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Integer Lattice (Lie Group)
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. Automorphism group The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2''n'' ''n''!. As a matrix group it is given by the set of all ''n'' × ''n'' signed permutation matrices. This group is isomorphic to the semidirect product :(\mathbb Z_2)^n \rtimes S_n where the symmetric group ''S''''n'' acts on (Z2)''n'' by permutation (this is a classic example of a wreath product). For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Pentagram Lattice Polygon Approximation
The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses in arts, entertainment, and media * Regular character, a main character who appears more frequently and/or prominently than a recurring character * Regular division of the plane, a series of drawings by the Dutch artist M. C. Escher which began in 1936 * ''Regular Show'', an animated television sitcom * ''The Regular Guys'', a radio morning show Language * Regular inflection, the formation of derived forms such as plurals in ways that are typical for the language ** Regular verb * Regular script, the newest of the Chinese script styles Mathematics There are an extremely large number of unrelated notions of "regularity" in mathematics. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. Definition and notation Given two groups (G, *) and (H, \odot), a ''group isomorphism'' from (G, *) to (H, \odot) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function f : G \to H such that for all u and v in G it holds that f(u * v) = f(u) \odot f(v). The two groups (G, *) and (H, \odot) are isomorphic if there exists an isomorphism from one to the other. This is written (G, *) \cong (H, \odot). Often shorter and simpler notations can be used. When the relevant group operations are understood, they are omitted and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pick Theorem Simple
Pick may refer to: Places * Pick City, North Dakota, a town in the United States * Pick Lake (Cochrane District, Ontario), a lake in Canada * Pick Lake (Thunder Bay District), a lake in Canada * Pick Mere, a lake in Pickmere, England People with the name * Pick (surname), a list of people with this name * nickname of Percy Charles Pickard (1915–1944), British Royal Air Force pilot * Pick Temple (1911–1991), American folk singer and children's television star * Pick Withers (born 1948), drummer for the English rock band Dire Straits Arts, entertainment, and media * Plectrum or pick, a device for strumming a stringed instrument :*Guitar pick, specific to guitars and similar instruments * The Picks, a vocal quartet which backed Buddy Holly and the Crickets in 1957 * Pick (TV channel), a British television channel * "The Pick", an episode of the television show ''Seinfeld'' * Odds and evens or pick, a hand game * Pick (film), short drama film, directed by Alicia K. Harris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coarse Structure
In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined. The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties. Properly, a coarse structure is not the large-scale analog o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Diophantine Graph
An Erdős–Diophantine graph is an object in the mathematical subject of Diophantine equations consisting of a set of integer points at integer distances in the plane that cannot be extended by any additional points. Equivalently, in geometric graph theory, it can be described as a complete graph with vertices located on the integer square grid \mathbb^2 such that all mutual distances between the vertices are integers, while all other grid points have a non-integer distance to at least one vertex. Erdős–Diophantine graphs are named after Paul Erdős and Diophantus of Alexandria. They form a subset of the set of Diophantine figures, which are defined as complete graphs in the Diophantine plane for which the length of all edges are integers (unit distance graph In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the resu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of ''n'' sets, also known as an ''n''-fold Cartesian product, which can be represented by an ''n''-dimensional array, where each element is an ''n''-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Examples A deck of cards ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Four theorems in Diophantine geometry which are of fundamental importance include: * Mordell–Weil Theorem * Roth's Theorem * Siegel's Theorem * Faltings's Theorem Background Serge Lang published a book ''Diophantine Geometry'' in the area in 1962, and by this book he coined the term "Diophantine Geometry". The traditional arrangement of material on Diophantine equations was by degree and number of variables, as in Mordell's ''Diophantine Equations'' (1969). Mordell's book starts with a remark on homogeneous equations ''f'' = 0 over the rational field, attributed to C. F. Gauss, that non-zero solutions in integers (even primitive lattice points) exist if non-zero rational solutions do, and notes a caveat of L. E. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Group
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron. The group of orientation-preserving symmetries is ''S''4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube. Details Chiral and full (or achiral) octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the crystallographic point groups of the cubic crystal system. As the hyperoctahedral group of dimension 3 the full octahedral group is the wreath product S_2 \wr S_3 \simeq S_2^3 \rtimes S_3,and a natural way to identify its elements is as pairs (m, n) with m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3- zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares. Orthogonal projections The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Spherical tiling The cube can also be represented as a spheric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. Usually, we take n \ge 3 here. The associated rotations and reflections make up the dihedral group \mathrm_n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and n/2 axes of symmetry connecting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilatera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
