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Kovner–Besicovitch Measure
In plane geometry the Kovner–Besicovitch measure is a number defined for any bounded convex set describing how close to being centrally symmetric it is. It is the fraction of the area of the set that can be covered by its largest centrally symmetric subset. Properties This measure is one for a set that is centrally symmetric, and less than one for sets whose closure is not centrally symmetric. It is invariant under affine transformations of the plane. If c is the center of symmetry of the largest centrally-symmetric set within a given convex body K, then the centrally-symmetric set itself is the intersection of K with its reflection across c. Minimizers The convex sets with the smallest possible Kovner–Besicovitch measure are the triangles, for which the measure is 2/3. The result that triangles are the minimizers of this measure is known as Kovner's theorem or the Kovner–Besicovitch theorem, and the inequality bounding the measure above 2/3 for all convex sets is the Kovner ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Measure Of Reuleaux Triangle
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logical system in which each result is '' proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Symmetry
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center. Terminology The term ''reflection'' is loose, and considered by some an abuse of language, with ''inversion'' preferred; however, ''point reflection'' is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. If is the point set of an affine space, then every affine transformation on can be repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve Of Constant Width
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. Standard examples are the circle and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the involutes of certain curves, or by intersecting circles centered on a partial curve. Every body of constant width is a convex set, its boundary crossed at most twice by any line, and if the line crosses perpendicularly it does so at both crossings, separated by the width. By Barbier's theorem, the body's perimeter is exactly times its width, but its area depends on its shape, with the Reuleaux triangle having the smallest possible area for its width and the circle the largest. Every superset of a body o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reuleaux Triangle
A Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the other two. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because its width is constant, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?" Reuleaux triangles have also been called spherical triangles, but that term more properly refers to triangles on the curved surface of a sphere. They are named after Franz Reuleaux,. a 19th-century German engineer who pioneered the study of machines for translating one type of motion into another, and who used Reuleaux triangles in his designs. However, these shapes were known before his time, for instance by the des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branko Grünbaum
Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentBranko Grünbaum Hrvatska enciklopedija LZMK. and a professor at the in . He received his Ph.D. in 1957 from Hebrew University of Jerusalem< ...
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Calculus Of Variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Lavrentyev
Mikhail Alekseevich Lavrentyev (or Lavrentiev, russian: Михаи́л Алексе́евич Лавре́нтьев) (November 19, 1900 – October 15, 1980) was a Soviet Union, Soviet mathematician and hydrodynamics, hydrodynamicist. Early years Lavrentiev was born in Kazan, where his father was an instructor at a college (he later became a professor at Kazan University, then Moscow University). Lavrentiev entered Kazan University, and, when his family moved to Moscow in 1921, he transferred to the Department of Physics and Mathematics of Moscow University. He graduated in 1922. He continued his studies in the university in 1923-26 as a graduate student of Nikolai Luzin. Although Luzin was alleged to plagiarize in science and indulge in anti-Sovietism by some of his students in 1936, Lavrentiev did not participate in the notorious political persecution of his teacher which is known as the Nikolai Luzin#The Luzin affair of 1936, Luzin case or Nikolai Luzin#The Luzin affair of 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazar Lyusternik
Lazar Aronovich Lyusternik (also Lusternik, Lusternick, Ljusternik; ; 31 December 1899, in Zduńska Wola, Congress Poland, Russian Empire – 23 July 1981, in Moscow, Soviet Union) was a Soviet mathematician. He is famous for his work in topology and differential geometry, to which he applied the variational principle. Using the theory he introduced, together with Lev Schnirelmann, he proved the theorem of the three geodesics, a conjecture by Henri Poincaré that every convex body in 3-dimensions has at least three simple closed geodesics. The ellipsoid with distinct but nearly equal axis is the critical case with exactly three closed geodesics. The ''Lusternik–Schnirelmann theory'', as it is called now, is based on the previous work by Poincaré, David Birkhoff, and Marston Morse. It has led to numerous advances in differential geometry and topology. For this work Lyusternik received the Stalin Prize in 1946. In addition to serving as a professor of mathematics at Moscow St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abram Samoilovitch Besicovitch
Abram Samoilovitch Besicovitch (or Besikovitch) (russian: link=no, Абра́м Само́йлович Безико́вич; 23 January 1891 – 2 November 1970) was a Russian Empire, Russian mathematician, who worked mainly in England. He was born in Berdyansk on the Sea of Azov (now in Ukraine) to a Karaite Judaism, Karaite Jewish family. Life and career Abram Besicovitch studied under the supervision of Andrey Markov at the St. Petersburg University, graduating with a PhD in 1912. He then began research in probability theory. He converted to Eastern Orthodoxy, joining the Russian Orthodox Church, on marrying in 1916. He was appointed professor at the Perm State University, University of Perm in 1917, and was caught up in the Russian Civil War over the next two years. In 1920 he took a position at the Saint Petersburg State University, Petrograd University. In 1924 he went to Copenhagen on a Rockefeller Fellowship, where he worked on almost periodic functions under Harald Bohr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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