Generalization to other dimensions
Main article: n-sphere In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number. It is an n-dimensionalSpheres can be generalized to spaces of any dimension In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two. For any natural number In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in 19th century, when formalization of mathematic started n, an n-sphere, often written as Sn, is the set of points in (n + 1)-dimensional Euclidean space which are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular:
- a 0-sphere is a pair of endpoints of an interval (−r, r) of the real line
- a 1-sphere is a circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius of radius r
- a 2-sphere is an ordinary sphere
- a 3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere is a two dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms is a sphere in 4-dimensional Euclidean space.
Spheres for n > 2 are sometimes called hyperspheres In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number. It is an n-dimensional.
The n-sphere of unit radius centred at the origin is denoted Sn and is often referred to as "the" n-sphere. Note that the ordinary sphere is a 2-sphere, because it is a 2-dimensional surface (which is embedded in 3-dimensional space).
The surface area of the (n − 1)-sphere of radius 1 is
where Γ(z) is Euler's Gamma function In mathematics, the Gamma function is an extension of the factorial function to real and complex numbers. For a complex number z with positive real part the Gamma function is defined by.
Another formula for surface area is
and the volume is the surface area times or
<<Table of Contents A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The | Next>> | Show All>>
