Seven-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 = 7, the set of all such locations is called 7-dimensional Euclidean space. Seven-dimensional elliptical and hyperbolic space
s are also studied, with constant positive and negative curvature.
Abstract seven-dimensional space occurs frequently in mathematics, and is a perfectly legitimate construct. Whether or not the real universe
in which we live is somehow seven-dimensional (or indeed higher) is a topic that is debated and explored in several branches of physics, including astrophysics
and particle physics
.
Formally, seven-dimensional Euclidean space
is generated by considering all real 7-tuple
s as 7-vectors in this space. As such it has the properties of all Euclidian spaces, so it is linear, has a metric
and a full set of vector operations. In particular the dot product
between two 7-vectors is readily defined, and can be used to calculate the metric. 7 × 7 matrices
can be used to describe transformations such as rotation
s which keep the origin fixed.
A distinctive property is that a cross product
can be defined only in three or seven dimensions (see seven-dimensional cross product). This is due to the existence of quaternion
s and octonion
s.
in seven dimensions is called a 7-polytope. The most studied are the regular polytope
s, 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.
The volume of the space bounded by this 6-sphere is
which is 4.72477 × r7, or 0.0369 of the 7-cube that contains the 6-sphere.
A key feature of string theory 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 a ten dimensional space, adding an extra six dimensions. These extra dimensions are required by the theory, but as they cannot be observed are thought to be quite different, perhaps compactified
so they form a six dimensional space with a particular geometry too small to be observable. In M-theory
, which unifies the five types of string theory, there is a seventh dimension involved.
constructed an exotic sphere
in 7 dimensions and showed that there are at least 7 differentiable structures on the 7-sphere. (The number is now known to be 28.)
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
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional Euclidean space. Seven-dimensional elliptical 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.
Abstract seven-dimensional space occurs frequently in mathematics, and is a perfectly legitimate construct. Whether or not 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 seven-dimensional (or indeed higher) 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...
.
Formally, seven-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 7-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 7-vectors in this space. As such it has the properties of all Euclidian spaces, so it is linear, 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. In particular 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 7-vectors is readily defined, and can be used to calculate the metric. 7 × 7 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.
A distinctive property is that a cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
can be defined only in three or seven dimensions (see seven-dimensional cross product). This is due to the existence of quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s and octonion
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
s.
7-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 seven dimensions is called a 7-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 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
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 7-demicube is a unique polytope from the D7 family, and 321, 231, and 132 polytopes from the E7 family.
| A6 | BC7 | D7 | E7 | |||
|---|---|---|---|---|---|---|
7-simplex |
7-cube |
7-orthoplex |
7-demicube |
321 |
231 |
132 |
6-sphere
The 6-sphere or hypersphere in seven dimensions 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 ofThe volume of the space bounded by this 6-sphere is
which is 4.72477 × r7, or 0.0369 of the 7-cube that contains the 6-sphere.
Non-associative Ashtekar gravity
There is a paper on Non-associative Ashtekar gravity in seven dimensions.String theory and M-theory
Compactification (physics)
In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic....
so they form a six dimensional space with a particular geometry too small to be observable. 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...
, which unifies the five types of string theory, there is a seventh dimension involved.
Cross product
As mentioned above, a cross product in seven dimensions analogous to the usual three can be defined, and in fact a cross product can only be defined in seven and three dimensions.Exotic sphere
In 1956, John MilnorJohn Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
constructed an exotic sphere
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
in 7 dimensions and showed that there are at least 7 differentiable structures on the 7-sphere. (The number is now known to be 28.)
See also
- Euclidean spaceEuclidean spaceIn 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...
- Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
- 7-polytope7-polytopeIn seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets....
- PolytopePolytopeIn 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...
- M-theoryM-theoryIn 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...
- List of geometry topics
- List of regular polytopes
- Exotic sphereExotic sphereIn differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...