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For any natural number , an -sphere of radius is defined as the set of points in (+) -dimensional Euclidean space that are at distance from some fixed point , where may be any positive real number and where may be any point in (+) -dimensional space.
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]
In one dimension it is packing line segments into a linear universe. [10] In dimensions higher than three, the densest lattice packings of hyperspheres are known up to 8 dimensions. [11] Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular.
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 ...
The given convex body can be approximated by a sequence of nested bodies, eventually reaching one of known volume (a hypersphere), with this approach used to estimate the factor by which the volume changes at each step of this sequence. Multiplying these factors gives the approximate volume of the original body.
The octonionic structure does give S 7 one important property: parallelizability. It turns out that the only spheres that are parallelizable are S 1, S 3, and S 7. By using a matrix representation of the quaternions, H, one obtains a matrix representation of S 3. One convenient choice is given by the Pauli matrices:
hypersphere volume and surface area graphs: Image title: Graphs of volumes and surface areas of n-spheres of radius 1 by CMG Lee. The apparent intersection is an artifact of the differing scales. In the SVG file, hover over a point to see its decimal value. Width: 100%: Height: 100%
The ratio of the volume of a sphere to the volume of its circumscribed cylinder is 2:3, as was determined by Archimedes. The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder.