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Icosahedron In geometry, an icosahedron (/ˌaɪkɒsəˈhiːdrən, kə, koʊ/ or /aɪˌkɒsəˈhiːdrən/[1]) is a polyhedron with 20 faces. The name comes from Greek εἴκοσι (eíkosi), meaning 'twenty', and ἕδρα (hédra), meaning 'seat'. The plural can be either "icosahedra" (/drə/) or "icosahedrons". There are many kinds of icosahedra, with some being more symmetrical than others [...More...]  "Icosahedron" on: Wikipedia Yahoo 

Prism In optics, a prism is a transparent optical element with flat, polished surfaces that refract light. At least two of the flat surfaces must have an angle between them. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use "prism" usually refers to this type. Some types of optical prism are not in fact in the shape of geometric prisms. Prisms can be made from any material that is transparent to the wavelengths for which they are designed. Typical materials include glass, plastic, and fluorite. A dispersive prism can be used to break light up into its constituent spectral colors (the colors of the rainbow) [...More...]  "Prism" on: Wikipedia Yahoo 

Right Angle In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees),[1] corresponding to a quarter turn.[2] If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.[3] The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles,[4] making the right angle basic to trigonometry.Contents1 In elementary geometry 2 Symbols 3 Euclid 4 Conversion to other units 5 Rule of 345 6 Thales' theorem 7 See also 8 ReferencesIn elementary geometry[edit] A rectangle is a quadrilateral with four right angles [...More...]  "Right Angle" on: Wikipedia Yahoo 

Vertex (geometry) In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet [...More...]  "Vertex (geometry)" on: Wikipedia Yahoo 

Edge (geometry) In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higherdimensional polytope.[1] In a polygon, an edge is a line segment on the boundary,[2] and is often called a side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet.[3] A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.Contents1 Relation to edges in graphs 2 Number of edges in a polyhedron 3 Incidences with other faces 4 Alternative terminology 5 See also 6 References 7 External linksRelation to edges in graphs[edit] In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment [...More...]  "Edge (geometry)" on: Wikipedia Yahoo 

Face (geometry) In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object;[1] a threedimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higherdimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).[2]Contents1 Polygonal face1.1 Number of polygonal faces of a polyhedron2 kface2.1 Cell or 3face 2.2 Facet or (n1)face 2.3 Ridge or (n2)face 2.4 Peak or (n3)face3 See also 4 References 5 External linksPolygonal face[edit] In elementary geometry, a face is a polygon on the boundary of a polyhedron.[2][3] Other names for a polygonal face include side of a polyhedron, and tile of a Euclidean plane tessellation. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2dimensional features of a 4polytope [...More...]  "Face (geometry)" on: Wikipedia Yahoo 

Coxeter Diagram In geometry, a Coxeter– Dynkin diagram Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge). An unlabeled branch implicitly represents order3. Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6 [...More...]  "Coxeter Diagram" on: Wikipedia Yahoo 

Geometry Geometry Geometry (from the Ancient Greek: γεωμετρία; geo "earth", metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes [...More...]  "Geometry" on: Wikipedia Yahoo 

Convex Set In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.[1] More specifically, in a Euclidean space, a convex region is a region where, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region.[2][3] For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all convex sets containing a given subset A of Euclidean space Euclidean space is called the convex hull of A. It is the smallest convex set containing A. A convex function is a realvalued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set [...More...]  "Convex Set" on: Wikipedia Yahoo 

Net (polyhedron) In geometry a net of a polyhedron is an arrangement of edgejoined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.[1] An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book Unterweysung der Messung mit dem Zyrkel und Rychtscheyd included nets for the Platonic solids and several of the Archimedean solids.[2]Contents1 Existence and uniqueness 2 Shortest path 3 Higherdimensional polytope nets 4 See also 5 References 6 External linksExistence and uniqueness[edit] Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated [...More...]  "Net (polyhedron)" on: Wikipedia Yahoo 

Alternation (geometry) In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.[1] Coxeter Coxeter labels an alternation by a prefixed by an h, standing for hemi or half. Because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all evensided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge. More generally any vertexuniform polyhedron or tiling with a vertex configuration consisting of all evennumbered elements can be alternated. For example, the alternation a vertex figure with 2a.2b.2c is a.3.b.3.c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons [...More...]  "Alternation (geometry)" on: Wikipedia Yahoo 

Truncated Octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces (8 regular hexagonal and 6 square), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces [...More...]  "Truncated Octahedron" on: Wikipedia Yahoo 

Dual Uniform Polyhedron A dual uniform polyhedron is the dual of a uniform polyhedron. Just like a uniform polyhedron is vertextransitive, a dual uniform polyhedron is facetransitive.Uniform set Dual uniform set5 Platonic solids 5 Platonic solids4 Kepler solids 4 Kepler solids13 Archimedean solids 13 Catalan solid∞ Prisms ∞ Bipyramids∞ Antiprisms ∞ Trapezohedra53 nonregular uniform star polyhedra 53 dual uniform star polyhedraExample[edit]List of uniform polyhedraReferences[edit]Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0521098599. Wenninger, Magnus (1983). Dual Models [...More...]  "Dual Uniform Polyhedron" on: Wikipedia Yahoo 

Dihedral Angle A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two halfplanes that have this line as a common edge [...More...]  "Dihedral Angle" on: Wikipedia Yahoo 

Hilbert's Third Problem The third on Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by Gauss,[1] Hilbert conjectured that this is not always possible [...More...]  "Hilbert's Third Problem" on: Wikipedia Yahoo 