A dimension

A dimension is an extension in some unique direction or sense; the word comes from the Latin for "measured out." The most common way to think of a dimension is as one of the three spatial dimensions in which we live. Mathematicians and science fiction writers alike have long imagined what it would be like in a world with a different number of spatial dimensions. Speculation has particularly focused on two-dimensional worlds and, to an even greater extent, on the fourth dimension. Time is also thought of as a dimension; indeed, in relativity theory and as a component of spacetime, it is treated almost exactly the same as a dimension of space. The universe may have additional spatial dimensions – a total of 10, 11, or 26 are especially favored – according to some theories of the subatomic world (see string theory and Kaluza-Klein theory), though the additional ones are "curled up" incredibly small and only become important at scales far smaller than those that can be experimentally probed today.

In mathematics, the term "dimension" is used in many different ways. Some of these correspond to the everyday idea of an extension in physical space or to some of the more esoteric meanings in physics. Others are purely abstract and exist only in certain types of theoretical, mathematical space. There are, for example, Hamel dimensions, Lebesgue covering dimensions, and Hilbert spaces. So-called Hausdorff dimensions are used to characterize fractals – mathematical objects that have fractional dimensions – by giving a precise meaning to the idea of how well something, such as an extremely "wriggly" curve or surface, fills up the space in which it is embedded.

Dimensions of physical quantities

As applied to a physical quantity, a dimension is an indication of the role the quantity plays in equations. The dimensions of a mechanical quantity, in terms of mass [M], length [L], and time [T], can be deduced from the units in which it its expressed. Thus velocity, measured in m/s, has dimensions [L]/[T]. Dimensions in this sense are purely conventional, having no real physical significance. This is clear in the case of electromagnetic quantities where dimensions vary according to the units system employed. Dimensions nevertheless find use in dimensional analysis.

Dimensional analysis

Dimensional analysis is the branch of applied mathematics concerned with the analysis of physical problems in terms of dimensions such as mass, length, and time. Its fundamental theorem is that the dimensions of the quantities appearing on opposite sides of an equation are the same.

Scaling the heights

If a 10-meter (33-foot) scale model of a blade of grass, it would promptly collapse. Similarly a flea the size of an elephant would not be able even to stand, let alone jump. This is because the weight of an object, like its volume, increases as the cube of its height, whereas its strength increases only as its square. Many related properties scale differently so it is quite difficult, for example, to calculate what thrust will propel an aircraft from the force needed to sustain a scale model in a wind tunnel. One way out of this difficulty is to think in terms of "dimensionless groups" like Reynold's number, valuable in many problems of gas and fluid flow. This is LVd/η, where L is a length (perhaps of a wing-section), V is a gas velocity, d is its density, and η its viscosity. This combination is a dimensionless ratio – a pure number – having the same value in any units.

Dimensionless ratios

Dimensionless ratios, free from arbitrary units, are fundamental entities. The ratio of the electrical to gravitational force between a proton and an electron, for example, is about 1039. That is approximately the ratio of the diameter of the knowable universe to that of the proton and of the estimated age of the universe to the time light takes to traverse a proton. The square of 1039, 1078, is about the number of particles in the knowable universe. Some cosmologists wonder if this ratio is trying to tell us something.