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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 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 ...
For a general discussion of the number of linear independent vector fields on a n-sphere, see the article vector fields on spheres. There is an interesting action of the circle group T on S 3 giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of S 3 as a subset of C 2, the action is given by
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 number of the above applications can be related to each other algebraically by considering the real, six-dimensional bivectors in four dimensions. These can be written Λ 2 R 4 {\displaystyle \Lambda ^{2}\mathbb {R} ^{4}} for the set of bivectors in Euclidean space or Λ 2 R 3 , 1 {\displaystyle \Lambda ^{2}\mathbb {R} ^{3,1}} for the set of ...
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 volume of the unit ball in Euclidean -space, and the surface area of the unit sphere, appear in many important formulas of analysis. The volume of the unit n {\displaystyle n} -ball, which we denote V n , {\displaystyle V_{n},} can be expressed by making use of the gamma function .
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.