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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 mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space , it is the set of points equidistant from a fixed central point.
The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation:
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 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
These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point P , given ( t , x , y , z ) depends only on the data on the sphere of radius ct that is intersected by the light cone drawn backwards from P .
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 ...
As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume. For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume.