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5cell In geometry , the 5CELL is a fourdimensional object bounded by 5 tetrahedral cells . It is also known as a C5, PENTACHORON, PENTATOPE, PENTAHEDROID, or TETRAHEDRAL PYRAMID . It is a 4SIMPLEX , the simplest possible convex regular 4polytope (fourdimensional analogue of a Platonic solid Platonic solid ), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base. The REGULAR 5CELL is bounded by regular tetrahedra , and is one of the six regular convex 4polytopes , represented by Schläfli symbol {3,3,3} [...More...]  "5cell" on: Wikipedia Yahoo 

Pentatope Number A PENTATOPE NUMBER is a number in the fifth cell of any row of Pascal\'s triangle starting with the 5term row 1 4 6 4 1 either from left to right or from right to left. The first few numbers of this kind are : 1 , 5 , 15 , 35 , 70 , 126 , 210, 330, 495 , 715, 1001 , 1365 (sequence A000332 in the OEIS ) A pentatope with side length 5 contains 70 3spheres. Each layer represents one of the first five tetrahedral numbers . For example the bottom (green) layer has 35 spheres in total. Pentatope numbers belong in the class of figurate numbers , which can be represented as regular, discrete geometric patterns. The formula for the nth pentatopic number is: ( n + 3 4 ) = n ( n + 1 ) ( n + 2 ) ( n + 3 ) 24 = n 4 4 ! . {displaystyle {n+3 choose 4}={frac {n(n+1)(n+2)(n+3)}{24}}={n^{overline {4}} over 4!}.} Two of every three pentatope numbers are also pentagonal numbers [...More...]  "Pentatope Number" 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 . In higher dimension, a dihedral angle represents the angle between two hyperplanes . CONTENTS * 1 Definitions * 2 Dihedral angles in stereochemistry * 3 Dihedral angles of proteins * 3.1 Converting from dihedral angles to Cartesian coordinates in chains * 4 Calculation of a dihedral angle * 5 Dihedral angles in polyhedra * 6 See also * 7 References * 8 External links DEFINITIONSA dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection [...More...]  "Dihedral Angle" on: Wikipedia Yahoo 

Golden Ratio In mathematics , two quantities are in the GOLDEN RATIO if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, a + b a = a b = def , {displaystyle {frac {a+b}{a}}={frac {a}{b}} {stackrel {text{def}}{=}} varphi ,} where the Greek letter phi ( {displaystyle varphi } or {displaystyle phi } ) represents the golden ratio. Its value is: = 1 + 5 2 = 1.6180339887 . {displaystyle varphi ={frac {1+{sqrt {5}}}{2}}=1.6180339887ldots .} The golden ratio is also called the GOLDEN MEAN or GOLDEN SECTION (Latin: sectio aurea). Other names include EXTREME AND MEAN RATIO, MEDIAL SECTION, DIVINE PROPORTION, DIVINE SECTION (Latin: sectio divina), GOLDEN PROPORTION, GOLDEN CUT, and GOLDEN NUMBER [...More...]  "Golden Ratio" on: Wikipedia Yahoo 

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 . Geometry Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid Euclid , whose treatment, Euclid\'s Elements , set a standard for many centuries to follow. Geometry Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC [...More...]  "Geometry" 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. 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 [...More...]  "Net (polyhedron)" on: Wikipedia Yahoo 

Isotoxal Figure In geometry , a polytope (for example, a polygon or a polyhedron ), or a tiling , is ISOTOXAL or EDGETRANSITIVE if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged. The term isotoxal is derived from the Greek τοξον meaning arc. CONTENTS * 1 Isotoxal polygons * 2 Isotoxal polyhedra and tilings * 3 See also * 4 References ISOTOXAL POLYGONSAn isotoxal polygon is an equilateral polygon , but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons . In general, an isotoxal 2ngon will have Dn (*nn) dihedral symmetry . A rhombus is an isotoxal polygon with D2 (*22) symmetry [...More...]  "Isotoxal Figure" on: Wikipedia Yahoo 

Isohedral Figure In geometry , a polytope of dimension 3 (a polyhedron ) or higher is ISOHEDRAL or FACETRANSITIVE when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit . In other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice . Isohedral polyhedra are called isohedra . They can be described by their face configuration . A form that is isohedral and has regular vertices is also edgetransitive (isotoxal) and is said to be a quasiregular dual : some theorists regard these figures as truly quasiregular because they share the same symmetries, but this is not generally accepted. A polyhedron which is isohedral has a dual polyhedron that is vertextransitive (isogonal) [...More...]  "Isohedral Figure" on: Wikipedia Yahoo 

Cartesian Coordinates A CARTESIAN COORDINATE SYSTEM is a coordinate system that specifies each point uniquely in a plane by a pair of numerical COORDINATES, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length . Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin , usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin [...More...]  "Cartesian Coordinates" on: Wikipedia Yahoo 

Hyperplane In geometry a HYPERPLANE is a subspace of one dimension less than its ambient space . If a space is 3dimensional then its hyperplanes are the 2dimensional planes , while if the space is 2dimensional, its hyperplanes are the 1dimensional lines . This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, the objects which are hyperplanes may have different properties. For instance, a hyperplane of an ndimensional affine space is a flat subset with dimension n − 1. By its nature, it separates the space into two half spaces . But a hyperplane of an ndimensional projective space does not have this property [...More...]  "Hyperplane" on: Wikipedia Yahoo 

3sphere In mathematics , a 3SPHERE is a higherdimensional analogue of a sphere . It consists of the set of points equidistant from a fixed central point in 4dimensional Euclidean space Euclidean space . Analogous to how an ordinary sphere (or 2sphere) is a twodimensional surface that forms the boundary of a ball in three dimensions, a 3sphere 3sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. A 3sphere 3sphere is an example of a 3manifold [...More...]  "3sphere" on: Wikipedia Yahoo 

Triangular Dipyramid In geometry , the TRIANGULAR BIPYRAMID (or DIPYRAMID) is a type of hexahedron , being the first in the infinite set of facetransitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, it can be constructed by joining two tetrahedra along one face. Although all its faces are congruent and the solid is facetransitive , it is not a Platonic solid Platonic solid because some vertices adjoin three faces and others adjoin four. The bipyramid whose six faces are all equilateral triangles is one of the Johnson solids , (J12). A Johnson solid Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids , Archimedean solids , prisms or antiprisms ). They were named by Norman Johnson , who first listed these polyhedra in 1966 [...More...]  "Triangular Dipyramid" on: Wikipedia Yahoo 

Stereographic Projection In geometry , the STEREOGRAPHIC PROJECTION is a particular mapping (function ) that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective . It is conformal , meaning that it preserves angles at which curves meet. It is neither isometric nor areapreserving: that is, it preserves neither distances nor the areas of figures. Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis , cartography , geology , and photography . In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a STEREOGRAPHIC NET, shortened to STEREONET, or WULFF NET [...More...]  "Stereographic Projection" on: Wikipedia Yahoo 

Dihedral Symmetry 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 of order n differs in geometry and abstract algebra . In geometry , Dn or Dihn refers to the symmetries of the ngon , a group of order 2n. In abstract algebra , Dn refers to the dihedral group of order n. The geometric convention is used in this article [...More...]  "Dihedral Symmetry" on: Wikipedia Yahoo 

Facet (geometry) 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 (geometry)" on: Wikipedia Yahoo 

Orthographic Projection ORTHOGRAPHIC PROJECTION (sometimes ORTHOGONAL PROJECTION), is a means of representing threedimensional objects in two dimensions . It is a form of parallel projection , in which all the projection lines are orthogonal to the projection plane , resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection , which is a parallel projection in which the projection lines are not orthogonal to the projection plane. The term orthographic is sometimes reserved specifically for depictions of objects where the principal axes or planes of the object are also parallel with the projection plane, but these are better known as multiview projections [...More...]  "Orthographic Projection" on: Wikipedia Yahoo 