Six-dimensional 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.
More generally, any space that can be described locally with six coordinates, not necessarily Euclidean ones, is six-dimensional.
This constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions.
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.
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.
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.
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.
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.
Quaternions, one of the ways to describe rotations in three dimensions, consist of a four-dimensional space.
[2] 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.
[3] A number of the above applications can be related to each other algebraically by considering the real, six-dimensional bivectors in four dimensions.
They can also be related to general transformations in three dimensions through homogeneous coordinates, which can be thought of as modified rotations in
They therefore have C42 = 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.
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.
