Eight-dimensional space

In

Euclidean metric

.

More generally the term may refer to an eight-dimensional vector space over any

8-sphere

, or a variety of other geometric constructions.

Geometry

8-polytope

A

Coxeter-Dynkin diagram. The 8-demicube is a unique polytope from the D₈ family, and 4₂₁, 2₄₁, and 1₄₂

polytopes from the E₈ family.

Regular and uniform polytopes in eight dimensions
(Displayed as orthogonal projections in each
Coxeter plane
of symmetry)
A₈	B₈				D₈
8-simplex {3,3,3,3,3,3,3}	8-cube {4,3,3,3,3,3,3}		8-orthoplex {3,3,3,3,3,3,4}		8-demicube h{4,3,3,3,3,3,3}
E₈
4₂₁ {3,3,3,3,3^2,1}		2₄₁ {3,3,3^4,1}		1₄₂ {3,3^4,2}

7-sphere

The 7-sphere or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point, e.g. the origin. It has symbol $S 7$ , with formal definition for the 7-sphere with radius r of

S^{7}=\left\{x\in \mathbb {R} ^{8}:\|x\|=r\right\}.

The volume of the space bounded by this 7-sphere is

V_{8}\,={\frac {\pi ^{4}}{24}}\,R^{8}

which is 4.05871 × r⁸, or 0.01585 of the 8-cube that contains the 7-sphere.

Kissing number problem

The

kissing number problem has been solved in eight dimensions, thanks to the existence of the 4₂₁ polytope and its associated lattice. The kissing number in eight dimensions is 240

.

Octonions

The octonions are a

normed division algebra

over the real numbers, the largest such algebra. Mathematically they can be specified by 8-tuplets of real numbers, so form an 8-dimensional vector space over the reals, with addition of vectors being the addition in the algebra. A normed algebra is one with a product that satisfies

\|xy\|\leq \|x\|\|y\|

for all x and y in the algebra. A normed division algebra additionally must be finite-dimensional, and have the property that every non-zero vector has a unique multiplicative inverse. Hurwitz's theorem prohibits such a structure from existing in dimensions other than 1, 2, 4, or 8.

Biquaternions

The complexified quaternions $\mathbb {C} \otimes \mathbb {H}$ , or "

isomorphic) to the Clifford algebra

C\ell _{2}(\mathbb {C} )

and the

Pauli algebra. It has also been proposed as a practical or pedagogical tool for doing calculations in special relativity, and in that context goes by the name Algebra of physical space (not to be confused with the Spacetime algebra

, which is 16-dimensional.)

References

H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
ISBN 978-0-471-01003-6 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Table of the Highest Kissing Numbers Presently Known maintained by Gabriele Nebe and Neil Sloane (lower bounds)
ISBN 1-56881-134-9. (Review
).

ISBN 978-1-4020-1338-6
(Second printing)

v t e Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes and shapes	Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category