Six-dimensional space

Last updated

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, , 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) 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)
A6B6D6E6
6-simplex t0.svg
6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{3,3,3,3,3}
6-cube t0.svg
6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{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
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{3,33,1} = h{4,3,3,3,3}
Up 2 21 t0 E6.svg
221
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.png
{3,3,32,1}
Up 1 22 t0 E6.svg
122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
{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 for the set of bivectors in Euclidean space or for the set of bivectors in spacetime. The Plücker coordinates are bivectors in while the electromagnetic tensor discussed in the previous section is a bivector in . Bivectors can be used to generate rotations in either or through the exponential map (e.g. applying the exponential map of all bivectors in generates all rotations in ). They can also be related to general transformations in three dimensions through homogeneous coordinates, which can be thought of as modified rotations in .

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

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.

<span class="mw-page-title-main">Spinor</span> Non-tensorial representation of the spin group; represents fermions in physics

In geometry and physics, spinors are elements of a complex number-based vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation, but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360°. It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors.

<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. Euclidean vectors can be added and scaled to form a vector space. 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

Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-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.

<span class="mw-page-title-main">Quaternion</span> Noncommutative extension of the complex 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. The algebra of quaternions is often denoted by H, or in blackboard bold by Although multiplication of quaternions is noncommutative, it gives a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form

<span class="mw-page-title-main">Hyperboloid</span> Unbounded quadric surface

In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

<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.

<span class="mw-page-title-main">Minkowski space</span> Spacetime used in theory of relativity

In mathematical physics, Minkowski space combines inertial space and time manifolds with a non-inertial reference frame of space and time into a four-dimensional model relating a position to the field.

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

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 a sign (as in the sign of an angle): 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.

<span class="mw-page-title-main">Bivector</span> Sum of directed areas in exterior 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. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is 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 vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations. They are also used in physics.

In abstract algebra, the biquaternions are the numbers w + xi + yj + zk, 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:

<span class="mw-page-title-main">Four-dimensional space</span> Geometric space with four dimensions

Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only 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. This concept of ordinary space is called Euclidean space because it corresponds to Euclid's geometry, which was originally abstracted from the spatial experiences of everyday life.

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. Each versor has the form

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

In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

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.

<span class="mw-page-title-main">Euclidean plane</span> Geometric model of the planar projection of the physical universe

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted or . It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement.

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 mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance. Eight-dimensional Euclidean space is eight-dimensional space equipped with the Euclidean metric.

In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space.

References