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Moulton Plane
In incidence geometry, the Moulton plane is an example of an affine plane in which Desargues's theorem does not hold. It is named after the American astronomer Forest Ray Moulton. The points of the Moulton plane are simply the points in the real plane R2 and the lines are the regular lines as well with the exception that for lines with a negative slope, the slope doubles when they pass the ''y''-axis. Formal definition The Moulton plane is an incidence structure \mathfrak M=\langle P, G,\textrm I\rangle, where P denotes the set of points, G the set of lines and \textrm I the incidence relation "lies on": : P:=\mathbb R^2 \, : G:=(\mathbb R \cup \) \times \mathbb R, \infty is just a formal symbol for an element \not\in\mathbb R. It is used to describe vertical lines, which you may think of as lines with an infinitely large slope. The incidence relation is defined as follows: For p = (x, y) \in P and g = (m, b) \in G we have : p\,\textrm I\,g\iff\begin x=b&\textm=\infty\\ y=\fracm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moulton Plane2
Moulton may refer to: Places in the United Kingdom ;In England *Moulton, Cheshire *Moulton, Lincolnshire **Moulton Windmill *Moulton St Mary, Norfolk *Moulton, Northamptonshire **Moulton College, agricultural college **Moulton Park, industrial estate *Moulton, Suffolk **Moulton Paddocks, racehorse training establishment *Moulton, North Yorkshire **Moulton Hall, 17th-century manor house ;In Wales *Moulton, Vale of Glamorgan Places in the U.S. *Moulton, Alabama *Moulton, Iowa *Moulton, Ohio *Moulton Township, Auglaize County, Ohio *Moulton, Texas Places in Antarctica *Moulton Escarpment *Mount Moulton Other uses *William Moulton Marston *Moulton (surname) *Moulton Bicycle *Moulton (crater), Moulton lunar crater *Moulton plane, a non-desarguesian plane geometry *Moulton (horse), a Thoroughbred racehorse See also *Molten (other) *Molton (other) {{disambiguation, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Geometry
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incidence structure'' is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries. Incidence structures arise naturally and have been studied in various areas of mathematics. Consequently, there are different terminologies to describe these objects. In graph theory they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Plane (incidence Geometry)
In geometry, an affine plane is a system of points and lines that satisfy the following axioms: * Any two distinct points lie on a unique line. * Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line. (Playfair's axiom) * There exist three non-collinear points (points not on a single line). In an affine plane, two lines are called ''parallel'' if they are equal or disjoint. Using this definition, Playfair's axiom above can be replaced by: * Given a point and a line, there is a unique line which contains the point and is parallel to the line. Parallelism is an equivalence relation on the lines of an affine plane. Since no concepts other than those involving the relationship between points and lines are involved in the axioms, an affine plane is an object of study belonging to incidence geometry. They are non-degenerate linear spaces satisfying Playfair's axiom. The familiar Euclidean plane is an affine pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desargues's Theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and those of the other by and . ''Axial perspectivity'' means that lines and meet in a point, lines and meet in a second point, and lines and meet in a third point, and that these three points all lie on a common line called the ''axis of perspectivity''. ''Central perspectivity'' means that the three lines and are concurrent, at a point called the ''center of perspectivity''. This intersection theorem is true in the usual Euclidean plane but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by adding points at infinity, following Jean-Victor Poncelet. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forest Ray Moulton
Forest Ray Moulton (April 29, 1872 – December 7, 1952) was an American astronomer. Biography He was born in Le Roy, Michigan, and was educated at Albion College. After graduating in 1894 (Bachelor of Arts, A.B.), he performed his graduate studies at the University of Chicago and gained a Doctor of Philosophy, Ph.D. in 1899. At the University of Chicago he was associate in astronomy (1898–1900), instructor (1900–03), assistant professor (1903–08), associate professor (1908–12), and professor after 1912. He is noted for being a proponent, along with Thomas Chrowder Chamberlin, Thomas Chamberlin, of the Chamberlin–Moulton planetesimal hypothesis that the planets coalesced from smaller bodies they termed planetesimals. Their hypothesis called for the close passage of another star to trigger this condensation, a concept that has since fallen out of favor. In the first decades of the twentieth century, some additional small satellites were discovered to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "''y'' = ''mx'' + ''c''". Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The ''steepness'', incline, or grade of a line is measured by the absolute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the relation of which points are on which lines for all points and lines. What is left is the incidence structure of the Euclidean plane. Incidence structures are most often considered in the geometrical context where they are abstracted from, and hence generalize, planes (such as affine, projective, and Möbius planes), but the concept is very broad and not limited to geometric settings. Even in a geometric setting, incidence structures are not limited to just points and lines; higher-dimensional objects (planes, solids, -spaces, conics, etc.) can be used. The study of finite structures is sometimes called finite geometry. Formal definition and terminology An incidence structure is a triple ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division Ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element usually denoted , such that . So, (right) ''division'' may be defined as , but this notation is avoided, as one may have . A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields". In some languages, such as French, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field). All division rings are simple. That is, they have no two-sided ideal besi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * '' Journal of the American Mathematical Society'' * ''Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
