Hypersphère

In mathematics, the hypersphère is a generalization of the Sphère to an Euclidean space of size higher than 3. It constitutes one of the simplest examples of variety and the sphere of dimension N , or N - sphere , is more precisely a Hypersurface Euclidean space \ mathbb R^ {n+1} , noted in general \ mathbb S^n.

Definition

That is to say E Euclidean space of dimension N +1. One calls hypersphère of center has and of ray reality R>0 the whole of the points M for which the distance AM is worth R .

Even if it means to carry out a Translation, which does not change anything with the geometrical properties, it is possible to be brought back to a hypersphère centered in the origin, whose equation is then written in an orthonormal frame of reference

R^2= \ sum_ {i=1} ^ {n+1} x_i^2. \,

For example

  • for the case N =0, the hypersphère consists of two points of respective X-coordinates R and - R .
  • for the case N =1, the hypersphère is a Cercle
  • for the case N =2, the hypersphère is a Sphère in the usual sense

Properties

Volume

The volume of the space delimited by a hypersphère of dimension n-1 and ray R , which is a swell Euclidean of dimension N , is worth:

V_n= {\ pi^ {n/2} R^n \ over \ Gamma (n/2+1)}\ \,

where \ Gamma indicates the Fonction gamma. In particular, for even N , \ Gamma (n/2+1) = (n/2)! .

The following table gives the values of the volume of the first 8 hypersphères of ray 1:

The volume of such a hypersphère is maximum for N =5. For N >5, the volume of the hypersphères is decreasing when N increases. In particular, the Limite of volume ad infinitum is null:

\ lim_ {N \ to \ infty} V_n = 0.

The Hypercube circumscribes with the hypersphère unit has edges length 2 and one volume 2 N ; the relationship between volumes of the hypersphère and the hypercube registered voter is decreasing according to N .

Surface

The surface of surface delimited by a hypersphère of dimension N and ray R can be given by taking of account the Dérivée from its volume compared to the ray:

S_n= \ frac {dV_n} {Dr.} = \ frac {N V_n} {R} = {2 \ pi^ {n/2} R^ {n-1} \ over \ Gamma (n/2)}.

\ Gamma indicates here also the Fonction gamma.

The following table gives the values of the surface of the first 8 hypersphères of ray 1 (the surface of the hypersphère of dimension 1 is however omitted):

The surface of such a hypersphère is maximum for N =7. For N >7, the surface of the hypersphère is decreasing when N increases. In particular, the Limite of the surface ad infinitum is null:

\ lim_ {N \ to \ infty} S_n = 0.

See too

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