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The dimension of a physical quantity is more fundamental than some scale or unit used to express the amount of that physical quantity. For example, mass is a dimension, while the kilogram is a particular reference quantity chosen to express a quantity of mass. The choice of unit is arbitrary, and its choice is often based on historical precedent.
For example, the dimension of a point is zero; the dimension of a line is one, as a point can move on a line in only one direction (or its opposite); the dimension of a plane is two, etc. 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.
The most basic example is the flat Euclidean plane, an idealization of a flat surface in physical space such as a sheet of paper or a chalkboard. On the Euclidean plane, any two points can be joined by a unique straight line along which the distance can be measured.
Scale analysis rules as follows: Rule1-First step in scale analysis is to define the domain of extent in which we apply scale analysis. Any scale analysis of a flow region that is not uniquely defined is not valid. Rule2-One equation constitutes an equivalence between the scales of two dominant terms appearing in the equation. For example,
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
Planck length; typical scale of hypothetical loop quantum gravity or size of a hypothetical string and of branes; according to string theory, lengths smaller than this do not make any physical sense. [1] Quantum foam is thought to exist at this scale. 10 −24: 1 yoctometer 142 ym Effective cross section radius of 1 MeV neutrinos [2] 10 −21
An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points (1, 0) and (-1, 0) can be joined by paths of length arbitrarily close to 2, but not by a path of length 2. An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between ...
The atomic length scale is ℓ a ~ 10 −10 m and is given by the size of hydrogen atom (i.e., the Bohr radius, approximately 53 pm).; The length scale for the strong interactions (or the one derived from QCD through dimensional transmutation) is around ℓ s ~ 10 −15 m, and the "radii" of strongly interacting particles (such as the proton) are roughly comparable.