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2 Mathematical dimensions |

Some theories predict that the space we live in has in fact many more dimensions (frequently 10, 11 or 26) but that the universe measured along these additional dimensions is subatomic in size. See string theory.

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.

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

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

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.

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.

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.

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.

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.

- 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)

- Thomas Banchoff, (1996)
*Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition*, Freeman - Clifford A. Pickover, (1999)
*Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons*, Oxford University Press - Rudy Rucker (1984),
*The Fourth Dimension*, Houghton-Mifflin\n