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Barbier's Theorem
In geometry, Barbier's theorem states that every curve of constant width has perimeter times its width, regardless of its precise shape. This theorem was first published by Joseph-Émile Barbier in 1860. Examples The most familiar examples of curves of constant width are the circle and the Reuleaux triangle. For a circle, the width is the same as the diameter; a circle of width ''w'' has perimeter ''w''. A Reuleaux triangle of width ''w'' consists of three arcs of circles of radius ''w''. Each of these arcs has central angle /3, so the perimeter of the Reuleaux triangle of width ''w'' is equal to half the perimeter of a circle of radius ''w'' and therefore is equal to ''w''. A similar analysis of other simple examples such as Reuleaux polygons gives the same answer. Proofs One proof of the theorem uses the properties of Minkowski sums. If ''K'' is a body of constant width ''w'', then the Minkowski sum of ''K'' and its 180° rotation is a disk with radius ''w'' and peri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reuleaux Polygons , the intersection of four spheres of equal radius centered at the vertices of a regular tetrahedron
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Reuleaux may refer to: * Franz Reuleaux (1829–1905), German mechanical engineer and lecturer * in geometry: ** Reuleaux polygon, a curve of constant width *** Reuleaux triangle, a Reuleaux polygon with three sides *** Reuleaux heptagon, a Reuleaux polygon with seven sides that provides the shape of some currency coins ** Reuleaux tetrahedron The Reuleaux tetrahedron is the intersection of four balls of radius ''s'' centered at the vertices of a regular tetrahedron with side length ''s''. The spherical surface of the ball centered on each vertex passes through the other three verti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Mathematics, senior instructor at Hebrew University and software consultant at Ben Gurion University. He wrote extensively about arithmetic, probability, algebra, geometry, trigonometry and mathematical games. He was known for his contribution to heuristics and mathematics education, creating and maintaining the mathematically themed educational website ''Cut-the-Knot'' for the Mathematical Association of America (MAA) Online. He was a pioneer in mathematical education on the internet, having started ''Cut-the-Knot'' in October 1996. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Plane Geometry
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasonin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoperimetric Inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. '' Isoperimetric'' literally means "having the same perimeter". Specifically, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that :4\pi A \le L^2, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length. The closely related ''Dido's problem'' asks for a region of the maximal area bounded by a straight line and a curvilinear arc whose endpoints belong to that line. It is named after Dido, the legendary founder and first queen of Carthage. The solution to the isoperimetric problem is given by a circle and was known already in Ancient Greece. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blaschke–Lebesgue Theorem
In plane geometry the Blaschke–Lebesgue theorem states that the Reuleaux triangle has the least area of all curves of given constant width. In the form that every curve of a given width has area at least as large as the Reuleaux triangle, it is also known as the Blaschke–Lebesgue inequality. It is named after Wilhelm Blaschke and Henri Lebesgue, who published it separately in the early 20th century. Statement The width of a convex set K in the Euclidean plane is defined as the minimum distance between any two parallel lines that enclose it. The two minimum-distance lines are both necessarily tangent lines to K, on opposite sides. A curve of constant width is the boundary of a convex set with the property that, for every direction of parallel lines, the two tangent lines with that direction that are tangent to opposite sides of the curve are at a distance equal to the width. These curves include both the circle and the Reuleaux triangle, a curved triangle formed from arcs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Body Of Constant Brightness
Body may refer to: In science * Physical body, an object in physics that represents a large amount, has mass or takes up space * Body (biology), the physical material of an organism * Body plan, the physical features shared by a group of animals * Human body, the entire structure of a human organism ** Dead body, cadaver, or corpse, a dead human body * (living) matter, see: Mind–body problem, the relationship between mind and matter in philosophy * Aggregates within living matter, such as inclusion bodies In arts and entertainment In film and television * ''Jism'' (2003 film) or ''Body'', a 2003 Indian film * ''Body'' (2015 Polish film), a 2015 Polish film * ''Body'' (2015 American film), a 2015 American film * "Body" (''Wonder Showzen'' episode), a 2006 episode of American sketch comedy television series ''Wonder Showzen'' * "Body", an episode of the Adult Swim television series, '' Off the Air'' In literature and publishing * body text, the text forming the main con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Of Revolution
A surface of revolution is a Surface (mathematics), surface in Euclidean space created by rotating a curve (the ''generatrix'') one full revolution (unit), revolution around an ''axis of rotation'' (normally not Intersection (geometry), intersecting the generatrix, except at its endpoints). The volume bounded by the surface created by this revolution is the ''solid of revolution''. Examples of surfaces of revolution generated by a straight line are cylinder (geometry), cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus). Properties The sections of the surface of revolution made by planes through the axis are called ''meridional sections''. Any meridional section can be consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Sphere
In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -sphere of unit radius in -dimensional Euclidean space; the unit circle is a special case, the unit -sphere in the Euclidean plane, plane. An (Open set, open) unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center. A sphere or ball with unit radius and center at the origin (mathematics), origin of the space is called ''the'' unit sphere or ''the'' unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of translation (geometry), translation and scaling (geometry), scaling, so the study of spheres in general can often be reduced to the study of the unit sphere. The unit sphere is often used as a model for spherical geometry because it has constant sectional cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Of Constant Width
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is the portion with which other materials first interact. The surface of an object is more than "a mere geometric solid", but is "filled with, spread over by, or suffused with perceivable qualities such as color and warmth". The concept of surface has been abstracted and formalized in mathematics, specifically in geometry. Depending on the properties on which the emphasis is given, there are several inequivalent such formalizations that are all called ''surface'', sometimes with a qualifier such as algebraic surface, smooth surface or fractal surface. The concept of surface and its mathematical abstractions are both widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buffon's Noodle
In geometric probability, the problem of Buffon's noodle is a variation on the well-known problem of Buffon's needle, named after Georges-Louis Leclerc, Comte de Buffon who lived in the 18th century. This approach to the problem was published by Joseph-Émile Barbier in 1860. Buffon's needle Suppose there exist infinitely many equally spaced parallel, horizontal lines, and we were to randomly toss a needle whose length is less than or equal to the distance between adjacent lines. What is the probability that the needle will lie across a line upon landing? To solve this problem, let \ell be the length of the needle and D be the distance between two adjacent lines. Then, let \theta be the acute angle the needle makes with the horizontal, and let x be the distance from the center of the needle to the nearest line. The needle lies across the nearest line if and only if x \le \frac . We see this condition from the right triangle formed by the needle, the nearest line, and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crofton Formula
In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), (also Cauchy-Crofton formula) is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it. Statement Suppose \gamma is a rectifiable plane curve. Given an oriented line ''ℓ'', let n_\gamma(''ℓ'') be the number of points at which \gamma and ''ℓ'' intersect. We can parametrize the general line ''ℓ'' by the direction \varphi in which it points and its signed distance p from the origin. The Crofton formula expresses the arc length of the curve \gamma in terms of an integral over the space of all oriented lines: :\operatorname (\gamma) = \frac14\iint n_\gamma(\varphi, p)\; d\varphi\; dp. The differential form :d\varphi\wedge dp is invariant under rigid motions of \R^2, so it is a natural integration measure for speaking of an "average" number of intersections. It is usually called the kinematic measure. The right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
