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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] |
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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article. Biography Early years Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed. General properties ''These properties apply to all regular polygons, whether convex or star.'' A regular ''n''-sided polygon has rotational symmetry of order ''n''. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Straightedge And Compass
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 formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert William Richmond
Herbert William Richmond (born on the 17 July 1863 in Tottenham, England) was a mathematician who studied the Cremona–Richmond configuration. One of his most popular works is an exact construction of the regular heptadecagon in 1893 (which was calculated before by Carl Friedrich Gauss). Herbert was elected as a Fellow of the Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ... in 1911. On the 22 April 1948, Herbert died in Cambridge, England. The Richmond surface is named after him. References * * {{DEFAULTSORT:Richmond, Herbert William English mathematicians 1863 births 1948 deaths People from Tottenham ... [...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 num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thales' Theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's '' Elements''. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras. History There is nothing extant of the writing of Thales. Work done in ancient Greece tended to be attributed to men of wisdom without respect to all the individuals involved in any particular intellectual constructions; this is true of Pythagoras especially. Attribution did tend to occur at a later time. Reference to Thales was made by Proclus, and by Diogenes Laërtius documenting Pamphila's statement that Thales "was the first to inscribe in a circle a right-angle triangle". Babylonian mathematicians knew this for special cases before Thales proved it. It is bel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Heptadecagon Inscribed In A 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
