N-dimensional space

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N-dimensional space

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an n-dimensional space is a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
whose dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
is n (where n is a fixed natural number
Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
). The archetypical example is n-dimensional Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
, which describes Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
in n dimensions.

Many familiar geometric objects can be generalized to any number of dimensions. For example, the two-dimensional triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
and the three-dimensional tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
can be seen as specific instances of the n-dimensional simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
.

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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an n-dimensional space is a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
whose dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
is n (where n is a fixed natural number
Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
). The archetypical example is n-dimensional Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
, which describes Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
in n dimensions.

Many familiar geometric objects can be generalized to any number of dimensions. For example, the two-dimensional triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
and the three-dimensional tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
can be seen as specific instances of the n-dimensional simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
. Also, the circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
and the sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
can be seen as specific instances of the n-dimensional hypersphere
Hypersphere

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real num...
. 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
Hausdorff dimension

In mathematics, the Hausdorff dimension is an Extended real number line non-negative real number associated to any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space....
in topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
and Kodaira dimension in algebraic geometry
Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry....
) that apply to even more general spaces.

Sometimes it is convenient in science to describe an object with n degrees of freedom
Degrees of freedom (physics and chemistry)

Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters....
as if it were a point in some n-dimensional space. For example, classical mechanics
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
describes the three-dimensional position and momentum of a point particle as a point in 6-dimensional phase space
Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space....
.

Rotation

Rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
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
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
, however, is very easily generalized to any natural number of dimensions. A rotation matrix
Rotation matrix

In matrix theory, a rotation matrix is a real number square matrix whose transpose is its invertible matrix and whose determinant is 1 The matrix is so-called because it geometrically corresponds to a linear map that sends vectors to a corresponding vector rotated about the origin by a fixed angle....
, 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:

where is the roll angle.

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

where 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:

See also

n-dimensional calculus
facet (mathematics)
Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
fourth dimension
Fourth dimension

In physics and mathematics, a vector of n real number can be understood as a Coordinate system in an n-dimensional Euclidean space. When n = 4, the set of all such locations is called 4-dimensional Euclidean space....
four-vector
Four-vector

In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations....
vector decomposition
Vector decomposition

Vector decomposition refers to decomposing a Vector of Rn into several vectors, each linearly independent ....
coordinate space
Coordinate space

In mathematics, specifically in linear algebra, the coordinate space, Fn, is the prototypical example of an n-dimensional vector space over a field F....
kissing number problem
Kissing number problem

In geometry, the kissing number is the maximum number of spheres of radius 1 that can simultaneously touch the unit sphere in n-dimensional Euclidean space....