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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 ...
The volume of the n-ball () can be computed by integrating the volume element in spherical coordinates. The spherical coordinate system has a radial coordinate r and angular coordinates φ 1 , …, φ n − 1 , where the domain of each φ except φ n − 1 is [0, π ) , and the domain of φ n − 1 is [0, 2 π ) .
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 interior of a 3-sphere is a 4-ball.
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 = {: ‖ ‖ =}.
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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.