In physics and mathematics, a sequence of n real numbers can also be understood as a location in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional Euclidean space. Eight-dimensional elliptical spaces and hyperbolic spaces are also studied, with constant positive and negative curvature.
Whether the real universe in which we live is somehow eight-dimensional is a topic that is debated and explored in several branches of physics, including astrophysics and particle physics. For example, the biquaternion algebra, which is based on C4, a four-dimensional space over the field of complex numbers, can be used to represent to the theory of special relativity.
Eight-dimensional Euclidean space is generated by considering all real 8-tuples as 8-vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, it has a metric and a full set of vector operations. The dot product between two 8-vectors is readily defined, and can be used to calculate the metric. 8 × 8 matrices can be used to describe transformations such as rotations which keep the origin fixed.
Three-dimensional space is the space we live in, the fourth dimension can be interpreted as spacetime a model for the universe used in special relativity and general relativity. The sixth dimension is not commonly used for modelling our universe, but in most string theories, six dimensions have curled up below the Planck length, making them unobservable. In M-theory, an extra dimension is added, making seven extra dimensions. In superspace, there are eight dimensions, four of ordinary spacetime and the other four of "superspace".
A polytope in eight dimensions is called a 8-polytope. The most studied are the regular polytopes, of which there are only three in eight dimensions: the 8-simplex, 8-cube, and 8-orthoplex. A broader family are the uniform 8-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 8-demicube is a unique polytope from the D8 family, and 421, 241, and 142 polytopes from the E8 family.
The 7-sphere or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point, e.g. the origin. It has symbol S7, with formal definition for the 7-sphere with radius r of
The volume of the space bounded by this 7-sphere is
which is 4.05871 × r8, or 0.01585 of the 8-cube that contains the 7-sphere.
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 division algebra is one with a product that satisfies
A key feature of superspace is that, though it is an attempt to model our physical universe, it takes place in a space with more dimensions than the four of spacetime that we are familiar with. In particular a number of string theories take place in an eight-dimensional space, adding an extra four dimensions. These extra dimensions are required by the theory as a form of "superspace" where every particle's superpartner lives.
Kissing number problem
Eighth dimension discussions are less common than fifth dimension and eleventh dimension theories popularized in science at the moment by mathematicians in quantum physics under quantum mechanics, string theory, and quantum gravity. This is possibly due in part to the stigma of time travel becoming a popular topic in science fiction, or simply that research has not needed to move in that direction lately.
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied 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)
- Conway, John Horton; Smith, Derek A. (2003), On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A. K. Peters, Ltd., ISBN 1-56881-134-9. (Review).
- Duplij, Steven; Siegel, Warren; Bagger, Jonathan, eds. (2005), Concise Encyclopedia of Supersymmetry And Noncommutative Structures in Mathematics and Physics, Berlin, New York: Springer, ISBN 978-1-4020-1338-6 (Second printing)