Six-dimensional (6D) 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. Formally, six-dimensional Euclidean space, R 6 {\displaystyle \mathbb {R} ^{6}} , is generated by considering all real 6-tuples as 6-vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined and can be used to calculate the metric. 6 × 6 matrices can be used to describe transformations such as rotations that keep the origin fixed. More generally, any space that can be described locally with six coordinates, not necessarily Euclidean ones, is six-dimensional. One example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional space (Euclidean) R 7 {\displaystyle \mathbb {R} ^{7}} that are a fixed distance from the origin. This constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions. Such non-Euclidean spaces are far more common than Euclidean spaces, and in six dimensions they have far more applications.
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.
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
S 5 = { x ∈ R 6 : ‖ x ‖ = r } . {\displaystyle S^{5}=\left\{x\in \mathbb {R} ^{6}:\|x\|=r\right\}.}
The volume of six-dimensional space bounded by this 5-sphere is
V 6 = π 3 r 6 6 {\displaystyle V_{6}={\frac {\pi ^{3}r^{6}}{6}}}
which is 5.16771 × r6, or 0.0807 of the smallest 6-cube that contains the 5-sphere.
Icosahedral lattice symmetry In six-dimensional Euclidean space, six mutually orthogonal axes may be drawn through any point. These form a group isomorphous with the six fivefold symmetry axes of the regular icosahedron, thus affording an icosahedrally symmetric projection into three-dimensional space. This projection plays a major role in the crystallography of icosahedral quasicrystals.
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 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.
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.
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