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Pappus Configuration
In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point. History and construction This configuration is named after Pappus of Alexandria. Pappus's hexagon theorem states that every two triples of collinear points and (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines , , , , , and , and their three intersection points , , and . These three points are the intersection points of the "opposite" sides of the hexagon . According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points , , and , called the ''Pappus line''. The Pappus configuration can also be derived from two triangles and that are in perspective with each other (the three lines through corresponding pairs of points meet at a single crossin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Symmetry In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Configurations (geometry)
Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice board, design system, or co-design platform, used in product design to capture customers' specifications * Configure script ("./configure" in Unix), the output of Autotools; used to detect system configuration * CONFIG.SYS, the primary configuration file for DOS and OS/2 operating systems Mathematics * Configuration (geometry), a finite set of points and lines with certain properties * Configuration (polytope), special kind of configuration for regular polytopes * Configuration space (mathematics), a space representing assignments of points to non-overlapping positions on a topological space Physics * Configuration space (physics), in classical mechanics, the vector space formed by the parameters of a system * Electron configuration, the distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orchard-planting Problem
In discrete geometry, the original orchard-planting problem (or the tree-planting problem) asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. There are also investigations into how many -point lines there can be. Hallard T. Croft and Paul Erdős proved t_k > \frac, where is the number of points and is the number of -point lines. Their construction contains some -point lines, where . One can also ask the question if these are not allowed. Integer sequence Define to be the maximum number of 3-point lines attainable with a configuration of points. For an arbitrary number of points, was shown to be \tfracn^2 - O(n) in 1974. The first few values of are given in the following table . Upper and lower bounds Since no two lines may share two distinct points, a trivial upper-bound for the number of 3-point lines determined by points is \left\lfloor \binom \Big/ \binom \right\rfloor = \left\lfloor \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inflection Point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function of differentiability class (its first derivative , and its second derivative , exist and are continuous), the condition can also be used to find an inflection point since a point of must be passed to change from a positive value (concave upward) to a negative value (concave downward) or vice versa as is continuous; an inflection point of the curve is where and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Plane Curve
In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equation. Here is a non-zero linear combination of the third-degree monomials : These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field . Each point imposes a single linear condition on , if we ask that pass through . Therefore, we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position; compare to two points determining a line and how five points determine a conic. If two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem. A cubic curve may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desmic System
Two desmic tetrahedra. The third tetrahedron of this system is not shown, but has one vertex at the center and the other three on the plane at infinity. In projective geometry, a desmic system () is a set of three tetrahedra in 3-dimensional projective space, such that any two are desmic (related such that each edge of one cuts a pair of opposite edges of the other). It was introduced by . The three tetrahedra of a desmic system are contained in a pencil of quartic surfaces. Every line that passes through two vertices of two tetrahedra in the system also passes through a vertex of the third tetrahedron. The 12 vertices of the desmic system and the 16 lines formed in this way are the points and lines of a Reye configuration In geometry, the Reye configuration, introduced by , is a Configuration (geometry), configuration of 12 Point (geometry), points and 16 Line (geometry), lines. Each point of the configuration belongs to four lines, and each line contains three p .... Ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reye Configuration
In geometry, the Reye configuration, introduced by , is a Configuration (geometry), configuration of 12 Point (geometry), points and 16 Line (geometry), lines. Each point of the configuration belongs to four lines, and each line contains three points. Therefore, in the notation of configurations, the Reye configuration is written as . It is symmetric (both point and line transitive) and has 576 automorphisms. Realization The Reye configuration can be realized in three-dimensional projective space by taking the lines to be the 12 edges and four long diagonals of a cube, and the points as the eight vertices of the cube, its center, and the three points where groups of four parallel cube edges meet the plane at infinity. Two regular tetrahedron, tetrahedra may be inscribed within a cube, forming a stella octangula; these two tetrahedra are perspective figures to each other in four different ways, and the other four points of the configuration are their centers of perspectivity. Thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desargues Configuration
In geometry, the Desargues configuration is a Configuration (geometry), configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues. The Desargues configuration can be constructed in two dimensions from the points and lines occurring in Desargues's theorem, in three dimensions from five planes in general position, or in four dimensions from the 5-cell, the four-dimensional regular simplex. It has a large group of symmetries, taking any point to any other point and any line to any other line. It is also self-dual, meaning that if the points are replaced by lines and vice versa using projective duality, the same configuration results. Graph (discrete mathematics), Graphs associated with the Desargues configuration include the Desargues graph (its graph of point-line incidences) and the Petersen graph (its graph of non-incident lines). The Desargues configuration is one of ten different configurations with ten p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *'' Memoirs of the American Mathematical Society'' *'' Notices of the American Mathematical Society'' *'' Proceedings of the Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hesse Configuration
In geometry, the Hesse configuration is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be denoted as (94 123) or configuration matrix \left begin9 & 4 \\ 3 & 12 \\ \end\right /math>. It is symmetric (point and line transitive) with 432 automorphisms. It can be realized in the complex projective plane as the set of inflection points of an elliptic curve, but it has no realization in the Euclidean plane. It was introduced by Colin Maclaurin and studied by , and is also known as Young's geometry, named after the later work of John Wesley Young on finite geometry. Description The Hesse configuration has the same incidence relations as the lines and points of the affine plane over the field of 3 elements. That is, the points of the Hesse configuration may be identified with ordered pairs of numbers modulo 3, and the lines of the configuration may correspondingly be identified with the triples of points satisfying a lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
