6-Dimensional+space

= Six-dimensional space =

From Wikipedia, the free encyclopedia
Revision as of 00:18, 28 January 2010 by [|JohnBlackburne] ([|talk] | [|contribs]) ( [|diff] ) [|← Previous revision] | Current revision (diff) | Newer revision → (diff) Jump to: [|navigation], [|search] **Six-dimensional space** is a term used to describe any space that has six dimensions, i.e. six degrees of freedom, and which needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but the ones that are of most interest are simpler ones which are models of some aspect of our environment. Of particular interest is six dimensional [|Euclidean space], in which 6-polytopes and the 5-sphere are constructed. 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 Euclidian spaces, so it it 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] which keep the origin fixed. More generally any space which can be described with six [|coordinates], not necessarily Euclidian ones, is described as six dimensional. One example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional Euclidian space ℝ 7 that is equidistant 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-Euclidian] spaces are far more common than Euclidian spaces, and in six dimensions they have far more applications. [ hide ]
 * This is the [|current revision] of this page, as edited by [|JohnBlackburne] ([|talk] | [|contribs]) at 00:18, 28 January 2010. The present address (URL) is a permanent link to this version. ** ||
 * This is the [|current revision] of this page, as edited by [|JohnBlackburne] ([|talk] | [|contribs]) at 00:18, 28 January 2010. The present address (URL) is a permanent link to this version. ** ||
 * == Contents ==
 * [|1 Geometry]
 * [|1.1 6-polytope]
 * [|1.2 5-sphere]
 * [|1.3 6-sphere]
 * [|2 Applications]
 * [|2.1 Transformations in three dimensions]
 * [|2.1.1 Homogeneous coordinates]
 * [|2.1.2 Screw theory]
 * [|2.1.3 Phase space]
 * [|2.2 Rotations in four dimensions]
 * [|2.3 Plücker coordinates]
 * [|2.4 Electromagnetism]
 * [|2.5 String theory]
 * [|3 Theoretical background]
 * [|3.1 Bivectors in four dimensions]
 * [|3.2 6-vectors]
 * [|3.3 Gibbs bivectors]
 * [|4 Footnotes]
 * [|5 References] ||

6-polytope
Main article: [|6-polytope] 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. ** Regular polytopes in six dimensions ** || [|6-simplex] || [|6-cube] || [|6-orthoplex] || ||   ||    ||    ||    ||    ||    ||    ||    ||

5-sphere
The 5-sphere, or hypersphere in six dimensions, is the five dimensional surface equidistant from a point, e.g. the origin. It has symbol S5, with formal definition for the 5-sphere with radius //r// of The volume of the 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.

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 S6, with formal definition for the 6-sphere with radius //r// of 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.

Transformations in three dimensions
In three dimensional space a generalised transformation has [|six degrees of freedom], three [|translations] along the three coordinate axes and three from the [|set of rotations] , 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.

** Homogeneous coordinates **
Main article: [|Homogeneous coordinates] Using four dimensional Homogeneous coordinates it is possible to describe a general transformation using a single 4 × 4 matrix. This matrix has six degrees of freedom which can identified with the six elements of the [|matrix] above the [|main diagonal], as all others are determined by these.

** Screw theory **
Main article: [|Screw theory] 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 **
Main article: [|Phase space] 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.[|[1]]

Rotations in four dimensions
Main article: [|SO(4)] 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, [|S3] × S3, 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 [|Euclidian space].

Plücker coordinates
Main article: [|Plücker coordinates] Plücker coordinates are a way of representing [|lines] in three dimensions using six homogeneous coordinates. As homogeneous coordinates they have only five degrees of freedom, corresponding to the five degrees of freedom of a general line, but they are treated as 6-vectors for some purposes. For example the check for the intersection of two lines is a 6-dimensional [|dot product] between two sets of Plücker coordinates, one of which has exchanged its displacement and moment parts.

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 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]]

String theory
[|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 [|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]]

Bivectors in four dimensions
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 Λ2 ℝ 4 for the set of bivectors in Euclidian space or Λ2 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 exponetial map of all bivectors in Λ2 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 which can are called //simple// and the rotations they generate are [|simple rotations]. Other rotations in four dimensions are [|double] and [|isoclinic] and correspond to non-simple bivectors that cannot be generated by single wedge product.[|[4]]

6-vectors
6-vectors are simply the vectors of six dimensional Euclidian 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** = (a1, a2, a3, a4, a5, a6). 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.

Gibbs bivectors
In 1901 [|J.W. Gibbs] published an influential work on vectors which included a six dimensional quantity which 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]]

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From Wikipedia, the free encyclopedia
Revision as of 00:18, 28 January 2010 by [|JohnBlackburne] ([|talk] | [|contribs]) ( [|diff] ) [|← Previous revision] | Current revision (diff) | Newer revision → (diff) Jump to: [|navigation], [|search] **Six-dimensional space** is a term used to describe any space that has six dimensions, i.e. six degrees of freedom, and which needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but the ones that are of most interest are simpler ones which are models of some aspect of our environment. Of particular interest is six dimensional [|Euclidean space], in which 6-polytopes and the 5-sphere are constructed. 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 Euclidian spaces, so it it 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] which keep the origin fixed. More generally any space which can be described with six [|coordinates], not necessarily Euclidian ones, is described as six dimensional. One example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional Euclidian space ℝ 7 that is equidistant 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-Euclidian] spaces are far more common than Euclidian spaces, and in six dimensions they have far more applications. [ hide ]
 * This is the [|current revision] of this page, as edited by [|JohnBlackburne] ([|talk] | [|contribs]) at 00:18, 28 January 2010. The present address (URL) is a permanent link to this version. ** ||
 * This is the [|current revision] of this page, as edited by [|JohnBlackburne] ([|talk] | [|contribs]) at 00:18, 28 January 2010. The present address (URL) is a permanent link to this version. ** ||
 * == Contents ==
 * [|1 Geometry]
 * [|1.1 6-polytope]
 * [|1.2 5-sphere]
 * [|1.3 6-sphere]
 * [|2 Applications]
 * [|2.1 Transformations in three dimensions]
 * [|2.1.1 Homogeneous coordinates]
 * [|2.1.2 Screw theory]
 * [|2.1.3 Phase space]
 * [|2.2 Rotations in four dimensions]
 * [|2.3 Plücker coordinates]
 * [|2.4 Electromagnetism]
 * [|2.5 String theory]
 * [|3 Theoretical background]
 * [|3.1 Bivectors in four dimensions]
 * [|3.2 6-vectors]
 * [|3.3 Gibbs bivectors]
 * [|4 Footnotes]
 * [|5 References] ||

6-polytope
Main article: [|6-polytope] 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. ** Regular polytopes in six dimensions ** || [|6-simplex] || [|6-cube] || [|6-orthoplex] || ||   ||    ||    ||    ||    ||    ||    ||    ||

5-sphere
The 5-sphere, or hypersphere in six dimensions, is the five dimensional surface equidistant from a point, e.g. the origin. It has symbol S5, with formal definition for the 5-sphere with radius //r// of The volume of the 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.

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 S6, with formal definition for the 6-sphere with radius //r// of 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.

Transformations in three dimensions
In three dimensional space a generalised transformation has [|six degrees of freedom], three [|translations] along the three coordinate axes and three from the [|set of rotations] , 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.

** Homogeneous coordinates **
Main article: [|Homogeneous coordinates] Using four dimensional Homogeneous coordinates it is possible to describe a general transformation using a single 4 × 4 matrix. This matrix has six degrees of freedom which can identified with the six elements of the [|matrix] above the [|main diagonal], as all others are determined by these.

** Screw theory **
Main article: [|Screw theory] 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 **
Main article: [|Phase space] 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.[|[1]]

Rotations in four dimensions
Main article: [|SO(4)] 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, [|S3] × S3, 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 [|Euclidian space].

Plücker coordinates
Main article: [|Plücker coordinates] Plücker coordinates are a way of representing [|lines] in three dimensions using six homogeneous coordinates. As homogeneous coordinates they have only five degrees of freedom, corresponding to the five degrees of freedom of a general line, but they are treated as 6-vectors for some purposes. For example the check for the intersection of two lines is a 6-dimensional [|dot product] between two sets of Plücker coordinates, one of which has exchanged its displacement and moment parts.

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 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]]

String theory
[|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 [|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]]

Bivectors in four dimensions
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 Λ2 ℝ 4 for the set of bivectors in Euclidian space or Λ2 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 exponetial map of all bivectors in Λ2 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 which can are called //simple// and the rotations they generate are [|simple rotations]. Other rotations in four dimensions are [|double] and [|isoclinic] and correspond to non-simple bivectors that cannot be generated by single wedge product.[|[4]]

6-vectors
6-vectors are simply the vectors of six dimensional Euclidian 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** = (a1, a2, a3, a4, a5, a6). 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.

Gibbs bivectors
In 1901 [|J.W. Gibbs] published an influential work on vectors which included a six dimensional quantity which 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]]

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> Wikipedia® is a registered trademark of the [|Wikimedia Foundation, Inc.], a non-profit organization. = Six-dimensional space =
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 * Text is available under the [|Creative Commons Attribution-ShareAlike License] ; additional terms may apply. See [|Terms of Use] for details.
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From Wikipedia, the free encyclopedia
Revision as of 00:18, 28 January 2010 by [|JohnBlackburne] ([|talk] | [|contribs]) ( [|diff] ) [|← Previous revision] | Current revision (diff) | Newer revision → (diff) Jump to: [|navigation], [|search] **Six-dimensional space** is a term used to describe any space that has six dimensions, i.e. six degrees of freedom, and which needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but the ones that are of most interest are simpler ones which are models of some aspect of our environment. Of particular interest is six dimensional [|Euclidean space], in which 6-polytopes and the 5-sphere are constructed. 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 Euclidian spaces, so it it 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] which keep the origin fixed. More generally any space which can be described with six [|coordinates], not necessarily Euclidian ones, is described as six dimensional. One example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional Euclidian space ℝ 7 that is equidistant 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-Euclidian] spaces are far more common than Euclidian spaces, and in six dimensions they have far more applications. [ hide ]
 * This is the [|current revision] of this page, as edited by [|JohnBlackburne] ([|talk] | [|contribs]) at 00:18, 28 January 2010. The present address (URL) is a permanent link to this version. ** ||
 * This is the [|current revision] of this page, as edited by [|JohnBlackburne] ([|talk] | [|contribs]) at 00:18, 28 January 2010. The present address (URL) is a permanent link to this version. ** ||
 * == Contents ==
 * [|1 Geometry]
 * [|1.1 6-polytope]
 * [|1.2 5-sphere]
 * [|1.3 6-sphere]
 * [|2 Applications]
 * [|2.1 Transformations in three dimensions]
 * [|2.1.1 Homogeneous coordinates]
 * [|2.1.2 Screw theory]
 * [|2.1.3 Phase space]
 * [|2.2 Rotations in four dimensions]
 * [|2.3 Plücker coordinates]
 * [|2.4 Electromagnetism]
 * [|2.5 String theory]
 * [|3 Theoretical background]
 * [|3.1 Bivectors in four dimensions]
 * [|3.2 6-vectors]
 * [|3.3 Gibbs bivectors]
 * [|4 Footnotes]
 * [|5 References] ||

6-polytope
Main article: [|6-polytope] 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. ** Regular polytopes in six dimensions ** || [|6-simplex] || [|6-cube] || [|6-orthoplex] || ||   ||    ||    ||    ||    ||    ||    ||    ||

5-sphere
The 5-sphere, or hypersphere in six dimensions, is the five dimensional surface equidistant from a point, e.g. the origin. It has symbol S5, with formal definition for the 5-sphere with radius //r// of The volume of the 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.

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 S6, with formal definition for the 6-sphere with radius //r// of 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.

Transformations in three dimensions
In three dimensional space a generalised transformation has [|six degrees of freedom], three [|translations] along the three coordinate axes and three from the [|set of rotations] , 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.

** Homogeneous coordinates **
Main article: [|Homogeneous coordinates] Using four dimensional Homogeneous coordinates it is possible to describe a general transformation using a single 4 × 4 matrix. This matrix has six degrees of freedom which can identified with the six elements of the [|matrix] above the [|main diagonal], as all others are determined by these.

** Screw theory **
Main article: [|Screw theory] 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 **
Main article: [|Phase space] 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.[|[1]]

Rotations in four dimensions
Main article: [|SO(4)] 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, [|S3] × S3, 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 [|Euclidian space].

Plücker coordinates
Main article: [|Plücker coordinates] Plücker coordinates are a way of representing [|lines] in three dimensions using six homogeneous coordinates. As homogeneous coordinates they have only five degrees of freedom, corresponding to the five degrees of freedom of a general line, but they are treated as 6-vectors for some purposes. For example the check for the intersection of two lines is a 6-dimensional [|dot product] between two sets of Plücker coordinates, one of which has exchanged its displacement and moment parts.

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 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]]

String theory
[|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 [|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]]

Bivectors in four dimensions
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 Λ2 ℝ 4 for the set of bivectors in Euclidian space or Λ2 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 exponetial map of all bivectors in Λ2 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 which can are called //simple// and the rotations they generate are [|simple rotations]. Other rotations in four dimensions are [|double] and [|isoclinic] and correspond to non-simple bivectors that cannot be generated by single wedge product.[|[4]]

6-vectors
6-vectors are simply the vectors of six dimensional Euclidian 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** = (a1, a2, a3, a4, a5, a6). 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.

Gibbs bivectors
In 1901 [|J.W. Gibbs] published an influential work on vectors which included a six dimensional quantity which 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]]

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From Wikipedia, the free encyclopedia
Revision as of 00:18, 28 January 2010 by [|JohnBlackburne] ([|talk] | [|contribs]) ( [|diff] ) [|← Previous revision] | Current revision (diff) | Newer revision → (diff) Jump to: [|navigation], [|search] **Six-dimensional space** is a term used to describe any space that has six dimensions, i.e. six degrees of freedom, and which needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but the ones that are of most interest are simpler ones which are models of some aspect of our environment. Of particular interest is six dimensional [|Euclidean space], in which 6-polytopes and the 5-sphere are constructed. 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 Euclidian spaces, so it it 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] which keep the origin fixed. More generally any space which can be described with six [|coordinates], not necessarily Euclidian ones, is described as six dimensional. One example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional Euclidian space ℝ 7 that is equidistant 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-Euclidian] spaces are far more common than Euclidian spaces, and in six dimensions they have far more applications. [ hide ]
 * This is the [|current revision] of this page, as edited by [|JohnBlackburne] ([|talk] | [|contribs]) at 00:18, 28 January 2010. The present address (URL) is a permanent link to this version. ** ||
 * This is the [|current revision] of this page, as edited by [|JohnBlackburne] ([|talk] | [|contribs]) at 00:18, 28 January 2010. The present address (URL) is a permanent link to this version. ** ||
 * == Contents ==
 * [|1 Geometry]
 * [|1.1 6-polytope]
 * [|1.2 5-sphere]
 * [|1.3 6-sphere]
 * [|2 Applications]
 * [|2.1 Transformations in three dimensions]
 * [|2.1.1 Homogeneous coordinates]
 * [|2.1.2 Screw theory]
 * [|2.1.3 Phase space]
 * [|2.2 Rotations in four dimensions]
 * [|2.3 Plücker coordinates]
 * [|2.4 Electromagnetism]
 * [|2.5 String theory]
 * [|3 Theoretical background]
 * [|3.1 Bivectors in four dimensions]
 * [|3.2 6-vectors]
 * [|3.3 Gibbs bivectors]
 * [|4 Footnotes]
 * [|5 References] ||

6-polytope
Main article: [|6-polytope] 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. ** Regular polytopes in six dimensions ** || [|6-simplex] || [|6-cube] || [|6-orthoplex] || ||   ||    ||    ||    ||    ||    ||    ||    ||

5-sphere
The 5-sphere, or hypersphere in six dimensions, is the five dimensional surface equidistant from a point, e.g. the origin. It has symbol S5, with formal definition for the 5-sphere with radius //r// of The volume of the 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.

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 S6, with formal definition for the 6-sphere with radius //r// of 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.

Transformations in three dimensions
In three dimensional space a generalised transformation has [|six degrees of freedom], three [|translations] along the three coordinate axes and three from the [|set of rotations] , 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.

** Homogeneous coordinates **
Main article: [|Homogeneous coordinates] Using four dimensional Homogeneous coordinates it is possible to describe a general transformation using a single 4 × 4 matrix. This matrix has six degrees of freedom which can identified with the six elements of the [|matrix] above the [|main diagonal], as all others are determined by these.

** Screw theory **
Main article: [|Screw theory] 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 **
Main article: [|Phase space] 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.[|[1]]

Rotations in four dimensions
Main article: [|SO(4)] 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, [|S3] × S3, 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 [|Euclidian space].

Plücker coordinates
Main article: [|Plücker coordinates] Plücker coordinates are a way of representing [|lines] in three dimensions using six homogeneous coordinates. As homogeneous coordinates they have only five degrees of freedom, corresponding to the five degrees of freedom of a general line, but they are treated as 6-vectors for some purposes. For example the check for the intersection of two lines is a 6-dimensional [|dot product] between two sets of Plücker coordinates, one of which has exchanged its displacement and moment parts.

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 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]]

String theory
[|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 [|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]]

Bivectors in four dimensions
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 Λ2 ℝ 4 for the set of bivectors in Euclidian space or Λ2 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 exponetial map of all bivectors in Λ2 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 which can are called //simple// and the rotations they generate are [|simple rotations]. Other rotations in four dimensions are [|double] and [|isoclinic] and correspond to non-simple bivectors that cannot be generated by single wedge product.[|[4]]

6-vectors
6-vectors are simply the vectors of six dimensional Euclidian 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** = (a1, a2, a3, a4, a5, a6). 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.

Gibbs bivectors
In 1901 [|J.W. Gibbs] published an influential work on vectors which included a six dimensional quantity which 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]]

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 * Text is available under the [|Creative Commons Attribution-ShareAlike License] ; additional terms may apply. See [|Terms of Use] for details.
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