Dimension counts how many coordinates a point needs
Informally, a space’s dimension is the fewest coordinates required to fix any point—one for a line, two for a plane, three for ordinary space. Spacetime’s four dimensions fuse events not absolutely simultaneous. A circle still has dimension one even when drawn in 3-D.
In mathematics, dimension counts the degrees of freedom of a point moving on an object: zero for a point, one for a line, where motion runs only forward or back, two for a plane. It belongs to the object itself rather than to its surroundings. A circle is one-dimensional because a single signed distance along it fixes any point, even though no curve other than a straight line fits in fewer than two dimensions, and a sphere's surface stays two-dimensional inside three-dimensional space; latitude and longitude are enough to locate a place on it.
Generalising beyond ordinary space forces a choice of definition. One approach notes that covering a ball in n-dimensional space with tiny balls of radius epsilon takes on the order of epsilon to the minus n of them, leading to the Minkowski and Hausdorff dimensions. Another observes that a ball's boundary looks locally like a space one dimension lower, giving the inductive dimension. These measures agree for Euclidean space but diverge elsewhere. For a vector space, dimension is the number of vectors in a basis, called the Hamel or algebraic dimension.
Physics stretches the idea. Classical mechanics kept absolute space and absolute time apart, but electromagnetism required four-dimensional spacetime, where the placing of events depends on an observer's motion. Minkowski space approximates a universe without gravity, while general relativity uses curved pseudo-Riemannian manifolds. Superstring theory works in 10 dimensions, supergravity and M-theory in 11, and the states of quantum mechanics live in an infinite-dimensional function space. Configuration spaces in Lagrangian and Hamiltonian mechanics are likewise abstract spaces with many dimensions.
Higher dimensions trace back to René Descartes, but real progress came in the nineteenth century. Hamilton's quaternions and John T. Graves's octonions in 1843, Ludwig Schläfli's 1852 work on multiple continuity and Bernhard Riemann's 1854 habilitation lecture launched higher-dimensional geometry, alongside contributions from Arthur Cayley. Mathematicians say a tesseract has dimension 4 rather than that it has four dimensions.
Source: Dimension
