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In geometry, a simplex (plural: simplices) or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher (i.e., a set of points such that no m-plane contains more than (m + 1) of them; such points are said to be in general position). Table of Geometry, from the 1728 Cyclopaedia. ...
Convex Hull: Elastic band analogy // Alternative definitions In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. (Note that X may be the union of any set of objects made of points). ...
In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as...
Point can refer to: Look up Point in Wiktionary, the free dictionary // Mathematics In mathematics: Point (geometry), an entity that has a location in space but no extent Fixed point (mathematics), a point that is mapped to itself by a mathematical function Point at infinity Point group Point charge, an...
In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
Two intersecting planes in R3 In mathematics, a plane is a fundamental two-dimensional object. ...
In geometry, general position for a set of points, or other configuration, means the general case situation, as opposed to some more special or coincidental cases that are possible. ...
For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron (in each case with interior). Point can refer to: Look up Point in Wiktionary, the free dictionary // Mathematics In mathematics: Point (geometry), an entity that has a location in space but no extent Fixed point (mathematics), a point that is mapped to itself by a mathematical function Point at infinity Point group Point charge, an...
In mathematics, a line segment is a part of a line that is bounded by two end points. ...
A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ...
A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
The pentachoron, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). ...
A regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length. A dodecahedron, one of the five Platonic solids. ...
The convex hull of any m of the n points is also a simplex, called an m-face. The 0-faces are called the vertices, the 1-faces are called the edges, the (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the binomial coefficient C(n + 1, m + 1). Consequently, the number of m-faces of an n-simplex may be found in column (m + 1) of row (n + 1) of Pascal's triangle. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ...
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first six rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ...
The standard simplex
The standard n-simplex is the subset of Rn+1 given by Removing the restriction ti ≥ 0 in the above gives an n-dimensional affine subspace of Rn+1 containing the standard n-simplex. The vertices of the standard n-simplex are the points An affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation. ...
- e0 = (1, 0, 0, …, 0),
- e1 = (0, 1, 0, …, 0),
 - en = (0, 0, 0, …, 1).
There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v0, …, vn) given by  The coefficients ti are called the barycentric coordinates of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving or reversing. In mathematics, barycentric coordinates are coordinates defined by the vertices of a simplex. ...
In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as...
In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...
Geometric properties The oriented volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is Volume, also called capacity, is a quantification of how much space an object occupies. ...
 where each column of the n × n determinant is the difference between two vertices. Any determinant which involves taking the difference between pairs of vertices, where the pairs connect the vertices as a simply connected graph will also give the (same) volume. Without the 1/n! it is the formula for the volume of an n-parallelepiped. One way to understand the 1/n! factor is as follows. If the coordinates of a point in a unit n-box are sorted, together with 0 and 1, and successive differences are taken, then since the results add to one, the result is a point in an n simplex spanned by the origin and the closest n vertices of the box. The taking of differences was an orthogonal (volume-preserving) transformation, but sorting compressed the space by a factor of n!. In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In geometry, a parallelepiped (pronounced ; meaning of parallel planes) or parallelopipedon is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...
The volume under a standard n-simplex (i.e. between the origin and the simplex) is Volume, also called capacity, is a quantification of how much space an object occupies. ...
 The volume of a regular n-simplex with unit side length is Volume, also called capacity, is a quantification of how much space an object occupies. ...
as can be seen by multiplying the previous formula by xn+1, to get the volume under the n-simplex as a function of its vertex distance x from the origin, differentiating w.r.t.  , at  (where the n-simplex side length is 1), and normalizing by the length  of the increment ( dx/(n+1),....dx/(n+1) ) along the normal vector.
Topology Topologically, an n-simplex is equivalent to an n-ball. Every n-simplex is therefore an n-dimensional manifold with boundary. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
This word should not be confused with homomorphism. ...
The solid interior of a sphere or circle; in mathematics, latter terms refer specifically to the (n-1)-dimensional surface of an n-dimensional solid ball. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ...
In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ...
Dividing a circle into areas. ...
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...
A finite set of k-simplexes embedded in an open subset of Rn is called an affine k-chain. The simplexes in a chain need not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ...
In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...
Note that each face of an n-simplex is an affine n-1-simplex, and thus the boundary of an n-simplex is an affine n-1-chain. Thus, if we denote one positively-oriented affine simplex as In topology, the boundary of a subset S of a topological space X is the sets closure minus its interior. ...
- σ = [v0,v1,v2,...vn]
with the vj denoting the vertices, then the boundary  of σ is the chain ![partialsigma = sum_{j=0}^n (-1)^j [v_0,...,v_{j-1},v_{j+1},...,v_n]](http://upload.wikimedia.org/math/8/f/2/8f27d29fd4e69e76238239bed198884f.png) . More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map . In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is, On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ...
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| f( | ∑ | aiσi) = | ∑ | aif(σi) | | i | | i | | where the ai are the integers denoting orientation and multiplicity. For the boundary operator  , one has:  where φ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation (by definition of a map). Partial plot of a function f. ...
A continuous map  to a topological space X is frequently referred to as a singular n-simplex. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ...
See also In mathematics, and computational geometry, the Delaunay triangulation or Delone triangularization for a set P of points in the plane is the triangulation DT(P) of P such that no point in P is inside the circumcircle of any triangle in DT(P). ...
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ...
Vertex figure: tetrahedron In geometry, the tesseract is the 4-dimensional analog of the cube. ...
In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ...
In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...
This page lists the regular polytopes in Euclidean space. ...
In mathematical optimization theory, the simplex algorithm of George Dantzig is the fundamental technique for numerical solution of the linear programming problem. ...
In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ...
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...
In mathematics, a simplicial set is a sequence of sets together with face maps and degeneracy maps for each and every . ...
References - Walter Rudin, Principles of Mathematical Analysis (Third Edition), (1976) McGraw-Hill, New York, ISBN 0-07-054235-X (See chapter 10).
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