2D

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

Physical dimensions

For example, the space in which we live appears to be 3-dimensional. We can move up-or-down, north-or-south, or east-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 anegative amount. Moving northwest is merely a combination of moving north and moving west.

Some theories predict that the space we live in has in fact many more dimensions (frequently 10, 11 or 26) but that theuniverse measured along these additional dimensions is subatomic in size. See also 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 notentirely equivalent to spatial dimensions. A tesseract is an example of afour-dimensional object.

Related topics:

• 2D geometric models
• 3D geometricmodels
• Stereoscopy (3-D imaging)
• 2D computer graphics
• 3D computer graphics
• 3-D films and video

Mathematical dimensions

In mathematics , no definition of dimension adequately captures the conceptin all situations where we would like to make use of it. Consequently, mathematicians have devised numerous definitions ofdimension for different types of spaces. All, however, are ultimately based on the concept of the dimension of Euclidean n-space E ⁿ. The pointE ⁰ is 0-dimensional. The line E ¹ is 1-dimensional. The planeE ² is 2-dimensional. And in general E ⁿ isn-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 thecardinality of a basis. See Hamel dimension for details.

Manifolds

A connected topological manifold is locally homeomorphic to Euclideann-space, and the number n is called the manifold's dimension. One can show that this yields a uniquely defineddimension 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 mostdifficult. 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 ben if n is the smallest integer for which the following holds: any open cover has a refinement (a second coverwhere each element is a subset of an element in the first cover) such that no point is included in more than n + 1elements. For manifolds, this coincides with the dimension mentioned above. If no such n exists, then the dimension isinfinite.

Hausdorff dimension

For sets which are of a complicated structure, especially fractals , the Hausdorff dimension is useful. The Hausdorff dimension is defined forall metric spaces and, unlike the Hamel dimension, can also attainnon-integer real values. The upper and lower boxdimensions are a variant of the same idea.

Hilbert spaces

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

Krull dimension of commutative rings

The Krull dimension of a commutative ring , named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasingchain of prime ideals in the ring.

More dimensions

• Dimension of an algebraicvariety
• Topological dimension
• Isoperimetric dimension
• Poset dimension
• Pointwisedimension
• Lyapunovdimension
• Kaplan-Yorkedimension
• Minkowski-Bouligand dimension
• Exteriordimension
• Hurst exponent
• q-dimension ; especially:
  • Informationdimension (corresponding to q=1)
  • Correlationdimension (corresponding to q=2)

Further reading

• Thomas Banchoff, (1996) Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, SecondEdition, Freeman

• Clifford A. Pickover , (1999) Surfing throughHyperspace: Understanding Higher Universes in Six Easy Lessons, Oxford University Press

• Rudy Rucker (1984), The Fourth Dimension, Houghton-Mifflin

• Edwin A. Abbott, (1884) Flatland