Information about Two Dimensional

In common usage, a dimension (Latin, "measured out") 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 dimension of a space is 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 (in that area) 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 (e.g., a city on a map of the Earth) 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 (2D). Locating the exact position of an aircraft in flight (relative to the Earth) requires another dimension (altitude), hence the position of the aircraft can be rendered in a three-dimensional space (3D). 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.

Time can be added as a 3rd or 4th dimension (to a 2D or 3D space, respectively). 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" (as a dimension) is a useful way of condensing (or translating) the more abstract time dimension, even if "speed" is not a dimension, but rather a calculation based on two dimensions. (Actually, it is possible and useful to consider "spaces" with extra dimensions for representing velocity, because it helps solving certain equations.)

Theoretical physicists often experiment with dimensions—adding more, or changing their properties—in order to describe unusual conceptual models of space, in order to help better describe concepts of quantum mechanics—i.e., the 'physics beneath the visible physical world.' This concept has been borrowed in science fiction as a metaphorical device, where an "alternate dimension" (i.e., 'alternate universe' or 'plane of existence') describes places, species, and cultures which function in various different and unusual ways from human culture.

The physical dimensions are the parameters required to answer the question where and when some event happened or will happen; for instance: When did Napoleon die?—On May 5 1821 at Saint Helena (15°56′ S 5°42′ W). 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.

Types of dimensions

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. (See Space and Cartesian coordinate system.)

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 in 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 (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 Poincaré and Einstein's special relativity (and extended to general relativity), which treats perceived space and time as parts of a four-dimensional manifold.

Additional dimensions

Theories such as string theory and M-theory predict that the space in general has in fact 11 dimensions, respectively, but that the universe, when measured along these additional dimensions, is subatomic in size. As a result, we perceive only the three spatial dimensions that have macroscopic size. We as humans can only perceive up to the third dimension while we have knowledge of our travel through the fourth. We cannot, however, see anything past the fourth.

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. Whereas outside of 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."

The rest of this section examines 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 (a second cover where each element is a subset of an element in the first cover) 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.^[1] The box dimension is a variant of the same idea. In general, there exist more definitions of fractal dimensions 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.

Krull dimension of commutative rings

The Krull dimension of a commutative ring, named after Wolfgang Krull (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

Negative dimension

The negative (fractal) dimension is introduced by Benoit Mandelbrot, in which, when it is positive gives the known definition, and when it is negative measures the degree of "emptiness" of empty sets.^[2]

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 (or higher) 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

See also

Degrees of freedom

• Zero dimensions
• Point
• Zero-dimensional space
• One dimension
• Line
• Two dimensions
• 2D geometric models
• 2D computer graphics
• Three dimensions
• 3D computer graphics
• 3-D films and video
• Stereoscopy (3-D imaging)
• Four dimensions
• Time (4th dimension)
• Fourth spatial dimension
• Tesseract (four dimensional shapes)
• Five dimensions
• Kaluza-Klein theory
• Fifth dimension
• Ten, eleven or twenty-six dimensions
• String theory
• M-theory
• Why 10 dimensions?
• Calabi-Yau spaces
• Infinitely many dimensions
• Hilbert space
• Special relativity
• General relativity

Other

• Dimension (data warehouse) and dimension tables
• Dimensional analysis
• Hyperspace (aka. subspace)

Further reading

• Thomas Banchoff, (1996) Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition, Freeman
• Clifford A. Pickover, (1999) Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Oxford University Press
• Rudy Rucker, (1984) The Fourth Dimension, Houghton-Mifflin
• Dimensionality, University of Winnipeg

References