Atwo-dimensional space is amathematical space with twodimensions, meaningpoints have twodegrees of freedom: their locations can belocally described with twocoordinates or they can move in two independent directions. Common two-dimensional spaces are often calledplanes, or, more generally,surfaces. These include analogs to physical spaces, like flat planes, and curved surfaces like spheres, cylinders, and cones, which can be infinite or finite. Some two-dimensional mathematical spaces are not used to represent physical positions, like anaffine plane orcomplex plane.
The most basic example is the flatEuclidean plane, an idealization of a flat surface inphysical space such as a sheet of paper or a chalkboard. On the Euclidean plane, any two points can be joined by a uniquestraight line along which thedistance can be measured. The space is flat because any two linestransversed by a third lineperpendicular to both of them areparallel, meaning they neverintersect and stay at uniform distance from each-other.
Two-dimensional spaces can also becurved, for example thesphere andhyperbolic plane, sufficiently small portions of which appear like the flat plane, but on which straight lines which are locally parallel do not stay equidistant from each-other but eventually converge or diverge, respectively. Two-dimensional spaces with a locally Euclidean concept of distance but which can have non-uniformcurvature are calledRiemannian surfaces. (Not to be confused withRiemann surfaces.) Some surfaces areembedded inthree-dimensional Euclidean space or some otherambient space, and inherit their structure from it; for example,ruled surfaces such as thecylinder andcone contain a straight line through each point, andminimal surfaces locally minimize their area, as is done physically bysoap films.
Lorentzian surfaces look locally like a two-dimensional slice ofrelativisticspacetime with one spatial and one time dimension; constant-curvature examples are the flat Lorentzian plane (a two-dimensional subspace ofMinkowski space) and the curvedde Sitter andanti-de Sitter planes.
Other types of mathematical planes and surfaces modify or do away with the structures defining the Euclidean plane. For example, theaffine plane has a notion of parallel lines but no notion of distance; however,signed areas can be meaningfully compared, as they can in a more generalsymplectic surface. Theprojective plane does away with both distance and parallelism. A two-dimensionalmetric space has some concept of distance but it need not match the Euclidean version. Atopological surface can be stretched, twisted, or bent without changing its essential properties. Analgebraic surface is a two-dimensional set of solutions of asystem of polynomial equations.
Some mathematical spaces have additional arithmetical structure associated with their points. Avector plane is an affine plane whose points, calledvectors, include a special designatedorigin or zero vector. Vectors can be added together orscaled by a number, and optionally have a Euclidean, Lorentzian, or Galilean concept of distance. Thecomplex plane,hyperbolic number plane, anddual number plane each have points which are considered numbers themselves, and can be added and multiplied. ARiemann surface orLorentz surface appear locally like the complex plane or hyperbolic number plane, respectively.
Mathematical spaces are often defined or represented using numbers rather thangeometric axioms. One of the most fundamental two-dimensional spaces is thereal coordinate space, denoted consisting of pairs ofreal-number coordinates. Sometimes the space represents arbitrary quantities rather than geometric positions, as in theparameter space of a mathematical model or theconfiguration space of a physical system.
More generally, other types of numbers can be used as coordinates. Thecomplex plane is two-dimensional when considered to be formed from real-number coordinates, butone-dimensional in terms ofcomplex-number coordinates. A two-dimensional complex space – such as the two-dimensionalcomplex coordinate space, thecomplex projective plane, or acomplex surface – has two complex dimensions, which can alternately be represented using four real dimensions. Atwo-dimensional lattice is an infinite grid of points which can be represented usinginteger coordinates. Some two-dimensional spaces, such asfinite planes, have only afinite set of elements.
