View source for Dimension (mathematics)

== Introduction ==

In [[Mathematics]], a ''dimension'' is a fundamental concept that describes the number of degrees of freedom available in a mathematical space. The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be abstract spaces, independent of the physical space we live in.

== Definition ==

The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the ''Hamel dimension'' or ''algebraic dimension'' to distinguish it from other definitions.

== Dimension in Geometry ==

In [[Geometry]], the dimension of an object is an intrinsic property, independent of the space in which the object may happen to be embedded. For example, a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane.

== Dimension in Topology ==

In [[Topology]], dimension can be defined in multiple ways. The most intuitive way is through covering dimension. A base B for a topological space X with the property that every open cover of X has a refinement base which is locally finite, is said to have dimension n if n is the smallest integer for which the following holds: any finite open cover of X has an open refinement such that no point is included in more than n+1 elements.

== Dimension in Physics ==

In [[Physics]], the dimension of a physical quantity is the exponent (or power) to which each of the base quantities is raised to represent that quantity. For example, speed has the dimension length/time, and the dimension of force is mass×length/time².

== Fractal Dimension ==

[[Fractal]]s are shapes that are self-similar at all scales. The concept of dimension can be extended to them using the idea of a ''fractal dimension'', which can take non-integer values. This is a measure of the space-filling capacity of a pattern that tells how a fractal scales differently than the space it is embedded in.

== Dimension in Algebraic Geometry ==

In [[Algebraic Geometry]], the dimension of an algebraic variety V in algebraically closed field k is defined to be the transcendence degree of the field extension k(V)/k.

== Dimension in Linear Algebra ==

In [[Linear Algebra]], the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called the ''Hamel dimension'' or the ''algebraic dimension'' to distinguish it from other types of dimension.

== Dimension in Set Theory ==

In [[Set Theory]], the dimension of a set is defined as the minimum number of coordinates needed to specify each point within it. This concept is used in the study of fractals and other complex geometric structures.

== See Also ==

* [[Hilbert Space]]
* [[Manifold]]
* [[Metric Space]]
* [[Vector Space]]

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[[Category:Mathematics]]
[[Category:Geometry]]
[[Category:Algebra]]