), a crater on the Moon * Koch, Iran (other), places in Iran * Koch, Łódź Voivodeship, a village in central Poland * Koch, Mississippi, United States * Koch, South Sudan, a village in Unity State, South Sudan * Koch Bihar, a princely state in north east India * Koch County, an administrative area in Unity State, South Sudan * Koch Kingdom, Assam, 13th-16th centuries Businesses * Koch Entertainment LP, now known as E1 En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyadic Rational
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations. Dyadic rationals also have applications in weights and measures, musical time signatures, and early mathematics education. They can accurately approximate any real number. The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula. However, division of one dyadic rational number by another does not always produce a dyadic rational result. Mathematically, this means that the dyadic rational numbers form a ring, lying between the ring of integers and the field of rational numbers. This ring may be denoted \Z tfrac12/math>. In advanced mathematics, the dyadic rational numbers are central to the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Space
In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space. Examples The Cantor set itself is a Cantor space. But the canonical example of a Cantor space is the countably infinite topological product of the discrete 2-point space . This is usually written as 2^\mathbb or 2ω (where 2 denotes the 2-element set with the discrete topology). A point in 2ω is an infinite binary sequence, that is a sequence that assumes only the values 0 or 1. Given such a sequence ''a''0, ''a''1, ''a''2,..., one can map it to the real number :\sum_^\infty \frac. This mapping gives a homeomorphism from 2ω onto the Cantor set, demonstrating that 2ω is indeed a Cantor space. Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, complete metri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Curve
In mathematics, a de Rham curve is a continuous fractal curve obtained as the image of the Cantor space, or, equivalently, from the base-two expansion of the real numbers in the unit interval. Many well-known fractal curves, including the Cantor function, Cesàro–Faber curve ( Lévy C curve), Minkowski's question mark function, blancmange curve, and the Koch curve are all examples of de Rham curves. The general form of the curve was first described by Georges de Rham in 1957.Georges de Rham, ''Sur quelques courbes definies par des equations fonctionnelles''. Univ. e Politec. Torino. Rend. Sem. Mat., 1957, 16, 101 –113 Construction Consider some complete metric space (M,d) (generally \mathbb2 with the usual euclidean distance), and a pair of contracting maps on M: :d_0:\ M \to M :d_1:\ M \to M. By the Banach fixed-point theorem, these have fixed points p_0 and p_1 respectively. Let ''x'' be a real number in the interval ,1/math>, having binary expansion :x = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in \R^n or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a simple curve in \R^n is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue measure#Construction of the Lebesgue measure, Lebesgue-measurable subset of \R^2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are ''d''-dimensional Hausdorff measures for any ''d'' ≥ 0, which is n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space-filling Curve
In mathematical analysis, a space-filling curve is a curve whose Range of a function, range reaches every point in a higher dimensional region, typically the unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the plane (mathematics), 2-dimensional plane are sometimes called ''Peano curves'', but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano. The closely related FASS curves (approximately space-Filling, self-Avoiding, Simple, and Self-similar curves) can be thought of as finite approximations of a certain type of space-filling curves. Definition Intuitively, a curve in two or three (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Camille Jordan, Jordan in 1887 introduced the following rigorous definition, which has since been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
