User:Jacobolus/2d

A two-dimensional space is a mathematical space with two dimensions, meaning locations of points can be specified with two coordinates. Two-dimensional spaces are often called planes or surfaces.

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. The space is flat because any two lines transversed by a third line perpendicular to both of them are parallel, meaning they never intersect and stay at uniform distance from each-other.

Two-dimensional spaces can also be curved, for example the sphere and hyperbolic plane, sufficiently small portions of which appear like the flat plane, but on which locally parallel straight lines eventually converge or diverge, respectively. Two-dimensional spaces with a locally Euclidean concept of distance but which can have non-uniform curvature are called Riemannian surfaces.

Other types of mathematical planes and surfaces modify or do away with the structures defining the Euclidean plane. For example, the affine plane has a notion of parallel lines but no notion of distance, and the projective plane does away with both. A two-dimensional metric space has some concept of distance but it need not match the Euclidean version. A topological surface can be stretched, twisted, or bent without changing its essential properties. An algebraic surface is a two-dimensional set of solutions of a system of polynomial equations.

Mathematical spaces are often defined or represented using numbers rather than geometric axioms. One of the most fundamental two-dimensional spaces is the real coordinate space, denoted $$\R^2,$$ consisting of pairs of real-number coordinates. Sometimes the space represents arbitrary quantities rather than geometric positions, as in the parameter space of a mathematical model or the configuration space of a physical system.

More generally, other types of numbers can be used as coordinates. The complex plane is two-dimensional when considered to be formed from real-number coordinates, but one-dimensional in terms of complex-number coordinates. A two-dimensional complex space, such as the two-dimensional complex coordinate space, the complex projective plane, or a complex surface has two complex dimensions (which can alternately be represented using four real dimensions). A two-dimensional lattice is an infinite grid of points which can be represented using integer coordinates. Some two-dimensional spaces, such as finite planes, have only a finite set of elements.