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5cell In geometry, the 5cell 5cell is a fourdimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid,[1] or tetrahedral pyramid. It is a 4simplex, the simplest possible convex regular 4polytope (fourdimensional analogue of a Platonic solid), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions [...More...]  "5cell" on: Wikipedia Yahoo 

Pentatope Number A pentatope number is a number in the fifth cell of any row of Pascal's triangle 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 [...More...]  "Pentatope Number" on: Wikipedia Yahoo 

Cartesian Coordinates A Cartesian coordinate system 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 (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0) [...More...]  "Cartesian Coordinates" 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 

Schlegel Diagram In geometry, a Schlegel diagram Schlegel diagram is a projection of a polytope from R d displaystyle R^ d into R d − 1 displaystyle R^ d1 through a point beyond one of its facets or faces. The resulting entity is a polytopal subdivision of the facet in R d − 1 displaystyle R^ d1 that is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimensions 3 and 4, a Schlegel diagram Schlegel diagram is a projection of a polyhedron into a plane figure and a projection of a 4polytope 4polytope to 3space, respectively [...More...]  "Schlegel Diagram" 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 

Configuration (polytope) In geometry, H. S. M. Coxeter H. S. M. Coxeter called a regular polytope a special kind of configuration. Other configurations in geometry are something different. These polytope configurations may more accurately called incidence matrices, where like elements are collected together in rows and columns. Regular polytopes will have one row and column per kface element, while other polytopes will have one row and column for each kface type by their symmetry classes. A polytope with no symmetry will have one row and column for every element, and the matrix will be filled with 0 if the elements are not connected, and 1 if they are connected. Elements of the same k will not be connected and will have a "*" table entry [...More...]  "Configuration (polytope)" on: Wikipedia Yahoo 

Fvector Polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higherdimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex) [...More...]  "Fvector" 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 [...More...]  "Golden Ratio" on: Wikipedia Yahoo 

Hyperplane In geometry a hyperplane is a subspace whose dimension is one less than that of 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 [...More...]  "Hyperplane" 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.[1] 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). The Catalan solids, the bipyramids and the trapezohedra are all isohedral [...More...]  "Isohedral Figure" 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.[1][2] 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.[3] 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 [...More...]  "Facet (geometry)" on: Wikipedia Yahoo 

Orthographic Projection 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,[1] 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,[1] but these are better known as multiview projections [...More...]  "Orthographic Projection" on: Wikipedia Yahoo 

Dihedral Symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon,[1][2] 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, Dn or Dihn refers to the symmetries of the ngon, a group of order 2n [...More...]  "Dihedral Symmetry" 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 [...More...]  "Stereographic Projection" on: Wikipedia Yahoo 

3sphere In mathematics, a 3sphere, or glome[1], is a higherdimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4dimensional 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 [...More...]  "3sphere" on: Wikipedia Yahoo 