Eight-dimensional space
Encyclopedia
In physics
and mathematics
, a sequence of n 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 space
s 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
, has long been used to give substance to the theory of special relativity
.
Eight-dimensional Euclidean space
is generated by considering all real 8-tuple
s 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 rotation
s 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
, while the fifth dimension can be used to model the curvature of spacetime as required in 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".
in eight dimensions is called a 8-polytope. The most studied are the regular polytope
s, 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 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.
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
for all x and y in the algebra; it is only possible to construct such an algebra in one, two, four and eight dimensions.
's superpartner lives.
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
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a sequence of n numbers
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
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 space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
s are also studied, with constant positive and negative curvature.
Whether the real universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
in which we live is somehow eight-dimensional is a topic that is debated and explored in several branches of physics, including astrophysics
Astrophysics
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...
and particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
. For example, the biquaternion algebra, which is based on C4, a four-dimensional space over the field of complex numbers
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
, has long been used to give substance to the theory of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
.
Eight-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
is generated by considering all real 8-tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
s as 8-vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, it has a metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
and a full set of vector operations. The dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
between two 8-vectors is readily defined, and can be used to calculate the metric. 8 × 8 matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
can be used to describe transformations such as rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s which keep the origin fixed.
Three-dimensional space
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
is the space we live in, the fourth dimension can be interpreted as spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
a model for the universe used in special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, while the fifth dimension can be used to model the curvature of spacetime as required in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. The sixth dimension
Sixth dimension
Six-dimensional space is any space that has six dimensions, that is, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of the...
is not commonly used for modelling our universe, but in most string theories
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, six dimensions have curled up below the Planck length, making them unobservable. In M-theory
M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
, an extra dimension is added, making seven extra dimensions. In superspace
Superspace
"Superspace" has had two meanings in physics. The word was first used by John Wheeler to describe the configuration space of general relativity; for example, this usage may be seen in his famous 1973 textbook Gravitation....
, there are eight dimensions, four of ordinary spacetime and the other four of "superspace".
8-polytope
A polytopePolytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
in eight dimensions is called a 8-polytope. The most studied are the regular polytope
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
s, 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
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
. The 8-demicube is a unique polytope from the D8 family, and 421, 241, and 142 polytopes from the E8 family.
| A8 | BC8 | D8 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
8-simplex |
8-cube |
8-orthoplex |
8-demicube |
||||||||
| E8 | |||||||||||
421 |
241 |
142 |
|||||||||
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 , with formal definition for the 7-sphere with radius r ofThe 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.
Octonions
The octonions are a normed division algebraNormed division algebra
In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:\|xy\| = \|x\| \|y\| for all x and y in A....
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
for all x and y in the algebra; it is only possible to construct such an algebra in one, two, four and eight dimensions.
Superspace
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 particleParticle
A particle is, generally, a small localized object to which can be ascribed physical properties. It may also refer to:In chemistry:* Colloidal particle, part of a one-phase system of two or more components where the particles aren't individually visible.In physics:* Subatomic particle, which may be...
's superpartner lives.
Kissing number problem
The kissing number problemKissing number problem
In geometry, a kissing number is defined as the number of non-overlapping unit spheres that touch another given unit sphere. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another...
has been solved in eight dimensions, thanks to the existence of the 421 polytope and its associated lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
. The kissing number in eight dimensions is 240
240 (number)
240 is the natural number following 239 and preceding 241.-In mathematics:With 20 divisors total , more than any previous number, 240 is a highly composite number, and it is also a refactorable number or tau number, since it has 20 divisors and 20 divides 240...
.