onedimensionality

One-dimensionality describes systems or spaces that can be described by a single independent coordinate. In Euclidean geometry, a line has length but no width or height, and in topology a one-dimensional manifold is locally homeomorphic to an open interval of the real line. More generally, a space is said to be one-dimensional if its topological dimension equals one.

Common examples include lines and smooth curves in the plane or space, wires, and strings. In physics, time is treated as a one-dimensional parameter, and the worldline of a particle in spacetime is a one-dimensional object embedded in four-dimensional space. One-dimensional models are often used to approximate extended systems when transverse structure is negligible.

Mathematically, one-dimensional objects can be described by a parameter t with coordinates given by r(t). A 1D manifold is locally modeled on R, and the length element ds induces a one-dimensional measure. For smooth curves, the length is the integral of |dr/dt| dt. The topological and Hausdorff dimensions agree and equal one for regular curves, whereas fractal curves can have Hausdorff dimension greater than one.

Applications and limitations: one-dimensional analysis involves functions of a single variable and single-variable calculus. In physics and engineering, 1D models simplify problems by reducing degrees of freedom. However, a truly one-dimensional object cannot enclose area or volume, and the embedding of 1D objects in higher-dimensional spaces can produce richer geometries and interactions.