Information AboutHypersphere |
| CATEGORIES ABOUT HYPERSPHERE | |
| 4-dimensional geometry | |
| multi-dimensional geometry | |
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: The above hypersphere in -dimensional Euclidean space is an example of an - Manifold . For example, an ordinary sphere in three dimensions is a 2-sphere, denoted by ; the 1-sphere being a Circle , and the 0-sphere is the end points of an Interval . Of course, translating the coordinates (i.e. moving the center around) doesn't change the analytic or geometric properties of the sphere. HYPERSPHERICAL VOLUME The hyperdimensional volume of the space which a (n-1)-sphere encloses (the n-ball) is: : where Γ is the Gamma Function . (For even , .) The "surface area" of this sphere is : The interior of a hypersphere, that is the set of all points whose distance from the centre is less than , is called a hyperball, or if the hypersphere itself is included, a closed hyperball. HYPERSPHERICAL VOLUME - SOME EXAMPLES For small values of , the volumes, , of the unit n-ball () are: If the dimension, , is not limited to integral values, the hypersphere volume is a Continuous Function of with a Global Maximum for the unit sphere in "dimension" ''n'' = 5.2569464... where the "volume" is 5.277768... Note that the hypercube circumscribed around the unit n-sphere has an edge length of 2 and hence a volume of 2''n''; the ratio of the volume of the hypersphere to its circumscribed hypercube decreases monotonically as the dimension increases. HYPERSPHERICAL COORDINATES We may define a coordinate system in an n-dimensional Euclidean space which is analogous to the Spherical Coordinate System defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate , and angular coordinates . If are the Cartesian coordinates, then we may define : : : : : : The hyperspherical volume element will be found from the Jacobian of the transformation: : |
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