mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a sequence of ''n'' real numbers can be understood as a location in ''n''-

dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...

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. 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

such as a

7-sphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call ...

, 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 In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

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

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

s and octonions in dimensions other than 2, 4, and 8. The first

exotic sphere In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of al ...

s 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.

6-sphere

The

6-sphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, ca ...

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

S^6 = \left\.

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

V_7\,=\frac\,r^7

which is 4.72477 × ''r''⁷, 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 In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...

and

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

product of two vectors, is defined in seven dimensions. Along with the more usual

in three dimensions it is the only such product, except for trivial products.

Exotic spheres

In 1956, John Milnor constructed an

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

* {{DEFAULTSORT:Seven-Dimensional Space Dimension Multi-dimensional geometry 7 (number)