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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.)
The spacetime in which we live appears to be 4-dimensional. It is conventional (and for most practical purposes entirely sensible) to consider this as three spatial dimensions and one of time. 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 a negative amount. Moving northwest is merely a combination of moving north and moving west.
Time is frequently referred to as the "fourth dimension". It is somewhat different to the three spatial dimensions in that there is only one of it, and movement is only possible in one direction.
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 also string theory.
In physics, the dimension of a quality is the expression of that quality in basic units: the dimension of speed, for example, is length divided by time. See Dimensional analysis.
In mathematics, 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.
A tesseract is an example of a four-dimensional object.
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 manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefor 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 topologyTopology Geometric topology In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. It has come over time to be almost synonymous with low-dimensional topology conce, 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 conjectureIn mathematics, the Poincare conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. It is widely considered to be the most important unsolved problem in topology. The Poincare conjecture is one of the se, where four different proof methods are applied.