|
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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartic Equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomial equation that can be solved by radicals in the general case (i.e., one in which the coefficients can take any value). History Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it couldn't be published immediately. The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book '' Ars Magna'' (1545). The proof that this was the highest order general polynomial for which such solutions could be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would ... [...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 In commutative algebra, the constructible topology on the spectrum \operatorname(A) of a commutative ring A is a topol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygons
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] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logical system in which each result is '' proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry ... [...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 formall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lill's Method
In mathematics, Lill's method is a visual method of finding the real roots of a univariate polynomial of any degree. It was developed by Austrian engineer Eduard Lill in 1867. A later paper by Lill dealt with the problem of complex roots. Lill's method involves drawing a path of straight line segments making right angles, with lengths equal to the coefficients of the polynomial. The roots of the polynomial can then be found as the slopes of other right-angle paths, also connecting the start to the terminus, but with vertices on the lines of the first path. Description of the method To employ the method a diagram is drawn starting at the origin. A line segment is drawn rightwards by the magnitude of the first coefficient (the coefficient of the highest-power term) (so that with a negative coefficient the segment will end left of the origin). From the end of the first segment another segment is drawn upwards by the magnitude of the second coefficient, then left by the magnitud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduard Lill
Eduard Lill (1830–1900) was an Austrian engineer and army officer. Life Lill was born 20 October 1830 in Brüx ( Bohemia). From 1848 to 1849 he studied mathematics at the Czech Technical University in Prague and in 1850 he joined the military engineering corps of the Austrian Empire. From 1852 to 1856 he continued his education at the military engineering academy at Klosterbruck near Znaim. Later he was stationed in Esseg, Kronstadt and Spalato until he retired from his military career in 1868 with the rank of captain (Hauptmann) of the engineering corps. In the same year he became an engineer for the Austrian Northwestern Railway and oversaw the railroad construction at Trautenau (Trutnov). A severe accident however restricted him soon to office work. From 1872-1875 he worked as a secretary for the director of construction of the railway company. Later he became a technical consultant for company's headquarters and in 1885 the head of its statistics department. He retired in 18 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Leslie (physicist)
Sir John Leslie, FRSE KH (10 April 1766 – 3 November 1832) was a Scottish mathematician and physicist best remembered for his research into heat. Leslie gave the first modern account of capillary action in 1802 and froze water using an air-pump in 1810, the first artificial production of ice. In 1804, he experimented with radiant heat using a cubical vessel filled with boiling water. One side of the cube is composed of highly polished metal, two of dull metal (copper) and one side painted black. He showed that radiation was greatest from the black side and negligible from the polished side. The apparatus is known as a Leslie cube. Early life Leslie was born the son of Robert Leslie, a joiner and cabinetmaker, and his wife Anne Carstairs, in Largo in Fife. He received his early education there and at Leven. In his thirteenth year, encouraged by friends who had even then remarked his aptitude for mathematical and physical science, he entered the University of St Andrews. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Howard Eves
Howard Whitley Eves (10 January 1911, New Jersey – 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics. Eves received his B.S. from the University of Virginia, an M.A. from Harvard University, and a Ph.D. in mathematics from Oregon State University Oregon State University (OSU) is a public land-grant, research university in Corvallis, Oregon. OSU offers more than 200 undergraduate-degree programs along with a variety of graduate and doctoral degrees. It has the 10th largest engineering c ... in 1948, the last with a dissertation titled ''A Class of Projective Space Curves'' written under Ingomar Hostetter. He then spent most of his career at the University of Maine, 1954–1976. In later life, he occasionally taught at University of Central Florida. Eves was a strong spokesman for the Mathematical Association of America, which he joined in 1942, and whose Northeast Section he founded. For 25 years he edited the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlyle Circle Original Problem
Carlyle may refer to: Places * Carlyle, Illinois, a US city * Carlyle, Kansas, an unincorporated place in the US * Carlyle, Montana, a ghost town in the US * Carlyle, Saskatchewan, a Canadian town ** Carlyle Airport ** Carlyle station * Carlyle Lake Resort, Saskatchewan, a Canadian hamlet * Carlyle Hotel, New York City * Carlyle Restaurant, New York City * The Carlyle, a residential condominium in Minneapolis, Minnesota * The Carlyle (Pittsburgh), a residential condominium in Pittsburgh, Pennsylvania Other uses * The Carlyle Group, a private equity company based in the US * Carlyle Works, a former bus bodybuilder in the UK *Carlyle (name) See also * Carlisle (other) * Carlile (other) * Carlyne Carlyne is both a given name that is a variant of Carly and Caroline. Notable people with the name include: *Arthur Carlyne Niven Dixey, full name of Arthur Dixey (1889 – 1954), British Member of Parliament * Carlyne Cerf de Dudzeele, French st ... {{disambigua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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– self-intersecting is 11796300°. Regular 65537-gon The area of a ''regular '' is (with ) :A = \frac t^2 \cot \frac A whole 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 algebraic numbers, and like any constructible numbers, they can be written in terms of square roots and no higher-order roots. Although it was known to Gauss by 1801 that the regular 65537-gon was constructible, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
