Encyclopedia
In common usage, a
dimension is a
parameter or measurement required to define the characteristics of an object—
i.e. length, width, and height or
size and shape.
In
mathematics,
dimensions are the parameters required to describe the position and relevant characteristics of any object within a conceptual
space —where the
dimensions of a space are the total number of different parameters used for all possible objects considered in the model.
Generalizations of this concept are possible and different fields of study will define their spaces by their own relevant dimensions, and use these spaces as frameworks upon which all other study is based. In specialized contexts,
units of measurement may sometimes be "dimensions"—
meters or
feet in
geographical space models, or
cost and
price in models of a local
economy.
For example, locating a point on a plane requires two parameters —
latitude and longitude. The corresponding space has therefore two
dimensions, its
dimension is two, and this space is said to be
2-dimensional .
Locating the exact position of an aircraft in flight requires another dimension , hence the position of the aircraft can be rendered in a three-dimensional space .
If
time is added as a 3rd or 4th dimension , then the aircraft's estimated "speed" may be calculated from a comparison between the times associated with any two positions. For common uses, simply using "speed" is a useful way of
condensing the more abstract
time dimension, even if "speed" is not a
dimension, but rather a
calculation based on two dimensions. Adding the three
Euler angles, for a total 6 dimensions, allows the current degrees of freedom —orientation and trajectory —of the aircraft to be known.
Theoretical physics often experiments with dimensions - adding more, or changing their properties - in order to describe unusual conceptual models of space, in order to help better describe concept of
quantum mechanics —ie. the 'physics beneath the visible physical world.' This concept has been borrowed in
science fiction as a metaphorical device, where an "alternate dimension" describes
extraterrestrial places, species, and cultures which function in various different and unusual ways from human culture.
The physical dimensions are the parameters required to answer to the question where and when happened or will happen some event; for instance: When did
Napoleon die? — On the 5 May 1821 at
Saint Helena . They play a fundamental role in our perception of the world around us. According to
Immanuel Kant, we actually do not perceive them but they form the frame in which we perceive events; they form the a priori background in which events are perceived.
Spatial dimensions
Classical physics theories describe three physical dimensions: from a particular point in space, the basic directions in which we can move are up/down, left/right, and forward/backward. 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 diagonally upward and forward is just as the name of the direction implies; i.e., moving in a linear combination of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions.
Time
Time is often referred to as the "
fourth dimension." It is, in essence, one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that movement seems to occur at a fixed rate and in one direction.
The equations used by physics to model reality often do not treat time in the same way that humans perceive it. In particular, the equations of
classical mechanics are
symmetric with respect to time, and equations of
quantum mechanics are typically symmetric if both time and other quantities are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics .
The best-known treatment of time as a dimension is
Poincaré and
Einstein's
special relativity , which treats perceived space and time as parts of a four-dimensional
manifold.
Additional dimensions
Theories such as
string theory predict that the space we live in has in fact many more dimensions , but that the universe measured along these additional dimensions is subatomic in size. As a result, we perceive only the three spatial dimensions that have macroscopic size.
Units
In the physical sciences and in engineering, the
dimension of a physical quantity is the expression of the class of
physical unit that such a quantity is measured against. The dimension of speed, for example, is length divided by time. In the SI system, the dimension is given by the seven exponents of the fundamental quantities. See Dimensional analysis.
Mathematical dimensions
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 section we examine some of the more important mathematical definitions of dimension.
Hamel dimension
For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis.
See Hamel dimension for details.
Manifolds
A connected topological
manifold 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 topology, is characterized 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 conjecture, where four different proof methods are applied.
Lebesgue covering dimension
For any topological space, the Lebesgue covering dimension is defined to be
n if
n is the smallest integer for which the following holds: any open cover has a refinement such that no point is included in more than
n + 1 elements. For manifolds, this coincides with the dimension mentioned above. If no such n exists, then the dimension is infinite.
Inductive dimension
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
n+1-dimensional balls have
n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.
Hausdorff dimension
For sets which are of a complicated structure, especially
fractals, the
Hausdorff dimension is useful. The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values . The upper and lower box dimensions are a variant of the same idea.
Hilbert spaces
Every Hilbert space admits an orthonormal basis, and any two such bases 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.
Krull dimension of commutative rings
The Krull dimension of a commutative ring, named after Wolfgang Krull , is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.
Science fiction
Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence. This usage is derived from the idea that in order to travel to parallel/alternate universes/planes of existence one must travel in a spatial direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth spatial dimension, not the standard ones.
Penrose's singularity theorem
In his book
,
scientist Sir
Roger Penrose explained his singularity theorem. It asserts that all theories that attribute more than three spatial dimensions and one
temporal dimension to the world of experience are unstable. The instabilities that exist in systems of such extra dimensions would result in their rapid collapse into a singularity. For that reason, Penrose wrote, the unification of
gravitation with other forces through extra dimensions cannot occur.
More dimensions
- Dimension of an algebraic variety
- Topological dimension
- Isoperimetric dimension
- Poset dimension
- Metric dimension
- Pointwise dimension
- Lyapunov dimension
- Kaplan-Yorke dimension
- Exterior dimension
- Hurst exponent
- q-dimension; especially:
- Information dimension
- Correlation dimension
See also
Degrees of freedom
- Zero dimensions
- Point
- Zero-dimensional space
- One dimension
- Two dimensions
- Three dimensions
- Four dimensions
- Five dimensions
- Ten, eleven or twenty-six dimensions
- Infinitely many dimensions
- Special relativity
- General relativity
Other
- Dimension and dimension tables
- Dimensional analysis
- Hyperspace
Further reading
- Thomas Banchoff, Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition, Freeman
- Clifford A. Pickover, Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Oxford University Press
- Rudy Rucker , The Fourth Dimension, Houghton-Mifflin
- Edwin A. Abbott,
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- Imagining the Tenth Dimension - Animated and easy explanation: http://www.tenthdimension.com/
