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The dimension of an object is a topological measure of the size of its covering properties. Roughly speaking, it is the number of coordinates needed to specify a point on the object. For example, a
rectangle
is two-dimensional, while a
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is three-dimensional. The dimension of an object is sometimes also called its "dimensionality."
The prefix "hyper-" is usually used to refer to the four- (and higher-) dimensional analogs of three-dimensional objects, e.g., Mini Woman Pleated Versace Lace Silk Size Versace Embellished Black 42 crepe trimmed Dress qcM3BYX hypercube , hyperplane .
The notion of dimension is important in mathematics because it gives a precise parameterization of the conceptual or visual complexity of any geometric object. In fact, the concept can even be applied to abstract objects which cannot be directly visualized. For example, the notion of time can be considered as one-dimensional, since it can be thought of as consisting of only "now," "before" and "after." Since "before" and "after," regardless of how far back or how far into the future they are, are extensions, time is like a line, a one-dimensional object.
To see how lower and higher dimensions relate to each other, take any geometric object (like a
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,
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,
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,
plane
, etc.), and "drag" it in an opposing direction (drag a
point
to trace out a
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, a
line
to trace out a box, a
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to trace out a
cylinder
, a
disk
to a solid
cylinder
, etc.). The result is an object which is qualitatively "larger" than the previous object, "qualitative" in the sense that, regardless of how you drag the original object, you always trace out an object of the same "qualitative size." The
point
could be made into a straight
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line
, a
circle
, a
helix
, or some other
curve
, but all of these objects are qualitatively of the same dimension. The notion of dimension was invented for the purpose of measuring this "qualitative" topological property.
Finite collections of objects (e.g., points in space) are considered 0-dimensional. Objects that are "dragged" versions of zero-dimensional objects are then called one-dimensional. Similarly, objects which are dragged one-dimensional objects are two-dimensional, and so on. Dimension is formalized in mathematics as the intrinsic dimension of a
topological space
. This dimension is called the
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(also known simply as the topological dimension). The archetypal example is
Euclidean
-space , which has topological dimension . The basic ideas leading up to this result (including the
dimension invariance theorem
,
domain invariance theorem
, and
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) were developed by Poincaré, Brouwer, Lebesgue, Urysohn, and Menger.
There are several branchings and extensions of the notion of topological dimension. Implicit in the notion of the
Lebesgue covering dimension
is that dimension, in a sense, is a measure of how an object fills space. If it takes up a lot of room, it is higher dimensional, and if it takes up less room, it is lower dimensional.
Hausdorff dimension
(also called
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) is a fine tuning of this definition that allows notions of objects with dimensions other than
integers
.
Fractals
are objects whose
Hausdorff dimension
is different from their topological dimension.
The concept of dimension is also used in
algebra
, primarily as the dimension of a
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over a
field
. This usage stems from the fact that
vector spaces
over the reals were the first
vector spaces
to be studied, and for them, their topological dimension can be calculated by purely algebraic means as the
cardinal number
of a maximal linearly independent subset. In particular, the dimension of a
subspace
of is equal to the number of
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vectors
needed to generate it (i.e., the number of
vectors
in its
basis
). Given a transformation of ,
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