Dimension

Dimension is, in essence, the number of degrees of freedom available for movement in a space.

For example, the space in which we live is 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.

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

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.

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

By a straightforward application of Zorn's Lemma, it can be seen that every vector space has a basis. A vector space may have many different bases, but each has the same cardinality, and this cardinality is called the dimension of the vector space, or the Hamel dimension when it is necessary to distinguish it from other types of dimension.

The Hamel dimension is a natural generalization of the dimension of Euclidean space, since E ⁿ is a vector space of dimension n over R (the reals). However, the Hamel dimension depends on the base field, so while R has dimension 1 when considered as a vector space over itself, it has dimension c (the cardinality of the continuum) when considered as a vector space over Q (the rationals).

Some simple formulae relate the Hamel dimension of a vector space with the cardinality of the base field and the cardinality of the space itself. If V is a vector space over a field K then, denoting the Hamel dimension of V by dimV, we have:

If dimV is finite, then |V| = |K|^dimV.

If dimV is infinite, then |V| = max(|K|, dimV).

More dimensions to be added later

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