twodimensionality

Two-dimensionality refers to the property of a space or system that requires two independent coordinates to specify a point. In mathematics, a two-dimensional space can be described by two coordinates, as in the Euclidean plane R^2, whose points are pairs (x, y). The standard distance between points is d((x1,y1),(x2,y2)) = sqrt((x2−x1)^2 + (y2−y1)^2). A two-dimensional space is locally flat and follows the familiar rules of plane geometry.

More generally, two-dimensionality is the attribute of a two-manifold: a topological space in which every point has a neighborhood homeomorphic to the plane. Surfaces embedded in higher dimensions, such as the surface of a sphere or a torus, are two-dimensional manifolds even though they exist in three (or more) dimensions.

Common representations of two-dimensional spaces include Cartesian coordinates (x, y) and polar coordinates (r, θ). Functions of two variables, z = f(x, y), define two-dimensional surfaces in three-dimensional space, illustrating how a two-dimensional domain can generate a three-dimensional graph.

Two-dimensionality has wide applications. In physics, electrons confined to a plane form a two-dimensional system with distinctive phenomena. In computer graphics, maps, and data visualization, two-dimensional representations are foundational. Planar maps and geographic information systems rely on 2D coordinates to model location, while engineering often uses two-dimensional sketches and simulations.

Two-dimensional spaces relate to higher dimensions as embedded or projected objects. Depth is either abstract or not represented in pure 2D models, and projections from three to two dimensions can introduce distortions. Some materials, such as graphene, are effectively two-dimensional at atomic scales.