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# N-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. [1]

66 relations: Affine sphere, Alexandroff extension, Ball (mathematics), Circle, Circle group, Closed set, Conformal geometry, Conformal map, Connected space, Constant curvature, Dimension, Disk (mathematics), Dover Publications, Euclidean space, Exotic sphere, Gamma function, Gegenbauer polynomials, Hausdorff measure, Hodge dual, Homeomorphism, Homology sphere, Homotopy groups of spheres, Homotopy sphere, Hopf fibration, Hyperbolic group, Hypercube, Integral, Inversive geometry, Jacobian matrix and determinant, Leech lattice, Line segment, Loop (topology), Manifold, Mathematics, Möbius transformation, Meridian (perimetry, visual field), Natural number, Normal distribution, Octonion, Open set, Orthogonal group, Polar coordinate system, Prentice Hall, Quasigroup, Quaternionic projective space, Real number, Real projective line, Riemann sphere, Sign (mathematics), Simply connected space, ... Expand index (16 more) »

## Affine sphere

In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point.

## Alexandroff extension

In mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact.

## Ball (mathematics)

In mathematics, a ball is the space inside a sphere.

## Circle

A circle is a simple shape in Euclidean geometry.

## Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C&times;, the multiplicative group of all nonzero complex numbers.

## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

## Conformal geometry

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

## Conformal map

In mathematics, a conformal map is a function that preserves angles locally.

## Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

## Constant curvature

In mathematics, constant curvature is a concept from differential geometry.

## Dimension

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.

## Disk (mathematics)

In geometry, a disk (also spelled disc).

## Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Exotic sphere

In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere.

## Gamma function

In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

## Gegenbauer polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(x) are orthogonal polynomials on the interval with respect to the weight function (1 &minus; x2)α–1/2.

## Hausdorff measure

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in to each set in Rn or, more generally, in any metric space.

## Hodge dual

In mathematics, the Hodge star operator or Hodge dual is an important linear map introduced in general by W. V. D. Hodge.

## Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

## Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1.

## Homotopy groups of spheres

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other.

## Homotopy sphere

In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere.

## Hopf fibration

In the mathematical field of topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere.

## Hyperbolic group

In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry.

## Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square (n.

## Integral

The integral is an important concept in mathematics.

## Inversive geometry

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion.

## Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

## Leech lattice

In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space.

## 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 end points.

## Loop (topology)

A loop in mathematics, in a topological space X is a continuous function f from the unit interval I.

## Manifold

In mathematics, a manifold is a topological space that resembles Euclidean space near each point.

## Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

## Möbius transformation

In geometry and complex analysis, a Möbius transformation of the plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

## Meridian (perimetry, visual field)

Meridian (plural: "meridians") is used in perimetry and in specifying visual fields.

## Natural number

In mathematics, the natural numbers (sometimes called the whole numbers): "whole number An integer, though sometimes it is taken to mean only non-negative integers, or just the positive integers." give definitions of "whole number" under several headwords: INTEGER … Syn. whole number.

## Normal distribution

In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution.

## Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H. The octonions are the largest such algebra, with eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

## Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

## Prentice Hall

Prentice Hall is a major educational publisher owned by Pearson PLC.

## Quasigroup

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible.

## Quaternionic projective space

In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted by and is a closed manifold of (real) dimension 4n.

## Real number

In mathematics, a real number is a value that represents a quantity along a continuous line.

## Real projective line

In geometry, a real projective line is an extension of the usual concept of line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity".

## Riemann sphere

In mathematics, the Riemann sphere, named after the 19th century mathematician Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

## Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number to be positive or negative.

## Simply connected space

In topology, a topological space is called simply-connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other such path while preserving the two endpoints in question (see below for an informal discussion).

## Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions).

## Sphere packing

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space.

## Spherical cap

In geometry, a spherical cap or spherical dome is a portion of a sphere cut off by a plane.

## Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

## Spherical harmonics

In mathematics, spherical harmonics are a series of special functions defined on the surface of a sphere used to solve some kinds of differential equations.

## Spherical shell

In geometry, a spherical shell is a generalization of an annulus to three dimensions.

## Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

## Suspension (topology)

In topology, the suspension SX of a topological space X is the quotient space: of the product of X with the unit interval I.

## Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study), is the study of topological spaces.

## Uniform distribution (continuous)

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable.

## Volume element

In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates.

## Volume form

In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing top-dimensionial form (i.e., a differential form of top degree).

## Volume of an n-ball

In geometry, a ball is a region in space consisting of all points within a fixed distance from a fixed point.

## Wendel's theorem

In geometric probability theory, Wendel's theorem, named after James G. Wendel, gives the probability that N points distributed uniformly at random on an ''n''-dimensional hypersphere all lie on the same "half" of the hypersphere.

## 3-sphere

In mathematics, a 3-sphere (also called a glome) is a higher-dimensional analogue of a sphere.

## References

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