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Dimension From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Dimension (disambiguation). 2-dimensional renderings (i.e., flat drawings) of a 0-dimensional point, a 1-dimensional line segment, a 2-dimensional square, a 3-dimensional cube, and a 4-dimensional tesseractIn common usage, a dimension (Latin, "measured out") is a parameter or measurement used to describe some relevant characteristic of an object. The most commonly used dimensions are the parameters describing the size of an object: length, width, and height, but dimensions can also be other physical parameters such as the mass and electric charge of an object, or even, in a context where cost is relevant, an economic parameter such as its price.

In mathematics, dimensions are the parameters required to describe the position 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. A system relating the dimensions to the positions in the space is called a coordinate system, and the dimensions are then called coordinates. 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, say) requires another dimension (altitude), hence the position of the aircraft can be rendered in a three-dimensional space (3D). A collection of such positions determines a trajectory in space. Adding the three Euler angles to the three positional parameters, for a total of six dimensions, allows the instantaneous six degrees of freedom—position and orientation—of the aircraft (or any rigid body) to be specified.

Time can be added as a further dimension: as a 3rd to a 2D space, or a 4th dimension to a 3D space. An aircraft's average "speed" between any two positions may be calculated from a comparison of the two positions together with their times. The concept of spacetime as used in relativity theory is four-dimensional.

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 Extraterrestrial places often supposed to exist outside the known physical universe and that are used as sources of 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 and where 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.

Contents [hide] 1 Physical dimensions 1.1 Spatial dimensions 1.2 Time 1.3 Additional dimensions 1.4 Units 2 Mathematical dimensions 2.1 Hamel dimension 2.2 Manifolds 2.3 Lebesgue covering dimension 2.4 Inductive dimension 2.5 Hausdorff dimension 2.6 Hilbert spaces 2.7 Krull dimension of commutative rings 2.8 Negative dimension 3 Science fiction 4 Penrose's singularity theorem 5 More dimensions 6 See also 6.1 Degrees of freedom 6.2 Other 7 Further reading 8 References

[edit] Physical dimensions

[edit] Spatial dimensions A three dimensional Cartesian coordinate system.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.)

[edit] 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.

Our movement through time is comparable to our movement through space in the sense that 1s through time is equivalent to 1 light second through space, using whatever value for light seconds are applicable in the current frame of reference

When speed through time and velocity through space are plotted on a graph as shown (relative to your speed, your speed is 0m/s, so the speed through time is the normal speed therefore, you perceive time passes normally. to find the value of time dilation in anything moving relative to you (to find the value of 1 light second at a certain velocity):

v=speed (through space) t=speed (through time, also c in that other frame of reference) c=speed of light (in your frame of reference)

v²+t²=c²

c²-v²=t²

divide by c² to find ratio between values of c in your frame of reference and the other frame of reference

1-v²/c²=t²/c²

(1-v²/c²)½=t/c

that is the equation for γ which is the ratio between the speed of light in another frame of reference and the speed of light in your frame of reference. γ is the most important value in relativity.