Seven-dimensional space

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.

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

7-polytope

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	B₇		D₇	E₇
7-simplex	7-cube	7-orthoplex	7-demicube	3₂₁	2₃₁	1₃₂

6-sphere

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 $S 6$ , 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

Cross product

Main article: Seven-dimensional cross product

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

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.

References

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

Hazewinkel, Michiel, ed. (2001), "Euclidean geometry", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Dimension

Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime

Other dimensions	Krull Homological Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom

Polytopes and shapes	Hyperplane Hypersurface Hypercube Hypersphere Hyperrectangle Demihypercube Cross-polytope Simplex

Dimensions by number	Zero One Two Three Four Five Six (degrees of freedom) Seven Eight n-dimensions

Category