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65537-gon
In geometry, a 65537-gon is a polygon with 65,537 (216 + 1) sides. The sum of the interior angles of any non–list of self-intersecting polygons, self-intersecting is 11796300°. Regular 65537-gon The area of a ''regular '' is (with ) :A = \frac t^2 \cot \frac A whole regular polygon, regular is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 15 parts per billion. Construction The regular 65537-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 65,537 is a Fermat prime, being of the form 22''n'' + 1 (in this case ''n'' = 4). Thus, the values \cos \frac and \cos \frac are 32768-Degree of a polynomial, degree algebraic numbers, and like any constructible numbers, they can be written in terms of square roots and no higher-order roots. Although it was ... [...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? Carl ... [...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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Gustav Hermes
Johann Gustav Hermes (20 June 1846 – 8 June 1912) was a German mathematician. Hermes is known for completion of a polygon with 65,537 sides. Early life On 20 June 1846, Hermes was born in Königsberg, a former German city (presently Kaliningrad, Russia). Hermes was educated at the Kneiphöfischen Gymnasium. He undertook his ''Abitur'' (final examination) at the school in 1866. After completing his secondary education, he studied mathematics from 1866 to 1870, mostly in Königsberg. His studies were interrupted due to his participation in the Franco-Prussian War between 1870 and 1871. Education On 14 December 1872, Hermes complete his studies and earned a degree in mathematics. On 5 April 1879, Hermes received a doctorate degree and his dissertation was on the "Reduction of the problem of cyclotomy on linear equations (for prime numbers of the form 2''m''+1)" (German: "''Zurückführung des Problems der Kreistheilung auf lineare Gleichungen (für Primzahlen von der For ... [...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 number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Dixon (mathematician)
Robert Dixon (born 1947) is a British mathematician and graphic artist, 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 friend Damien stole my skull idea" ''The Times ''The Times'' is a British daily national newspaper based in London. It began in 1785 under the title ''The Daily Universal Register'', adopting its current name on 1 January 1788. ''The Times'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 constructible, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptadecagon
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 Fermat prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''star pentagon'') is called a pentagram. Regular pentagons A '' regular pentagon'' has Schläfli symbol and interior angles of 108°. A '' regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, \\ W &= \sqr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac a^2, *The perimeter is p=3a\,\! *The radius of the circumscribed circle is R = \frac *The radius of the inscribed circle is r=\frac a or r=\frac *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac a Denoting the radius of the circumscribed circle as ''R'', we can determine using trigonometry that: *The area of the triangle is \mathrm=\fracR^2 Many of these quantities have simple r ... [...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] |
Regular 65537-gon First Carlyle Circle
The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses in arts, entertainment, and media * Regular character, a main character who appears more frequently and/or prominently than a recurring character * Regular division of the plane, a series of drawings by the Dutch artist M. C. Escher which began in 1936 * ''Regular Show'', an animated television sitcom * ''The Regular Guys'', a radio morning show Language * Regular inflection, the formation of derived forms such as plurals in ways that are typical for the language ** Regular verb * Regular script, the newest of the Chinese script styles Mathematics There are an extremely large number of unrelated notions of "regularity" in mathematics. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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