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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 ...
In general, for any natural number n, there are regular n-pointed stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m} = {n/(n − m)}) and m and n are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars).
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 ...
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 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 ...
For N = 6, electrons reside at vertices of a regular octahedron. [5] The configuration may be imagined as four electrons residing at the corners of a square about the equator and the remaining two residing at the poles. For N = 12, electrons reside at the vertices of a regular icosahedron. [6]
A table of virial coefficients for up to eight dimensions can be found on the page Hard sphere: virial coefficients. [ 1 ] Phase diagram of hard sphere system (Solid line - stable branch, dashed line - metastable branch): Pressure P {\displaystyle P} as a function of the volume fraction (or packing fraction) η {\displaystyle \eta }