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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 ''ABC'' and ''abc'' (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines ''Ab'', ''aB'', ''Ac'', ''aC'', ''Bc'', and ''bC'', and their three intersection points , , and . These three points are the intersection points of the "opposite" sides of the hexagon ''AbCaBc''. According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points ''X'', ''Y'', and ''Z'', called the ''Pappus line''. The Pappus configuration can also be derived from two triangles ''XcC'' and ''YbB'' that are in perspective with e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Graph
In the mathematical field of graph theory, a graph is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices and of , there is an automorphism :f : V(G) \rightarrow V(G) such that :f(u_1) = u_2 and f(v_1) = v_2. In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called -transitive or flag-transitive. By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex-transitive. Since the definition above maps one edge to another, a symmetric graph must also be edge-transitive. However, an edge-transitive graph need not be symmetric, since might map to , but not to . Star graphs are a simple example of being edge-transitive without being vertex-transitive or symmetric. As a further example, semi-symmetric graphs are edge-transitive and regular, but not vertex-transitiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orchard-planting Problem
In discrete geometry, the original orchard-planting problem asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. It is also called the tree-planting problem or simply the orchard problem. There are also investigations into how many ''k''-point lines there can be. Hallard T. Croft and Paul Erdős proved ''t''''k'' > ''c'' ''n''2 / ''k''3, where ''n'' is the number of points and ''t''''k'' is the number of ''k''-point lines. Their construction contains some ''m''-point lines, where ''m'' > ''k''. One can also ask the question if these are not allowed. Integer sequence Define ''t''3orchard(''n'') to be the maximum number of 3-point lines attainable with a configuration of ''n'' points. For an arbitrary number of points, ''n'', ''t''3orchard(''n'') was shown to be (1/6)''n''2 − O(n) in 1974. The first few values of ''t''3orchard(''n'') are given in the following table . Upp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orchard-planting Problem
In discrete geometry, the original orchard-planting problem asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. It is also called the tree-planting problem or simply the orchard problem. There are also investigations into how many ''k''-point lines there can be. Hallard T. Croft and Paul Erdős proved ''t''''k'' > ''c'' ''n''2 / ''k''3, where ''n'' is the number of points and ''t''''k'' is the number of ''k''-point lines. Their construction contains some ''m''-point lines, where ''m'' > ''k''. One can also ask the question if these are not allowed. Integer sequence Define ''t''3orchard(''n'') to be the maximum number of 3-point lines attainable with a configuration of ''n'' points. For an arbitrary number of points, ''n'', ''t''3orchard(''n'') was shown to be (1/6)''n''2 − O(n) in 1974. The first few values of ''t''3orchard(''n'') are given in the following table . Upp ... [...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 (British English: 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 (''f'', its first derivative ''f, and its second derivative ''f'''', exist and are continuous), the condition ''f'' = 0'' can also be used to find an inflection point since a point of ''f'' = 0'' must be passed to change ''f'''' from a positive value (concave upward) to a negative value (concave downward) or vice versa as ''f'''' is continuous; an inflection point of the curve is where ''f'' = 0'' and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but do ... [...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 ha ... [...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 of 12 points and 16 lines. Each point of the configuration belongs to four lines, and each line contains three points. Therefore, in the notation of configurations, the Reye .... Ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reye Configuration
In geometry, the Reye configuration, introduced by , is a configuration of 12 points and 16 lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts .... 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 . 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desargues Configuration
In geometry, the Desargues configuration is a 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. 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 points and lines, three points per line, and three lines ... [...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 each possible ... [...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 denoting ... [...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 geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
