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257-gon
In geometry, a 257-gon is a polygon with 257 sides. The sum of the interior angles of any non- self-intersecting 257-gon is 45,900°. Regular 257-gon The area of a regular 257-gon is (with ) :A = \frac t^2 \cot \frac\approx 5255.751t^2. A whole regular 257-gon is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 24 parts per million. Construction The regular 257-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 257 is a Fermat prime, being of the form 22''n'' + 1 (in this case ''n'' = 3). Thus, the values \cos \frac and \cos \frac are 128-degree algebraic numbers, and like all constructible numbers they can be written using square roots and no higher-order roots. Although it was known to Gauss by 1801 that the regular 257-gon was construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known. Conditions for constructibility Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969). This led to the question being posed: is it possible to construct ''all'' regular polygons with compass and straightedge? If not, which ''n''-gons (that is, polygons with ''n'' edges) are constructible and which are not? Ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlyle Circle
In mathematics, a Carlyle circle (named for Thomas Carlyle) is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons. Definition Given the quadratic equation :''x''2 − ''sx'' + ''p'' = 0 the circle in the coordinate plane having the line segment joining the points ''A''(0, 1) and ''B''(''s'', ''p'') as a diameter is called the Carlyle circle of the quadratic equation.E. John Hornsby, Jr.''Geometrical and Graphical Solutions of Quadratic Equations'' The College Mathematics Journal, Vol. 21, No. 5 (Nov., 1990), pp. 362–369JSTORJSTOR Defining property The defining property of the Carlyle circle can be established thus: the equation of the circle having th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its ''edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygons By The Number Of Sides
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its ''edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compass And Straightedge Construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedrich Julius Richelot
Friedrich Julius Richelot (6 November 1808 – 31 March 1875) was a German mathematician, born in Königsberg. He was a student of Carl Gustav Jacob Jacobi. He was promoted in 1831 at the Philosophical Faculty of the University of Königsberg with a dissertation on the division of the circle into 257 equal parts (see references) and was a professor there. Richelot authored numerous publications in German, French and Latin, among them — with his 1832 dissertation — the first known guide to the Euclidean construction of the regular 257-gon with compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali .... In 1825 he joined the Corps Masovia.Kösener Korps-Listen 1910, 141, 8 He died in Königsberg in 1875. See also * Timeline of abelian varieties References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a '' disc''. A circle may also be defined as a special ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which alw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Polygons
Constructibility or constructability may refer to: * Constructability or construction feasibility review, a process in construction design whereby plans are reviewed by others familiar with construction techniques and materials to assess whether the design is actually buildable * Constructible strategy game, a tabletop strategy game employing pieces assembled from components Mathematics * Compass-and-straightedge construction * Constructible point, a point in the Euclidean plane that can be constructed with compass and straightedge * Constructible number, a complex number associated to a constructible point * Constructible polygon, a regular polygon that can be constructed with compass and straightedge * Constructible sheaf, a certain kind of sheaf of abelian groups * Constructible set (topology), a finite union of locally closed sets * Constructible topology, a topology on the spectrum of a commutative ring y in which every closed set is the image of Spec(''B'') in Spec(''A'') f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Eugene Dickson
Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory, '' History of the Theory of Numbers''. Life Dickson considered himself a Texan by virtue of having grown up in Cleburne, where his father was a banker, merchant, and real estate investor. He attended the University of Texas at Austin, where George Bruce Halsted encouraged his study of mathematics. Dickson earned a B.S. in 1893 and an M.S. in 1894, under Halsted's supervision. Dickson first specialised in Halsted's own specialty, geometry.A. A. Albert (1955Leonard Eugene Dickson 1874–1954from National Academy of Sciences Both the University of Chicago and Harvard University welcomed Dickson as a Ph.D. student, and Dickson initially accepted Harvard's offer, but chose to attend Chicago ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Dixon (mathematician)
Robert Dixon (born 1947) is a British mathematician and graphic artist A graphic designer is a professional within the graphic design and graphic arts industry who assembles together images, typography, or motion graphics to create a piece of design. A graphic designer creates the graphics primarily for published, p ..., known primarily for his book ''Mathographics'' and for his plagiarism dispute with Damien Hirst. Dixon was a research associate at the Royal College of Art. He complained in 2004 that a circular pattern Hirst produced for a children's colouring book was a copy of one of his works. In 2006, Dixon said that Hirst's print ''Valium'' had "unmistakable similarities" to one of his own designs. Hirst's manager contested this by explaining the origin of Hirst's piece was from a book ''The Penguin Dictionary of Curious and Interesting Geometry'' (1991)—not realising this was one place where Dixon's design had been published.Alberge, Dalya. (27 June 2007)"My old ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
17-gon
In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon. Regular heptadecagon A '' regular heptadecagon'' is represented by the Schläfli symbol . Construction As 17 is a Fermat prime, the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796 at the age of 19.Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, ''Abstract Algebra and Famous Impossibilities'', Springer, 1991, p. 178./ref> This proof represented the first progress in regular polygon construction in over 2000 years. Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions, and secondly on his proof that this can be done if the odd prime factors of N, the number of sides of the regular polygon, are distinct Ferma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
