Six-dimensional space

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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.

Contents

Formally, six-dimensional Euclidean space, ℝ6, 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, S6. This is the set of all points in seven-dimensional space (Euclidean) ℝ7 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.

Geometry

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: 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 D6 family, and 221 and 122 polytopes from the E6 family.

Uniform polytopes in six dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
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6-simplex t0.svg
6-simplex
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{3,3,3,3,3}
6-cube t0.svg
6-cube
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{4,3,3,3,3}
6-cube t5.svg
6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{3,3,3,3,4}
6-demicube t0 D6.svg
6-demicube
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{3,33,1} = h{4,3,3,3,3}
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221
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{3,32,2}

5-sphere

The 5-sphere, or hypersphere in six dimensions, is the five-dimensional surface equidistant from a point. It has symbol S5, 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 × r6, 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 S6, 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 × r7, 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 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

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

Phase portrait of the Van der Pol oscillator Limitcycle.svg
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

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 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 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) Calabi-Yau.png
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]

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 bivectors in four dimensions. These can be written Λ24 for the set of bivectors in Euclidean space or Λ23,1 for the set of bivectors in spacetime. The Plücker coordinates are bivectors in ℝ4 while the electromagnetic tensor discussed in the previous section is a bivector in ℝ3,1. Bivectors can be used to generate rotations in either ℝ4 or ℝ3,1 through the exponential map (e.g. applying the exponential map of all bivectors in Λ24 generates all rotations in ℝ4). They can also be related to general transformations in three dimensions through homogeneous coordinates, which can be thought of as modified rotations in ℝ4.

The bivectors arise from sums of all possible wedge products between pairs of 4-vectors. They therefore have C4
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

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 = (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 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]

Footnotes

  1. Arthur Besier (1969). Perspectives of Modern Physics. McGraw-Hill.
  2. Lounesto (2001), pp. 109–110
  3. Aharony (2000)
  4. Lounesto (2001), pp. 86–89
  5. 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.

Related Research Articles

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<span class="mw-page-title-main">Euclidean vector</span> Geometric object that has length and direction

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 pointA with a terminal pointB, and denoted by .

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<span class="mw-page-title-main">3-sphere</span> Mathematical object

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.

<span class="mw-page-title-main">Quaternion</span> Noncommutative extension of the real numbers

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.

<span class="mw-page-title-main">Cross product</span> Mathematical operation on vectors in 3D space

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.

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<span class="mw-page-title-main">Rotation (mathematics)</span> Motion of a certain space that preserves at least one point

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<span class="mw-page-title-main">Bivector</span> Oriented area in geometric algebra

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.

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<span class="mw-page-title-main">Three-dimensional space</span> Geometric model of the physical space

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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 Rp,q to null vectors in Rp+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.

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