Seven-dimensional space

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In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional space. Often such a space are studied as a vector space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean metric, which is defined by the dot product.[disputed ]

More generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions. It may also refer to a seven-dimensional manifold such as a 7-sphere, or a variety of other geometric constructions.

Seven-dimensional spaces have a number of special properties, many of them related to the octonions. An especially distinctive property is that a cross product can be defined only in three or seven dimensions. This is related to Hurwitz's theorem, which prohibits the existence of algebraic structures like the quaternions and octonions in dimensions other than 2, 4, and 8. The first exotic spheres ever discovered were seven-dimensional.

Geometry[edit]

7-polytope[edit]

Main article: Uniform 7-polytope

A polytope in seven dimensions is called a 7-polytope. The most studied are the regular polytopes, of which there are only three in seven dimensions: the 7-simplex, 7-cube, and 7-orthoplex. A wider family are the uniform 7-polytopes, constructed from fundamental symmetry domains of reflection, each domain defined by a Coxeter group. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram. The 7-demicube is a unique polytope from the D7 family, and 321, 231, and 132 polytopes from the E7 family.

Regular and uniform polytopes in seven dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A6 BC7 D7 E7
altN=7-simplex
7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
altN=7-cube
7-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
altN=7-orthoplex
7-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-demicube t0 D7.svg
7-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Up2 3 21 t0 E7.svg
321
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 2 31 t0 E7.svg
231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Up2 1 32 t0 E7.svg
132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

6-sphere[edit]

The 6-sphere or hypersphere in seven-dimensional Euclidean space is the six-dimensional surface equidistant from a point, e.g. the origin. It has symbol S6, with formal definition for the 6-sphere with radius r of

S^6 = \left\{ x \in \mathbb{R}^7 : \|x\| = r\right\}.

The volume of the space bounded by this 6-sphere is

V_7\,=\frac{16 \pi^3}{105}\,r^7

which is 4.72477 × r7, or 0.0369 of the 7-cube that contains the 6-sphere.

Applications[edit]

Cross product[edit]

As mentioned above, a cross product in seven dimensions analogous to the usual three can be defined, and in fact a cross product can only be defined in three and seven dimensions.

Exotic spheres[edit]

Main article: Exotic sphere

In 1956, John Milnor constructed an exotic sphere in 7 dimensions and showed that there are at least 7 differentiable structures on the 7-sphere. In 1963 he showed that the exact number of such structures is 28.

See also[edit]

References[edit]

  • H.S.M. Coxeter: Regular Polytopes. Dover, 1973
  • J.W. Milnor: On manifolds homeomorphic to the 7-sphere. Annals of Mathematics 64, 1956

External links[edit]