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Schläfli Symbol In geometry , the SCHLäFLI SYMBOL is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations . The Schläfli symbol Schläfli symbol is named after the 19thcentury Swiss mathematician Ludwig Schläfli , who made important contributions in geometry and other areas [...More...]  "Schläfli Symbol" on: Wikipedia Yahoo 

Octahedral Symmetry A regular octahedron has 24 rotational (or orientationpreserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual of an octahedron. The group of orientationpreserving symmetries is S4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four pairs of opposite sides of the octahedron [...More...]  "Octahedral Symmetry" on: Wikipedia Yahoo 

Facet (mathematics) In geometry , a FACET is a feature of a polyhedron , polytope , or related geometric structure, generally of dimension one less than the structure itself. * In threedimensional geometry a FACET of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face . To FACET a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to stellation and may also be applied to higherdimensional polytopes . * In polyhedral combinatorics and in the general theory of polytopes , a FACET of a polytope of dimension n is a face that has dimension n − 1. Facets may also be called (n − 1)faces. In threedimensional geometry, they are often called "faces" without qualification. * A FACET of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex [...More...]  "Facet (mathematics)" on: Wikipedia Yahoo 

Icosahedral Symmetry A regular icosahedron has 60 rotational (or orientationpreserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is the dual of the icosahedron. The set of orientationpreserving symmetries forms a group referred to as A5 (the alternating group on 5 letters), and the full symmetry group (including reflections) is the product A5 × Z2. The latter group is also known as the Coxeter group H3, and is also represented by Coxeter notation , and Coxeter diagram [...More...]  "Icosahedral Symmetry" on: Wikipedia Yahoo 

4polytope In geometry , a 4POLYTOPE (sometimes also called a POLYCHORON, POLYCELL, or POLYHEDROID) is a fourdimensional polytope . It is a connected and closed figure, composed of lowerdimensional polytopal elements: vertices , edges , faces (polygons ), and cells (polyhedra ). Each face is shared by exactly two cells. The twodimensional analogue of a 4polytope 4polytope is a polygon , and the threedimensional analogue is a polyhedron . Topologically 4polytopes are closely related to the uniform honeycombs , such as the cubic honeycomb , which tessellate 3space; similarly the 3D cube is related to the infinite 2D square tiling . Convex 4polytopes can be cut and unfolded as nets in 3space [...More...]  "4polytope" on: Wikipedia Yahoo 

Angular Defect In geometry , the (ANGULAR) DEFECT (or DEFICIT or DEFICIENCY) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the excess . Classically the defect arises in two ways: * the defect of a vertex of a polyhedron; * the defect of a hyperbolic triangle ;and the excess also arises in two ways: * the excess of a toroidal polyhedron . * the excess of a spherical triangle ;In the plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex on average add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to less than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to more than 360°) [...More...]  "Angular Defect" on: Wikipedia Yahoo 

Polyhedron In geometry , a POLYHEDRON (plural POLYHEDRA or POLYHEDRONS) is a solid in three dimensions with flat polygonal faces , straight edges and sharp corners or vertices . The word polyhedron comes from the Classical Greek πολύεδρον, as poly (stem of πολύς, "many") + hedron (form of ἕδρα, "base" or "seat"). A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3dimensional example of the more general polytope in any number of dimensions [...More...]  "Polyhedron" on: Wikipedia Yahoo 

Tetrahedral Symmetry A regular tetrahedron has 12 rotational (or orientationpreserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientationpreserving symmetries forms a group referred to as the alternating subgroup A4 of S4 [...More...]  "Tetrahedral Symmetry" on: Wikipedia Yahoo 

Coxeter Group In mathematics , a COXETER GROUP, named after H. S. M. Coxeter H. S. M. Coxeter , is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors ). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups ; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934 ) as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 (Coxeter 1935 ) [...More...]  "Coxeter Group" on: Wikipedia Yahoo 

Euclidean Space In geometry , EUCLIDEAN SPACE encompasses the twodimensional Euclidean plane , the threedimensional space of Euclidean geometry , and certain other spaces. It is named after the Ancient Greek mathematician Euclid Euclid of Alexandria . The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry . Euclidean spaces also generalize to higher dimensions . Classical Greek geometry defined the Euclidean plane and Euclidean threedimensional space using certain postulates , while the other properties of these spaces were deduced as theorems . Geometric constructions are also used to define rational numbers . When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean space Euclidean space using Cartesian coordinates Cartesian coordinates and the ideas of analytic geometry [...More...]  "Euclidean Space" on: Wikipedia Yahoo 

Hyperbolic Space In mathematics , HYPERBOLIC SPACE is a homogeneous space that has a constant negative curvature , where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions , and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry Euclidean geometry , and elliptic geometry that have a constant positive curvature. When embedded to a Euclidean space Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point . Another distinctive property is the amount of space covered by the nball in hyperbolic nspace: it increases exponentially with respect to the radius of the ball for large radii, rather than polynomially [...More...]  "Hyperbolic Space" on: Wikipedia Yahoo 

Reflection Symmetry REFLECTION SYMMETRY, LINE SYMMETRY, MIRROR SYMMETRY, MIRRORIMAGE SYMMETRY, is symmetry with respect to reflection . That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric . CONTENTS * 1 Symmetric function * 2 Symmetric geometrical shapes * 3 Mathematical equivalents * 4 Advanced types of reflection symmetry * 5 In nature * 6 In architecture * 7 See also * 8 References * 9 Bibliography * 9.1 General * 9.2 Advanced * 10 External links SYMMETRIC FUNCTION A normal distribution bell curve is an example symmetric function In formal terms, a mathematical object is symmetric with respect to a given operation such as reflection, rotation or translation , if, when applied to the object, this operation preserves some property of the object [...More...]  "Reflection Symmetry" on: Wikipedia Yahoo 

Convex Polyhedron A CONVEX POLYTOPE is a special case of a polytope , having the additional property that it is also a convex set of points in the ndimensional space Rn. Some authors use the terms "convex polytope" and "CONVEX POLYHEDRON" interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope. In addition, some texts require a polytope to be a bounded set , while others (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex npolytope as a surface or (n1)manifold. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming . A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum [...More...]  "Convex Polyhedron" on: Wikipedia Yahoo 

Symmetry Group In group theory , the SYMMETRY GROUP of an object (image , signal , etc.) is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric , it is a subgroup of the isometry group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry Euclidean geometry , but the concept may also be studied in more general contexts as expanded below. CONTENTS * 1 Introduction * 2 One dimension * 3 Two dimensions * 4 Three dimensions * 5 Symmetry Symmetry groups in general * 6 See also * 7 Further reading * 8 External links INTRODUCTIONThe "objects" may be geometric figures, images, and patterns, such as a wallpaper pattern . The definition can be made more precise by specifying what is meant by image or pattern, e.g., a function of position with values in a set of colors [...More...]  "Symmetry Group" on: Wikipedia Yahoo 

Angle Defect In geometry , the (ANGULAR) DEFECT (or DEFICIT or DEFICIENCY) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the excess . Classically the defect arises in two ways: * the defect of a vertex of a polyhedron; * the defect of a hyperbolic triangle ;and the excess also arises in two ways: * the excess of a toroidal polyhedron . * the excess of a spherical triangle ;In the plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex on average add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to less than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to more than 360°) [...More...]  "Angle Defect" on: Wikipedia Yahoo 

Euclidean Geometry EUCLIDEAN GEOMETRY is a mathematical system attributed to the Alexandrian Greek mathematician Euclid Euclid , which he described in his textbook on geometry : the Elements . Euclid's method consists in assuming a small set of intuitively appealing axioms , and deducing many other propositions (theorems ) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system . The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof . It goes on to the solid geometry of three dimensions . Much of the Elements states results of what are now called algebra and number theory , explained in geometrical language [...More...]  "Euclidean Geometry" on: Wikipedia Yahoo 