Editing Dimension (section)

== In mathematics ==

In [[mathematics]], the dimension of an object is, roughly speaking, the number of [[degrees of freedom]] of a point that moves on this object. In other words, the dimension is the number of independent [[parameter]]s or [[coordinates]] that are needed for defining the position of a point that is constrained to be on the object. For example, the dimension of a point is [[Zero-dimensional space|zero]]; the dimension of a [[line (geometry)|line]] is [[One-dimensional space|one]], as a point can move on a line in only one direction (or its opposite); the dimension of a [[plane (geometry)|plane]] is [[Two-dimensional space|two]], etc.

The dimension is an intrinsic property of an object, in the sense that it is independent of the dimension of the space in which the object is or can be [[Embedding (mathematics)|embedded]]. For example, a [[curve]], such as a [[circle]], is of dimension one, because the position of a point on a curve is determined by its signed distance along the curve to a fixed point on the curve. This is independent from the fact that a curve cannot be embedded in a [[Euclidean space]] of dimension lower than two, unless it is a line. Similarly, a [[surface (mathematics)|surface]] is of dimension two, even if embedded in [[three-dimensional space]].

The dimension of [[Euclidean space|Euclidean {{math|''n''}}-space]] {{math|'''E'''<sup>''n''</sup> }}is {{math|''n''}}. When trying to generalize to other types of spaces, one is faced with the question "what makes {{math|'''E'''<sup>''n''</sup> }} {{math|''n''}}-dimensional?" One answer is that to cover a fixed [[Ball (mathematics)|ball]] in {{math|'''E'''<sup>''n''</sup> }} by small balls of radius {{math|''ε''}}, one needs on the order of {{math|''ε''<sup>−''n''</sup>}} such small balls. This observation leads to the definition of the [[Minkowski dimension]] and its more sophisticated variant, the [[Hausdorff dimension]], but there are also other answers to that question. For example, the boundary of a ball in {{math|'''E'''<sup>''n''</sup> }} looks locally like {{math|'''E'''<sup>''n''-1</sup> }} and this leads to the notion of the [[inductive dimension]]. While these notions agree on {{math|'''E'''<sup>''n''</sup>}}, they turn out to be different when one looks at more general spaces.

A [[tesseract]] is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract ''has four dimensions''", mathematicians usually express this as: "The tesseract ''has dimension 4''", or: "The dimension of the tesseract ''is'' 4".

Although the notion of higher dimensions goes back to [[René Descartes]], substantial development of a higher-dimensional geometry only began in the 19th century, via the work of [[Arthur Cayley]], [[William Rowan Hamilton]], [[Ludwig Schläfli]] and [[Bernhard Riemann]]. Riemann's 1854 [[Habilitationsschrift]], Schläfli's 1852 ''[[Theorie der vielfachen Kontinuität]]'', and Hamilton's discovery of the [[quaternion]]s and [[John T. Graves]]' discovery of the [[octonion]]s in 1843 marked the beginning of higher-dimensional geometry.

The rest of this section examines some of the more important mathematical definitions of dimension.

===Vector spaces===
{{Main|Dimension (vector space)}}
The dimension of a [[vector space]] is the number of vectors in any [[Basis (linear algebra)|basis]] for the space, [[Id est|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 notions of dimension.

For the non-[[free module|free]] case, this generalizes to the notion of the [[length of a module]].

===Manifolds===
<!--Linked from [[Ball (mathematics)]]-->
The uniquely defined dimension of every [[Connectedness|connected]] topological [[manifold]] can be calculated. A connected topological manifold is [[Local property|locally]] [[homeomorphic]] to Euclidean {{math|''n''}}-space, in which the number {{math|''n''}} is the manifold's dimension.

For connected [[differentiable manifold]]s, the dimension is also the dimension of the [[Tangent space|tangent vector space]] at any point.

In [[geometric topology]], the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, the '''high-dimensional''' cases {{nowrap|{{math|''n'' > 4}}}} are simplified by having extra space in which to "work"; and the cases {{math|''n'' {{=}} 3}} and {{math|4}} are in some senses the most difficult. This state of affairs was highly marked in the various cases of the [[Poincaré conjecture]], in which four different proof methods are applied.

====Complex dimension====
{{main|Complex dimension}}
[[File:Riemann Sphere.gif|right|thumb|The complex plane can be mapped to the surface of a sphere, called the Riemann sphere, with the complex number 0 mapped to one pole, the unit circle mapped to the equator, and a [[point at infinity]] mapped to the other pole.]]
The dimension of a manifold depends on the base field with respect to which Euclidean space is defined. While analysis usually assumes a manifold to be over the [[real numbers]], it is sometimes useful in the study of [[complex manifold]]s and [[dimension of an algebraic variety|algebraic varieties]] to work over the [[complex numbers]] instead. A complex number (''x'' + ''iy'') has a [[real part]] ''x'' and an [[imaginary part]] ''y'', in which x and y are both real numbers; hence, the complex dimension is half the real dimension.

Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional [[sphere|spherical surface]], when given a complex metric, becomes a [[Riemann sphere]] of one complex dimension.<ref>{{cite book |first1=Shing-Tung |last1=Yau |first2=Steve |last2=Nadis |chapter=4. Too Good to be True |title=The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions |chapter-url=https://books.google.com/books?id=vlA4DgAAQBAJ&pg=PT60 |date=2010 |publisher=Basic Books |isbn=978-0-465-02266-3 |pages=60–}}</ref>

===Varieties===
{{Main|Dimension of an algebraic variety}}
The dimension of an [[algebraic variety]] may be defined in various equivalent ways. The most intuitive way is probably the dimension of the [[tangent space]] at any [[Regular point of an algebraic variety]]. Another intuitive way is to define the dimension as the number of [[hyperplane]]s that are needed in order to have an intersection with the variety that is reduced to a finite number of points (dimension zero). This definition is based on the fact that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety.

An [[algebraic set]] being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains <math>V_0\subsetneq V_1\subsetneq \cdots \subsetneq V_d</math> of sub-varieties of the given algebraic set (the length of such a chain is the number of "<math>\subsetneq</math>").

Each variety can be considered as an [[stack (mathematics)|algebraic stack]], and its dimension as variety agrees with its dimension as stack. There are however many stacks which do not correspond to varieties, and some of these have negative dimension. Specifically, if ''V'' is a variety of dimension ''m'' and ''G'' is an [[algebraic group]] of dimension ''n'' [[Group action (mathematics)|acting on ''V'']], then the [[quotient stack]] [''V''/''G''] has dimension ''m''&nbsp;−&nbsp;''n''.<ref>{{citation |last=Fantechi|first=Barbara|chapter=Stacks for everybody|title=European Congress of Mathematics Volume I|volume=201|pages=349–359|series=Progr. Math.|publisher=Birkhäuser|year=2001|chapter-url=http://www.mathematik.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf|url-status=live|archive-url=https://web.archive.org/web/20060117052957/http://www.mathematik.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf|archive-date=2006-01-17}}
</ref>

===Krull dimension===
The [[Krull dimension]] of a [[commutative ring]] is the maximal length of chains of [[prime ideal]]s in it, a chain of length ''n'' being a sequence <math>\mathcal{P}_0\subsetneq \mathcal{P}_1\subsetneq \cdots \subsetneq\mathcal{P}_n </math> of prime ideals related by inclusion. It is strongly related to the dimension of an algebraic variety, because of the natural correspondence between sub-varieties and prime ideals of the ring of the polynomials on the variety.

For an [[algebra over a field]], the dimension as [[vector space]] is finite if and only if its Krull dimension is 0.

===Topological spaces===

For any [[normal topological space]] {{math|''X''}}, the [[Lebesgue covering dimension]] of {{math|''X''}} is defined to be the smallest [[integer]] ''n'' for which the following holds: any [[open cover]] has an open refinement (a second open cover in which each element is a subset of an element in the first cover) such that no point is included in more than {{math|''n'' + 1}} elements. In this case dim {{math|''X'' {{=}} ''n''}}. For {{math|''X''}} a manifold, this coincides with the dimension mentioned above. If no such integer {{math|''n''}} exists, then the dimension of {{math|''X''}} is said to be infinite, and one writes dim {{math|''X'' {{=}} ∞}}. Moreover, {{math|''X''}} has dimension −1, i.e. dim {{math|''X'' {{=}} −1}} if and only if {{math|''X''}} is empty. This definition of covering dimension can be extended from the class of normal spaces to all [[Tychonoff space]]s merely by replacing the term "open" in the definition by the term "'''functionally open'''".

An [[inductive dimension]] may be defined [[Mathematical induction|inductively]] as follows. Consider a [[Isolated point|discrete set]] of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a ''new direction'', one obtains a 2-dimensional object. In general, one obtains an ({{math|''n'' + 1}})-dimensional object by dragging an {{math|''n''}}-dimensional object in a ''new'' direction. The inductive dimension of a topological space may refer to the ''small inductive dimension'' or the ''large inductive dimension'', and is based on the analogy that, in the case of metric spaces, {{nowrap|({{math|''n'' + 1}})-dimensional}} balls have {{math|''n''}}-dimensional [[boundary (topology)|boundaries]], permitting an inductive definition based on the dimension of the boundaries of open sets. Moreover, the boundary of a discrete set of points is the empty set, and therefore the empty set can be taken to have dimension −1.<ref>{{cite book |title=Dimension Theory (PMS-4), Volume 4 |first1=Witold |last1=Hurewicz |first2=Henry |last2=Wallman |publisher=[[Princeton University Press]] |year=2015 |isbn=978-1-4008-7566-5 |page=24 |url=https://books.google.com/books?id=_xTWCgAAQBAJ}} [https://books.google.com/books?id=_xTWCgAAQBAJ&pg=PA24 Extract of page 24]</ref>

Similarly, for the class of [[CW complexes]], the dimension of an object is the largest {{mvar|n}} for which the [[n-skeleton|{{mvar|n}}-skeleton]] is nontrivial. Intuitively, this can be described as follows: if the original space can be [[homotopy|continuously deformed]] into a collection of [[simplex|higher-dimensional triangles]] joined at their faces with a complicated surface, then the dimension of the object is the dimension of those triangles.{{citation needed|date=June 2018}}

{{See also|dimension of a scheme}}

===Hausdorff dimension===
The [[Hausdorff dimension]] is useful for studying structurally complicated sets, especially [[fractal]]s. The Hausdorff dimension is defined for all [[metric space]]s and, unlike the dimensions considered above, can also have non-integer real values.<ref name="Hausdorff dimension">[http://math.bu.edu/DYSYS/chaos-game/node6.html Fractal Dimension] {{webarchive|url=https://web.archive.org/web/20061027003440/http://math.bu.edu/DYSYS/chaos-game/node6.html |date=2006-10-27 }}, Boston University Department of Mathematics and Statistics</ref> The [[box-counting dimension|box dimension]] or [[Minkowski dimension]] is a variant of the same idea. In general, there exist more definitions of [[fractal dimension]]s that work for highly irregular sets and attain non-integer positive real values.

===Hilbert spaces===
Every [[Hilbert space]] admits an [[orthonormal basis]], and any two such bases for a particular space 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.