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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 perimete ... [...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-born 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.Interview with Alexander ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Plane Geometry
In mathematics, a theorem is a statement that has been proved, or can be proved. 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 the mainstream of 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, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. 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'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoperimetric Inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by its volume \operatorname(S), :\operatorname(S)\geq n \operatorname(S)^ \, \operatorname(B_1)^, where B_1\subset\R^n is a unit sphere. The equality holds only when S is a sphere in \R^n. On a plane, i.e. when n=2, the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. '' Isoperimetric'' literally means "having the same perimeter". Specifically in \R ^2, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that : L^2 \ge 4\pi A, 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 ... [...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 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Body Of Constant Brightness
In convex geometry, a body of constant brightness is a three-dimensional convex set all of whose two-dimensional projections have equal area. A sphere is a body of constant brightness, but others exist. Bodies of constant brightness are a generalization of curves of constant width, but are not the same as another generalization, the surfaces of constant width. The name comes from interpreting the body as a shining body with isotropic luminance, then a photo (with focus at infinity) of the body taken from any angle would have the same total light energy hitting the photo. Properties A body has constant brightness if and only if the reciprocal Gaussian curvatures at pairs of opposite points of tangency of parallel supporting planes have almost-everywhere-equal sums. According to an analogue of Barbier's theorem, all bodies of constant brightness that have the same projected area A as each other also have the same surface area, \textstyle\sqrt. This can be proved by the Crofton f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Of Revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are 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 considered to be the generatrix in the plane determined by it and the axis. The sections of the surface of revolution made by planes that are perpendicular to the axis are circles. Some special cases of hyperboloids (of either one or two sheets) and elliptic paraboloids are su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ball is the closed set of points of distance less than or equal to 1 from a fixed central point. Usually the center is at the origin of the space, so one speaks of "the unit ball" or "the unit sphere". Special cases are the unit circle and the unit disk. The importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of translation and scaling. In this way the properties of spheres in general can be reduced to the study of the unit sphere. Unit spheres and balls in Euclidean space In Euclidean space of ''n'' dimensions, the -dimensional unit sphere is the set of all points (x_1, \ldots, x_n) which satisfy the equation : x_1^2 + x_2^2 + \cdots + x_n ^2 = 1. The ''n''-dimensional open unit ball ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Of Constant Width
In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the width of the surface in a given direction to be the perpendicular distance between the parallels perpendicular to that direction. Thus, a surface of constant width is the three-dimensional analogue of a curve of constant width, a two-dimensional shape with a constant distance between pairs of parallel tangent lines. Definition More generally, any compact convex body ''D'' has one pair of parallel supporting planes in a given direction. A supporting plane is a plane that intersects the boundary of ''D'' but not the interior of ''D''. One defines the width of the body as before. If the width of ''D'' is the same in all directions, then one says that the body is of constant width and calls its boundary a surface of constant width, and the bo ... [...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 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 the line of len ... [...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 affin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crofton Formula
In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), 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-hand side in the Crofton formula is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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