Dimension

In

In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.

The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the

In mathematics

In

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 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.

The dimension of

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" or: 4D.

Although the notion of higher dimensions goes back to

The dimension of a

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

Manifolds

The uniquely defined dimension of every

For connected differentiable manifolds, the dimension is also the dimension of the 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 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, in which four different proof methods are applied.

Complex dimension

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

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 spherical surface, when given a complex metric, becomes a Riemann sphere of one complex dimension.^[3]

Varieties

The dimension of an

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 ${\displaystyle V_{0}\subsetneq V_{1}\subsetneq \cdots \subsetneq V_{d}}$ of sub-varieties of the given algebraic set (the length of such a chain is the number of "${\displaystyle \subsetneq }$").

Each variety can be considered as an

The Krull dimension of a commutative ring is the maximal length of chains of prime ideals in it, a chain of length n being a sequence ${\displaystyle {\mathcal {P}}_{0}\subsetneq {\mathcal {P}}_{1}\subsetneq \cdots \subsetneq {\mathcal {P}}_{n}}$ 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

An inductive dimension may be defined inductively as follows. Consider a 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 (n + 1)-dimensional object by dragging an 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, (n + 1)-dimensional balls have n-dimensional 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.^[5]

Similarly, for the class of

Hausdorff dimension

The

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 distance. 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. (See Space and Cartesian coordinate system.)

Number of dimensions	Example co-ordinate systems
1	Number line Angle
2	Cartesian (two-dimensional) Polar Latitude and longitude
3	Cartesian (three-dimensional) Cylindrical Spherical

Time

A temporal dimension, or time dimension, is a dimension of time. Time is often referred to as the "fourth dimension" for this reason, but that is not to imply that it is a spatial dimension^{[citation needed]}. A temporal dimension is 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 we cannot move freely in time but subjectively move in one direction.

The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it. 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 (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy).

The best-known treatment of time as a dimension is

In addition to small and curled up extra dimensions, there may be extra dimensions that instead are not apparent because the matter associated with our visible universe is localized on a (3 + 1)-dimensional subspace. Thus the extra dimensions need not be small and compact but may be large extra dimensions. D-branes are dynamical extended objects of various dimensionalities predicted by string theory that could play this role. They have the property that open string excitations, which are associated with gauge interactions, are confined to the brane by their endpoints, whereas the closed strings that mediate the gravitational interaction are free to propagate into the whole spacetime, or "the bulk". This could be related to why gravity is exponentially weaker than the other forces, as it effectively dilutes itself as it propagates into a higher-dimensional volume.

Some aspects of brane physics have been applied to cosmology. For example, brane gas cosmology^[10][11] attempts to explain why there are three dimensions of space using topological and thermodynamic considerations. According to this idea it would be since three is the largest number of spatial dimensions in which strings can generically intersect. If initially there are many windings of strings around compact dimensions, space could only expand to macroscopic sizes once these windings are eliminated, which requires oppositely wound strings to find each other and annihilate. But strings can only find each other to annihilate at a meaningful rate in three dimensions, so it follows that only three dimensions of space are allowed to grow large given this kind of initial configuration.

Extra dimensions are said to be

In computer graphics and spatial data

Several types of digital systems are based on the storage, analysis, and visualization of geometric shapes, including

• Point (0-dimensional), a single coordinate in a Cartesian coordinate system.
• Line or Polyline (1-dimensional), usually represented as an ordered list of points sampled from a continuous line, whereupon the software is expected to interpolate the intervening shape of the line as straight or curved line segments.
• Polygon (2-dimensional), usually represented as a line that closes at its endpoints, representing the boundary of a two-dimensional region. The software is expected to use this boundary to partition 2-dimensional space into an interior and exterior.
• Surface (3-dimensional), represented using a variety of strategies, such as a polyhedron consisting of connected polygon faces. The software is expected to use this surface to partition 3-dimensional space into an interior and exterior.

Frequently in these systems, especially GIS and Cartography, a representation of a real-world phenomena may have a different (usually lower) dimension than the phenomenon being represented. For example, a city (a two-dimensional region) may be represented as a point, or a road (a three-dimensional volume of material) may be represented as a line. This dimensional generalization correlates with tendencies in spatial cognition. For example, asking the distance between two cities presumes a conceptual model of the cities as points, while giving directions involving travel "up," "down," or "along" a road imply a one-dimensional conceptual model. This is frequently done for purposes of data efficiency, visual simplicity, or cognitive efficiency, and is acceptable if the distinction between the representation and the represented is understood, but can cause confusion if information users assume that the digital shape is a perfect representation of reality (i.e., believing that roads really are lines).

More dimensions

List of topics by dimension

See also

References

Further reading

• Murty, Katta G. (2014). "1. Systems of Simultaneous Linear Equations" (PDF). Computational and Algorithmic Linear Algebra and n-Dimensional Geometry. World Scientific Publishing.
  ISBN 978-981-4366-62-5
  .
• Abbott, Edwin A. (1884). Flatland: A Romance of Many Dimensions
  . London: Seely & Co.
  • —. Flatland: ... Project Gutenberg.
  • —;
    ISBN 978-0-7867-2183-2
    .
• ISBN 978-0-7167-6015-3
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• ISBN 978-0-19-992381-6
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• ISBN 978-0-486-77978-2. Google preview
• ISBN 978-0-19-286189-4
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• ISBN 978-0-670-03395-9
  .

External links

Wikiquote has quotations related to Dimension.

Wikimedia Commons has media related to Dimensions.

Wikisource has the text of the 1911 Encyclopædia Britannica article "Dimension".

• Copeland, Ed (2009). "Extra Dimensions". Sixty Symbols. Brady Haran for the University of Nottingham.