Dimension

Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space. (In common usage, the dimensions of an object are the measurements that define its shape and size. That usage is related to, but different from, what this article is about.)

Physical dimensions

For example, the space in which we live appears to be 3-dimensional. We can move up, north or west, and movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative amount. Moving northwest is merely a combination of moving north and moving west.

Time is frequently referred to as the "fourth dimension"; time is not the fourth dimension of space, but rather of spacetime. This does not have a Euclidean geometry, so temporal directions are not entirely equivalent to spatial dimensions. A tesseract is an example of a four-dimensional object.

For 3-D images see Stereoscopy.

For 3-D films and video see 3-D.

Mathematical dimensions

In mathematics, we find that no definition of dimension adequately captures the concept in all situations where we would like to make use of it. Consequently, mathematicians have devised numerous definitions of dimension for different types of spaces. All, however, are ultimately based on the concept of the dimension of Euclidean n-space E n. The point E 0 is 0-dimensional. The line E 1 is 1-dimensional. The plane E 2 is 2-dimensional. And in general E n is n-dimensional.

In the rest of this article we examine some of the more important mathematical definitions of dimension.

Hamel Dimension

For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis. See Hamel dimension for details.

Manifolds

A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold.

The theory of manifolds, in the field of geometric topology, is characterised by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to 'work'; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied.

Lebesgue covering dimension

For any topological space, the Lebesgue covering dimension is defined to be n if any open cover has a refinement (a second cover where each element is a subset of an element in the first cover) such that no point is included in more than n+1 elements. For manifolds, this coincides with the dimension mentioned above.

Hausdorff dimension

For sets which are of a complicated structure, especially fractals, the Hausdorff dimension is useful. The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values.

Hilbert spaces

Every Hilbert space admits an orthonormal basis, and any two such bases have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide.

Krull dimension of commutative rings

The Krull dimension of a commutative ring is defined to be the maximal length of a strictly increasing chain of prime ideals in the ring.

More dimensions

• Dimension of an algebraic variety
• Poset dimension
• Pointwise dimension
• Lyapunov dimension
• Kaplan-Yorke dimension
• Minkowski-Bouligand dimension
• Exterior dimension
• Hurst exponent
• q-dimension; especially:
• Information dimension (corresponding to q=1)
• Correlation dimension (corresponding to q=2)