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{{Short description|Geometric space with six dimensions}}
{{Redirect|Sixth dimension||The Sixth Dimension (disambiguation){{!}}The Sixth Dimension}}
'''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 {{anchor|6D EUCLIDEAN SPACE}}six-dimensional [[Euclidean space]], in which 6-polytopes and the 5-sphere are constructed. Six-dimensional [[elliptical space]] and [[hyperbolic space]]s are also studied, with constant positive and negative curvature.

Formally, six-dimensional Euclidean space, <math>\mathbb{R}^{6}</math>, is generated by considering all [[Real number|real]] 6-[[tuple]]s as 6-[[Euclidean vector|vector]]s in this space. As such it has the properties of all Euclidean spaces, so it is linear, has a [[Metric (mathematics)|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 [[matrix (mathematics)|matrices]] can be used to describe transformations such as [[rotation]]s that keep the origin fixed.

More generally, any space that can be [[manifold|described locally]] with six [[coordinates]], not necessarily Euclidean ones, is six-dimensional. One example is the surface of the 6-sphere, ''S''<sup>6</sup>. This is the set of all points in [[seven-dimensional space]] (Euclidean) <math>\mathbb{R}^{7}</math> 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 geometry|non-Euclidean]] spaces are far more common than Euclidean spaces, and in six dimensions they have far more applications.

==Geometry==

===6-polytope===
{{Main article|6-polytope}}

A [[polytope]] in six dimensions is called a 6-polytope. The most studied are the [[regular polytope]]s, of which there are only [[List of regular polytopes#Convex 4|three in six dimensions]]: the [[6-simplex]], [[6-cube]], and [[6-orthoplex]]. A wider family are the [[uniform 6-polytope]]s, 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<sub>6</sub> family, and [[2 21 polytope|2<sub>21</sub>]] and [[1 22 polytope|1<sub>22</sub>]] polytopes from the E<sub>6</sub> family.

{| class=wikitable
|+ Uniform polytopes in six dimensions<BR>(Displayed as orthogonal projections in each [[Coxeter plane]] of symmetry)
|-
!A<sub>6</sub>
!colspan=2|B<sub>6</sub>
!D<sub>6</sub>
!colspan=2|E<sub>6</sub>
|- align=center
|[[File:6-simplex t0.svg|altN=6-simplex|120px]]<BR>[[6-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}<BR>{3,3,3,3,3}
|[[File:6-cube t0.svg|altN=6-cube|120px]]<BR>[[6-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}<BR>{4,3,3,3,3}
|[[File:6-cube t5.svg|altN=6-orthoplex|120px]]<BR>[[6-orthoplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}<BR>{3,3,3,3,4}
|[[File:6-demicube t0 D6.svg|120px]]<BR>[[6-demicube]]<BR>{{CDD|nodes_10ru|split2|node|3|node||3|node||3|node}} = {{CDD|node_h|4|node|3|node|3|node|3|node|3|node}}<BR>{3,3<sup>3,1</sup>} = h{4,3,3,3,3}
|[[File:up 2 21 t0 E6.svg|120px]]<BR>[[2 21 polytope|2<sub>21</sub>]]<BR>{{CDD|nodea_1|3a|nodea||3a|branch|3a|nodea||3a|nodea|}} = {{CDD|node|3|node|split1|nodes|3ab|nodes_10l}}<BR>{3,3,3<sup>2,1</sup>}
|[[File:up 1 22 t0 E6.svg|120px]]<BR>[[1 22 polytope|1<sub>22</sub>]]<BR>{{CDD|nodea||3a|nodea||3a|branch_01lr|3a|nodea||3a|nodea|}} = {{CDD|node_1|3|node|split1|nodes|3ab|nodes}}<BR>{3,3<sup>2,2</sup>}
|}

===5-sphere===
The 5-sphere, or hypersphere in six dimensions, is the five-dimensional surface equidistant from a point. It has symbol ''S''<sup>5</sup>, and the equation for the 5-sphere, radius ''r'', centre the origin is

:<math>S^5 = \left\{ x \in \mathbb{R}^6 : \|x\| = r\right\}.</math>

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

: <math> V_6 = \frac{\pi^3 r^6 }{6} </math>

which is 5.16771 × ''r''<sup>6</sup>, or 0.0807 of the smallest [[6-cube]] that contains the 5-sphere.

===6-sphere===
The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point. It has symbol ''S''<sup>6</sup>, and the equation for the 6-sphere, radius ''r'', centre the origin is

:<math>S^6 = \left\{ x \in \mathbb{R}^7 : \|x\| = r\right\}.</math>

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

: <math> V_7 = \frac{16\pi^3 r^7 }{105} </math>

which is 4.72477 × ''r''<sup>7</sup>, or 0.0369 of the smallest [[7-cube]] that contains the 6-sphere.

==Applications==

===Transformations in three dimensions===
In three dimensional space a [[rigid transformation]] has [[six degrees of freedom]], three [[translation (geometry)|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.

====Screw theory====
{{main article|Screw theory}}

In screw theory [[angular velocity|angular]] and [[velocity|linear]] velocity are combined into one six-dimensional object, called a '''twist'''. A similar object called a '''wrench''' combines [[force]]s and [[torque]]s 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====
{{main article|Phase space}}

[[File:Limitcycle.svg|thumb|200px|right|Phase portrait of the [[Van der Pol oscillator]]]]
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.<ref>{{cite book
| title = Perspectives of Modern Physics
| author = Arthur Besier
| year = 1969
| publisher = McGraw-Hill}}</ref>

===Rotations in four dimensions===
{{main article|Rotations in 4-dimensional Euclidean space}}

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 [[SO(4)#Relation to quaternions|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, [[3-sphere|S<sup>3</sup>]] × S<sup>3</sup>, is a [[double cover (topology)|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]].

===Electromagnetism===

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 field]]s, 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:

: <math> \partial \mathbf{F} = \mathbf{J} \,</math>

where {{math|'''F'''}} is the bivector form of the electromagnetic tensor, {{math|'''J'''}} is the [[four-current]] and {{math|∂}} is a suitable [[differential operator]].<ref>Lounesto (2001), pp. 109–110</ref>

===String theory===
[[Image:Calabi-Yau.png|right|thumb|200px|[[Calabi–Yau manifold]] ([[3D projection]])]]

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 [[Compactification (physics)|compactified]] to form a six-dimensional space with a [[Calabi–Yau manifold|particular geometry]] too small to be observable.

Since 1997 another string theory has come to light that works in six dimensions. [[Little string theory|Little string theories]] are non-gravitational string theories in five and six dimensions that arise when considering limits of ten-dimensional string theory.<ref>Aharony (2000)</ref>

==Theoretical background==

===Bivectors in four dimensions===

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

The bivectors arise from sums of all possible [[wedge product]]s between pairs of 4-vectors. They therefore have [[combination|'''C'''{{SubSup||2|4}}]] = 6 components, and can be written most generally as

: <math>\mathbf{B} = B_{12}\mathbf{e}_{12} + B_{13}\mathbf{e}_{13} + B_{14}\mathbf{e}_{14} + B_{23}\mathbf{e}_{23} + B_{24}\mathbf{e}_{24} + B_{34}\mathbf{e}_{34}</math>

They are the first bivectors that cannot all be generated by products of pairs of vectors. Those that can are [[Bivector#Simple bivectors|simple bivectors]] and the rotations they generate are [[SO(4)#Simple rotations|simple rotations]]. Other rotations in four dimensions are [[SO(4)#Double rotations|double]] and [[SO(4)#Isoclinic rotations|isoclinic]] rotations and correspond to non-simple bivectors that cannot be generated by single wedge product.<ref>Lounesto (2001), pp. 86–89</ref>

==6-vectors==

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 {{nowrap |'''a''' {{=}} (a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>, a<sub>5</sub>, a<sub>6</sub>)}}. Written like this the six [[basis vector]]s are {{nowrap |(1, 0, 0, 0, 0, 0)}}, {{nowrap |(0, 1, 0, 0, 0, 0)}}, {{nowrap |(0, 0, 1, 0, 0, 0)}}, {{nowrap |(0, 0, 0, 1, 0, 0)}}, {{nowrap |(0, 0, 0, 0, 1, 0)}} and {{nowrap |(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

:<math>\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 + a_4b_4 + a_5b_5 + a_6b_6.</math>

It can be used to find the angle between two vectors and the [[Norm (mathematics)|norm]],

:<math>\left | \mathbf{a} \right \vert = \sqrt{\mathbf{a} \cdot \mathbf{a}} = \sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 + {a_5}^2 + {a_6}^2}.</math>

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 {{nowrap |(1, 1, 1, 1, 1, 1)}}, the norm of which is

:<math>\sqrt{1 + 1 + 1 + 1 + 1 +1} = \sqrt{6} = 2.4495,</math>

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

===Gibbs bivectors===

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.<ref>{{cite book
|title=Vector analysis: a text-book for the use of students of mathematics and physics
|author=Josiah Willard Gibbs, Edwin Bidwell Wilson
|publisher=Yale University Press |year= 1901 |page=481ff}}</ref>

==Footnotes==
{{reflist}}

==References==
* {{Cite book
| last=Lounesto
| first=Pertti
| title=Clifford algebras and spinors
| publisher=[[Cambridge University Press]]
| location=Cambridge
| isbn=978-0-521-00551-7
| year=2001}}

* {{cite journal
| last = Aharony
| first  = Ofer
| title = A brief review of "little string theories"
| year = 2000
| journal= Classical and Quantum Gravity
| volume = 17
| issue = 5
| doi = 10.1088/0264-9381/17/5/302
| bibcode=2000CQGra..17..929A
| arxiv= hep-th/9911147
}}

{{Dimension topics}}

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[[Category:Dimension]]
[[Category:6 (number)|dimensional space]]