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 is 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 – discuss]}

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

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 {3,3,3,3,3,3}	7-cube {4,3,3,3,3,3}	7-orthoplex {3,3,3,3,3,4}	7-demicube = h{4,3,3,3,3,3} = {3,3^4,1}	3₂₁ {3,3,3,3^2,1}	2₃₁ {3,3,3^3,1}	1₃₂ {3,3^3,2}

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

A cross product, that is a vector-valued, bilinear, anticommutative and orthogonal product of two vectors, is defined in seven dimensions. Along with the more usual cross product in three dimensions it is the only such product, except for trivial products.

Exotic spheres

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

"Euclidean geometry", Encyclopedia of Mathematics, EMS Press, 2001 [1994]