Strip a plane down to topology and it becomes an infinite rubber sheet
A mathematical plane is an endless flat surface in two dimensions, but mathematicians keep changing how much structure it carries. Remove distances and straightness and it becomes an idealised rubber sheet; add multiplication of points and it becomes the complex plane, where only two symmetries leave the real line alone.
The plane sits in a ladder of dimensions after the point and the line and before three-dimensional space. When all the work happens in two dimensions, people say the Euclidean plane, meaning the entire space. That version obeys Euclid's parallel postulate. Add points at infinity where parallel lines would meet and you get a projective plane, where any two lines cross exactly once; give the real projective plane a metric and it becomes elliptic. A hyperbolic plane has negative curvature instead.
At the barest level sits the topological plane, equivalent to an open disc. It keeps a sense of which points are close and allows paths, but has no distances and no straight lines, and its symmetries are all continuous one-to-one mappings. It is the basic building block for surfaces in low-dimensional topology and the natural home of planar graphs, including the four colour theorem. Many everyday tasks in geometry, trigonometry, graph theory and graphing also take place in the plane.
Intermediate levels add a little at a time. Seen as an affine space, the plane still has no distances, but lines stay lines and ratios along a line are preserved under its symmetries, which combine translations with invertible linear maps. Differential geometry treats it as a two-dimensional real manifold with a smooth structure, so paths can be differentiable even without a metric.
Going the other way, giving the plane the arithmetic of a field produces the complex plane and all of complex analysis. Viewed as a one-dimensional complex manifold, the complex line, its conformal symmetries shrink to multiplying by a complex number and then translating. The geometry need not be flat either. Stereographic projection, like setting a ball on the floor, removing its top point and projecting from there, gives the plane constant positive curvature, and the same projection is used to map parts of the Earth.
Source: Plane (mathematics)
