N-dimensional space

N-dimensional space

In mathematics, an "n"-dimensional space is a topological space whose dimension is "n" (where "n" is a fixed natural number). The archetypical example is "n"-dimensional Euclidean space, which describes Euclidean geometry in "n" dimensions.

Many familiar geometric objects can be generalized to any number of dimensions. For example, the two-dimensional triangle and the three-dimensional tetrahedron can be seen as specific instances of the "n"-dimensional simplex. Also, the circle and the sphere can be seen as specific instances of the "n"-dimensional hypersphere. More generally, an "n"-dimensional manifold is a space that locally looks like "n"-dimensional Euclidean space, but whose global structure may be non-Euclidean.

There are also notions of dimension (such as Hausdorff dimension in topology and Kodaira dimension in algebraic geometry) that apply to even more general spaces.

Sometimes it is convenient in science to describe an object with "n" degrees of freedom as if it were a point in some "n"-dimensional space. For example, classical mechanics describes the three-dimensional position and momentum of a point particle as a point in 6-dimensional phase space.

Rotation

Rotation is not as easily generalized to higher dimensions. Rotation is motion of points in a circular path. A point cannot trace out a spherical path as it moves through space, so rotation is inherently a two-dimensions-at-a-time phenomenon. The sine and cosine functions, which describe circular motion, are a pair of functions that cannot readily be expanded to form a triple.

A matrix, however, is very easily generalized to any natural number of dimensions. A rotation matrix, which is a type of square matrix, requires at least two rows and at least two columns. When a rotation matrix has an odd number of rows and columns, and describes rotation around a coordinate axis, one row and one column are uninvolved. That row and that column contain all zero entries except for a single one. (See the 3 x 3 matrix below.) This is consistent with the fact that three-dimensional objects rotate around a linear axis, and the points that are on that axis rotate "in place," moving in a circle with a radius of zero, as it were.

This 3 × 3 matrix represents rotation around the "x"-axis:

: mathcal{R}( heta_R):= egin{pmatrix} 1 & 0 & 0 \ 0 & cos{ heta_R} & - sin{ heta_R} \ 0 & sin{ heta_R} & cos{ heta_R} end{pmatrix} where heta_R is the roll angle.

Analogously, this 4 × 4 matrix represents rotation around the "wx"-plane:

: mathcal{R}( heta):= egin{pmatrix} 1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & cos{ heta} & - sin{ heta} \ 0 & 0 & sin{ heta} & cos{ heta} end{pmatrix} where heta is one of six angles that are analogous to the roll angle.

By examining rotation matrices of various sizes, several things can be inferred: First, that linear axes of rotation can be found in any odd number of dimensions, including three. Secondly, that objects in four dimensions can rotate around a two-dimensional axis, in which case the points that are in that plane rotate "in place." Thirdly, objects in four dimensions can also rotate at two different speeds at the same time, since a two-dimensional axis of rotation offers enough space for rotation to begin at a speed that is different from that of the rotation that is already taking place. More generally, objects in an even number of dimensions can rotate at up to half that number of different speeds simultaneously.

This is a 4 × 4 matrix that represents rotation at two speeds at once:

: mathcal{R}( heta):= egin{pmatrix} cos{ heta} & - sin{ heta} & 0 & 0\ sin{ heta} & cos{ heta} & 0 & 0\ 0 & 0 & cos{ heta} & - sin{ heta} \ 0 & 0 & sin{ heta} & cos{ heta} end{pmatrix}

See also

* n-dimensional calculus
* facet (mathematics)
* fourth dimension
* four-vector
* vector decomposition
* coordinate space
* kissing number problem


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