**Six-dimensional space** is any space that has six dimensions, 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 environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature.

- Geometry
- 6-polytope
- 5-sphere
- 6-sphere
- Applications
- Transformations in three dimensions
- Rotations in four dimensions
- Electromagnetism
- String theory
- Theoretical background
- Bivectors in four dimensions
- 6-vectors
- Gibbs bivectors
- Footnotes
- References

Formally, six-dimensional Euclidean space, ℝ^{6}, is generated by considering all real 6-tuples as 6-vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined and can be used to calculate the metric. 6 × 6 matrices can be used to describe transformations such as rotations that keep the origin fixed.

More generally, any space that can be described locally with six coordinates, not necessarily Euclidean ones, is six-dimensional. One example is the surface of the 6-sphere, *S*^{6}. This is the set of all points in seven-dimensional space (Euclidean) ℝ^{7} that are a fixed distance from the origin. This constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions. Such non-Euclidean spaces are far more common than Euclidean spaces, and in six dimensions they have far more applications.

A polytope in six dimensions is called a 6-polytope. The most studied are the regular polytopes, of which there are only three in six dimensions: the 6-simplex, 6-cube, and 6-orthoplex. A wider family are the uniform 6-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 6-demicube is a unique polytope from the D_{6} family, and 2_{21} and 1_{22} polytopes from the E_{6} family.

A_{6} | B_{6} | D_{6} | E_{6} | ||
---|---|---|---|---|---|

6-simplex {3,3,3,3,3} | 6-cube {4,3,3,3,3} | 6-orthoplex {3,3,3,3,4} | 6-demicube = {3,3 ^{3,1}} = h{4,3,3,3,3} | 2 _{21} = {3,3,3 ^{2,1}} | 1 _{22} = {3,3 ^{2,2}} |

The 5-sphere, or hypersphere in six dimensions, is the five-dimensional surface equidistant from a point. It has symbol *S*^{5}, and the equation for the 5-sphere, radius *r*, centre the origin is

The volume of six-dimensional space bounded by this 5-sphere is

which is 5.16771 × *r*^{6}, or 0.0807 of the smallest 6-cube that contains the 5-sphere.

The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point. It has symbol *S*^{6}, and the equation for the 6-sphere, radius *r*, centre the origin is

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

which is 4.72477 × *r*^{7}, or 0.0369 of the smallest 7-cube that contains the 6-sphere.

In three dimensional space a rigid transformation has six degrees of freedom, three translations along the three coordinate axes and three from the rotation group SO(3). Often these transformations are handled separately as they have very different geometrical structures, but there are ways of dealing with them that treat them as a single six-dimensional object.

In screw theory angular and linear velocity are combined into one six-dimensional object, called a **twist**. A similar object called a **wrench** combines forces and torques in six dimensions. These can be treated as six-dimensional vectors that transform linearly when changing frame of reference. Translations and rotations cannot be done this way, but are related to a twist by exponentiation.

Phase space is a space made up of the position and momentum of a particle, which can be plotted together in a phase diagram to highlight the relationship between the quantities. A general particle moving in three dimensions has a phase space with six dimensions, too many to plot but they can be analysed mathematically.^{ [1] }

The rotation group in four dimensions, SO(4), has six degrees of freedom. This can be seen by considering the 4 × 4 matrix that represents a rotation: as it is an orthogonal matrix the matrix is determined, up to a change in sign, by e.g. the six elements above the main diagonal. But this group is not linear, and it has a more complex structure than other applications seen so far.

Another way of looking at this group is with quaternion multiplication. Every rotation in four dimensions can be achieved by multiplying by a pair of unit quaternions, one before and one after the vector. These quaternion are unique, up to a change in sign for both of them, and generate all rotations when used this way, so the product of their groups, S^{3} × S^{3}, is a double cover of SO(4), which must have six dimensions.

Although the space we live in is considered three-dimensional, there are practical applications for four-dimensional space. Quaternions, one of the ways to describe rotations in three dimensions, consist of a four-dimensional space. Rotations between quaternions, for interpolation, for example, take place in four dimensions. Spacetime, which has three space dimensions and one time dimension is also four-dimensional, though with a different structure to Euclidean space.

In electromagnetism, the electromagnetic field is generally thought of as being made of two things, the electric field and magnetic field. They are both three-dimensional vector fields, related to each other by Maxwell's equations. A second approach is to combine them in a single object, the six-dimensional electromagnetic tensor, a tensor- or bivector-valued representation of the electromagnetic field. Using this Maxwell's equations can be condensed from four equations into a particularly compact single equation:

where **F** is the bivector form of the electromagnetic tensor, **J** is the four-current and ∂ is a suitable differential operator.^{ [2] }

In physics string theory is an attempt to describe general relativity and quantum mechanics with a single mathematical model. Although it is an attempt to model our 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 to form a six-dimensional space with a particular geometry too small to be observable.

Since 1997 another string theory has come to light that works in six dimensions. Little string theories are non-gravitational string theories in five and six dimensions that arise when considering limits of ten-dimensional string theory.^{ [3] }

A number of the above applications can be related to each other algebraically by considering the real, six-dimensional bivectors in four dimensions. These can be written Λ^{2}ℝ^{4} for the set of bivectors in Euclidean space or Λ^{2}ℝ^{3,1} for the set of bivectors in spacetime. The Plücker coordinates are bivectors in ℝ^{4} while the electromagnetic tensor discussed in the previous section is a bivector in ℝ^{3,1}. Bivectors can be used to generate rotations in either ℝ^{4} or ℝ^{3,1} through the exponential map (e.g. applying the exponential map of all bivectors in Λ^{2}ℝ^{4} generates all rotations in ℝ^{4}). They can also be related to general transformations in three dimensions through homogeneous coordinates, which can be thought of as modified rotations in ℝ^{4}.

The bivectors arise from sums of all possible wedge products between pairs of 4-vectors. They therefore have **C**4

2 = 6 components, and can be written most generally as

They are the first bivectors that cannot all be generated by products of pairs of vectors. Those that can are simple bivectors and the rotations they generate are simple rotations. Other rotations in four dimensions are double and isoclinic rotations and correspond to non-simple bivectors that cannot be generated by single wedge product.^{ [4] }

6-vectors are simply the vectors of six-dimensional Euclidean space. Like other such vectors they are linear, can be added subtracted and scaled like in other dimensions. Rather than use letters of the alphabet, higher dimensions usually use suffixes to designate dimensions, so a general six-dimensional vector can be written **a** = (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}). Written like this the six basis vectors are (1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0) and (0, 0, 0, 0, 0, 1).

Of the vector operators the cross product cannot be used in six dimensions; instead, the wedge product of two 6-vectors results in a bivector with 15 dimensions. The dot product of two vectors is

It can be used to find the angle between two vectors and the norm,

This can be used for example to calculate the diagonal of a 6-cube; with one corner at the origin, edges aligned to the axes and side length 1 the opposite corner could be at (1, 1, 1, 1, 1, 1), the norm of which is

which is the length of the vector and so of the diagonal of the 6-cube.

In 1901 J.W. Gibbs published a work on vectors that included a six-dimensional quantity he called a *bivector*. It consisted of two three-dimensional vectors in a single object, which he used to describe ellipses in three dimensions. It has fallen out of use as other techniques have been developed, and the name bivector is now more closely associated with geometric algebra.^{ [5] }

- ↑ Arthur Besier (1969).
*Perspectives of Modern Physics*. McGraw-Hill. - ↑ Lounesto (2001), pp. 109–110
- ↑ Aharony (2000)
- ↑ Lounesto (2001), pp. 86–89
- ↑ Josiah Willard Gibbs, Edwin Bidwell Wilson (1901).
*Vector analysis: a text-book for the use of students of mathematics and physics*. Yale University Press. p. 481ff.

In mathematics, a **geometric algebra** is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions.

In mathematics, physics, and engineering, a **Euclidean vector** or simply a **vector** is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a *directed line segment*, or graphically as an arrow connecting an *initial point**A* with a *terminal point**B*, and denoted by .

**Vector calculus**, or **vector analysis**, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow.

In mathematics, a **3-sphere** is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere, the boundary of a ball in four dimensions is a 3-sphere. A 3-sphere is an example of a 3-manifold and an *n*-sphere.

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 quaternion as the quotient of two *directed lines* in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.

In mathematics, the **cross product** or **vector product** is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space, and is denoted by the symbol . Given two linearly independent vectors **a** and **b**, the cross product, **a** × **b**, is a vector that is perpendicular to both **a** and **b**, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.

Unit quaternions, known as *versors*, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.

**Rotation** in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have sign : a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (*n* − 1)-dimensional flat of fixed points in a n-dimensional space.

In mathematics, a **bivector** or **2-vector** is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector can be thought of as being of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any number of dimensions, and are a useful tool for classifying such rotations. They are also used in physics, tying together a number of otherwise unrelated quantities.

In abstract algebra, the **biquaternions** are the numbers *w* + *x***i** + *y***j** + *z***k**, where *w*, *x*, *y*, and z are complex numbers, or variants thereof, and the elements of {**1**, **i**, **j**, **k**} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

A **four-dimensional space** (**4D**) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called *dimensions*, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height.

In mathematics, the **seven-dimensional cross product** is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors **a**, **b** in a vector **a** × **b** also in . Like the cross product in three dimensions, the seven-dimensional product is anticommutative and **a** × **b** is orthogonal both to **a** and to **b**. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions.

In mathematics, a **versor** is a quaternion of norm one. The word is derived from Latin *versare* = "to turn" with the suffix *-or* forming a noun from the verb. It was introduced by William Rowan Hamilton in the context of his quaternion theory.

In multilinear algebra, a **multivector**, sometimes called **Clifford number**, is an element of the exterior algebra Λ(*V*) of a vector space V. This algebra is graded, associative and alternating, and consists of linear combinations of **simple***k*-vectors of the form

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

In geometry, various **formalisms** exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In mathematics, a sequence of *n* real numbers can be understood as a location in *n*-dimensional 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.

In geometry, a **plane of rotation** is an abstract object used to describe or visualize rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as two, four or more dimensions.

In geometry, a (globally) **projective polyhedron** is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.

**Conformal geometric algebra** (**CGA**) is the geometric algebra constructed over the resultant space of a map from points in an *n*-dimensional base space **R**^{p,q} to null vectors in **R**^{p+1,q+1}. This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations.

- Lounesto, Pertti (2001).
*Clifford algebras and spinors*. Cambridge: Cambridge University Press. ISBN 978-0-521-00551-7.

- Aharony, Ofer (2000). "A brief review of "little string theories"".
*Classical and Quantum Gravity*.**17**(5). arXiv: hep-th/9911147 . Bibcode:2000CQGra..17..929A. doi:10.1088/0264-9381/17/5/302.

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