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Simplex
In geometry , a SIMPLEX (plural: SIMPLEXES or SIMPLICES) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions . Specifically, a K-SIMPLEX is a k-dimensional polytope which is the convex hull of its k + 1 vertices . More formally, suppose the k + 1 points u 0 , , u k R k {displaystyle u_{0},dots ,u_{k}in mathbb {R} ^{k}} are affinely independent , which means u 1 u 0 , , u k u 0 {displaystyle u_{1}-u_{0},dots ,u_{k}-u_{0}} are linearly independent . Then, the simplex determined by them is the set of points C = { 0 u 0 + + k u k i = 0 k i = 1 and i 0 for all i } {displaystyle C=left{theta _{0}u_{0}+dots +theta _{k}u_{k}~{bigg }~sum _{i=0}^{k}theta _{i}=1{mbox{ and }}theta _{i}geq 0{mbox{ for all }}iright}} . For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a 5-cell . A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices. 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
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Simplex (other)
SIMPLEX may refer to: CONTENTS * 1 Mathematics * 2 Companies and trade names * 3 Technology * 4 Other uses MATHEMATICS * Simplex
Simplex
, a term in geometry meaning an n-dimensional analogue of a triangle * Pascal\'s simplex , a version of Pascal's triangle of more than three dimensions * Simplex
Simplex
algorithm , a popular algorithm for numerical solution of linear programming problems * Simplex
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Tetrahedron
In geometry , a TETRAHEDRON (plural: TETRAHEDRA or TETRAHEDRONS), also known as a TRIANGULAR PYRAMID, is a polyhedron composed of four triangular faces , six straight edges , and four vertex corners . The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3-SIMPLEX. The tetrahedron is one kind of pyramid , which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra , a tetrahedron can be folded from a single sheet of paper. It has two such nets . For any tetrahedron there exists a sphere (called the circumsphere ) on which all four vertices lie, and another sphere (the insphere ) tangent to the tetrahedron's faces
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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 arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . 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 , whose treatment, Euclid\'s _Elements_ , set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists preserved Greek ideas and expanded on them during the Middle Ages . By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat . Since then, and into modern times, geometry has expanded into non- Euclidean geometry and manifolds , describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry
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Triangle
A TRIANGLE is a polygon with three edges and three vertices . It is one of the basic shapes in geometry . A triangle with vertices _A_, _B_, and _C_ is denoted A B C {displaystyle triangle ABC} . In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space ). This article is about triangles in Euclidean geometry except where otherwise noted
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Dimensions
In physics and mathematics , the DIMENSION of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube , a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics , space and time are different categories and refer to absolute space and time . That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism . The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer . Minkowski space first approximates the universe without gravity ; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity
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Polytope
In elementary geometry , a POLYTOPE is a geometric object with "flat" sides. It is a generalisation in any number of dimensions, of the three-dimensional polyhedron . Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or N-POLYTOPE. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k-1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations , decompositions or tilings of curved manifolds including spherical polyhedra , and set-theoretic abstract polytopes . Polytopes in more than three dimensions were first discovered by Ludwig Schläfli
Ludwig Schläfli
. The German term polytop was coined by the mathematician Reinhold Hoppe , and was introduced to English mathematicians as polytope by Alicia Boole Stott
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Convex Hull
In mathematics , the CONVEX HULL or CONVEX ENVELOPE of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X. Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X. With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces ; they may also be generalized further, to oriented matroids . The algorithmic problem of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces is one of the fundamental problems of computational geometry . CONTENTS * 1 Definitions * 2 Convex hull
Convex hull
of a finite point set * 3 Computation of convex hulls * 4 Minkowski addition and convex hulls * 5 Relations to other structures * 6 Applications * 7 See also * 8 Notes * 9 References * 10 External links DEFINITIONSA set of points is defined to be convex if it contains the line segments connecting each pair of its points
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Vertex (geometry)
In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place. OF A POLYTOPEA vertex is a corner point of a polygon , polyhedron , or other higher-dimensional polytope , formed by the intersection of edges , faces or facets of the object. In a polygon, a vertex is called "convex " if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians ( 180°, two right angles ) ; otherwise, it is called "concave" or "reflex"
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Affinely Independent
In mathematics , an AFFINE SPACE is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments . A Euclidean space is an affine space over the reals , equipped with a metric , the Euclidean distance . Therefore, in Euclidean geometry , an AFFINE PROPERTY is a property that may be proved in affine spaces. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors , also called translation vectors or simply translations, between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be considered as an affine space, and this amounts to forgetting the special role played by the zero vector
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Linearly Independent
In the theory of vector spaces , a set of vectors is said to be LINEARLY DEPENDENT if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be LINEARLY INDEPENDENT. These concepts are central to the definition of dimension . A vector space can be of finite-dimension or infinite-dimension depending on the number of linearly independent basis vectors . The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining a basis for a vector space
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5-cell
In geometry , the 5-CELL is a four-dimensional object bounded by 5 tetrahedral cells . It is also known as a C5, PENTACHORON, PENTATOPE, PENTAHEDROID, or TETRAHEDRAL PYRAMID . It is a 4-SIMPLEX , the simplest possible convex regular 4-polytope (four-dimensional 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 5-CELL is bounded by regular tetrahedra , and is one of the six regular convex 4-polytopes , represented by Schläfli symbol {3,3,3}. CONTENTS * 1 Alternative names * 2 Geometry
Geometry
* 2.1 Construction * 2.2 Boerdijk–Coxeter helix * 2.3 Projections * 3 Irregular 5-cell
5-cell
* 4 Compound * 5 Related polytopes and honeycomb * 6 References * 7 External links ALTERNATIVE NAMES * Pentachoron * 4-simplex * Pentatope * Pentahedroid (Henry Parker Manning) * Pen (Jonathan Bowers: for pentachoron) * Hyperpyramid, tetrahedral pyramidGEOMETRYThe 5-cell
5-cell
is self-dual , and its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism . Its dihedral angle is cos−1(1/4), or approximately 75.52°
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Point (geometry)
In modern mathematics , a POINT refers usually to an element of some set called a space . More specifically, in Euclidean geometry , a point is a primitive notion upon which the geometry is built, meaning that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms , that it must satisfy. In particular, the geometric points do not have any length , area , volume or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space . CONTENTS * 1 Points in Euclidean geometry * 2 Dimension of a point * 2.1 Vector space dimension * 2.2 Topological dimension * 2.3 Hausdorff dimension * 3 Geometry without points * 4 Point masses and the Dirac delta function * 5 See also * 6 References * 7 External links POINTS IN EUCLIDEAN GEOMETRY A finite set of points (blue) in two-dimensional Euclidean space . Points, considered within the framework of Euclidean geometry , are one of the most fundamental objects. Euclid
Euclid
originally defined the point as "that which has no part"
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Line Segment
In geometry , a LINE SEGMENT is a part of a line that is bounded by two distinct end points , and contains every point on the line between its endpoints. A CLOSED LINE SEGMENT includes both endpoints, while an OPEN LINE SEGMENT excludes both endpoints; a HALF-OPEN LINE SEGMENT includes exactly one of the endpoints. Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron , the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or otherwise a diagonal . When the end points both lie on a curve such as a circle , a line segment is called a chord (of that curve)
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