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Pappian Plane
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of Collinearity, collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line (mathematics), line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the ''Pappus line''. These three points are the points of intersection of the "opposite" sides of the hexagon AbCaBc. It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes. If one restricts the projective plane such that the Pappus line u is the line at infinity, one gets the ''affine version'' of Pappus's theorem shown in the second diagram. If the Pappus line u and the lines g,h have a point in common, one gets the so-called little version of Pappus's theorem. The duality (projective geometry), dual of this intersection theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Configuration
In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points. Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book ''Geometrie der Lage'', in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book ''Anschauliche Geometrie'', reprinted in English as . Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be ''realizable'' in that geometry), or as a type of abstract incidence geometry. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomsen's Theorem
Thomsen's theorem, named after Gerhard Thomsen, is a theorem in elementary geometry. It shows that a certain path constructed by line segments being parallel to the edges of a triangle always ends up at its starting point. Consider an arbitrary triangle ''ABC'' with a point ''P''1 on its edge ''BC''. A sequence of points and parallel lines is constructed as follows. The parallel line to ''AC'' through ''P''1 intersects ''AB'' in ''P''2 and the parallel line to BC through ''P''2 intersects AC in ''P''3. Continuing in this fashion the parallel line to AB through ''P''3 intersects BC in ''P''4 and the parallel line to ''AC'' through ''P''4 intersects ''AB'' in ''P''5. Finally the parallel line to ''BC'' through ''P''5 intersects AC in ''P''6 and the parallel line to ''AB'' through ''P''6 intersects ''BC'' in ''P''7. Thomsen's theorem now states that ''P''7 is identical to ''P''1 and hence the construction always leads to a closed path ''P''1''P''2''P''3''P''4''P''5''P''6''P''1 Ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pencil (mathematics)
In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane. Although the definition of a pencil is rather vague, the common characteristic is that the pencil is completely determined by any two of its members. Analogously, a set of geometric objects that are determined by any three of its members is called a bundle. Thus, the set of all lines through a point in three-space is a bundle of lines, any two of which determine a pencil of lines. To emphasize the two dimensional nature of such a pencil, it is sometimes referred to as a ''flat pencil''. Any geometric object can be used in a pencil. The common ones are lines, planes, circles, conics, spheres, and general curves. Even points can be used. A pencil of points is the set of all points on a given line. A more common term for this set is a ''range'' of points. Penci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-Desarguesian Plane
In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective spaces of dimension not 2; in other words, the only projective spaces of dimension not equal to 2 are the classical projective geometries over a field (or division ring). However, David Hilbert found that some projective planes do not satisfy it. The current state of knowledge of these examples is not complete. Examples There are many examples of both finite and infinite non-Desarguesian planes. Some of the known examples of infinite non-Desarguesian planes include: *The Moulton plane. *Moufang planes over alternative division rings that are not associative, such as the projective plane over the octonions. Since all finite alternative division rings are fields ( Artin–Zorn theorem), the only non-Desarguesian Moufang planes are infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desarguesian
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] |
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] |
Gerhard Hessenberg
Gerhard Hessenberg (16 August 1874 – 16 November 1925) was a German mathematician who worked in projective geometry, differential geometry, and set theory. Career Hessenberg received his Ph.D. from the University of Berlin in 1899 under the guidance of Hermann Schwarz and Lazarus Fuchs. His name is usually associated with projective geometry, where he is known for proving that Desargues' theorem is a consequence of Pappus's hexagon theorem, and differential geometry where he is known for introducing the concept of a connection. He was also a set theorist: the Hessenberg sum and product of ordinals are named after him. However, Hessenberg matrices are named for Karl Hessenberg, a near relative. In 1908 Gerhard Hessenberg was an Invited Speaker of the International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance-regular Graph
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and , the number of vertices at distance from and at distance from depends only upon , , and the distance between and . Every distance-transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group. Intersection arrays It turns out that a graph G of diameter d is distance-regular if and only if there is an array of integers \ such that for all 1 \leq j \leq d , b_j gives the number of neighbours of u at distance j+1 from v and c_j gives the number of neighbours of u at distance j - 1 from v for any pair of vertices u and v at distance j on G . The array of integers characterizing a distance-regular graph is known as its intersection array. Cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Graph
In the mathematical field of graph theory, the Pappus graph is a bipartite 3- regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek mathematician who is believed to have discovered the "hexagon theorem" describing the Pappus configuration. All the cubic distance-regular graphs are known; the Pappus graph is one of the 13 such graphs. The Pappus graph has rectilinear crossing number 5, and is the smallest cubic graph with that crossing number . It has girth 6, diameter 4, radius 4, chromatic number 2, chromatic index 3 and is both 3- vertex-connected and 3- edge-connected. It has book thickness 3 and queue number 2. The Pappus graph has a chromatic polynomial equal to: (x-1)x(x^-26x^+325x^-2600x^+14950x^-65762x^+229852x^-653966x^9+1537363x^8-3008720x^7+4904386x^6-6609926x^5+7238770x^4-6236975x^3+3989074x^2-1690406x+356509). The name "Pappus graph" has also bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
