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In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they ...
In particular, for any fixed value of R the volume tends to a limiting value of 0 as n goes to infinity. Which value of n maximizes V n (R) depends upon the value of R; for example, the volume V n (1) is increasing for 0 ≤ n ≤ 5, achieves its maximum when n = 5, and is decreasing for n ≥ 5. [2]
A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a ...
A hypersphere in 5-space (also called a 4-sphere due to its surface being 4-dimensional) consists of the set of all points in 5-space at a fixed distance r from a central point P, that is rotationally symmetrical. The hypervolume enclosed by this hypersurface is: =
The 7-sphere or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point, e.g. the origin. It has symbol S 7, with formal definition for the 7-sphere with radius r of = {: ‖ ‖ =}.
The 5-sphere, or hypersphere in six dimensions, is the five-dimensional surface equidistant from a point. It has symbol S 5, and the equation for the 5-sphere, radius r, centre the origin is = {: ‖ ‖ =}. The volume of six-dimensional space bounded by this 5-sphere is
A polytope in seven dimensions is called a 7-polytope. The most studied are the regular polytopes , of which there are only three in seven dimensions : the 7-simplex , 7-cube , and 7-orthoplex . A wider family are the uniform 7-polytopes , constructed from fundamental symmetry domains of reflection, each domain defined by a Coxeter group .
It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. For example, when traveling on a 3-sphere, you can go north and south, east and west, or along a 3rd set of cardinal directions. This means that a 3-sphere is an example of a 3-manifold.