fivedimension

Five-dimension, often written as five-dimensional or fivedimension, refers to a space with five independent coordinates. In mathematics it is the Euclidean space R^5 with coordinates (x1, x2, x3, x4, x5). Distances are defined by the Euclidean metric, so the distance between two points is the square root of the sum of the squared differences along all five coordinates. Geometric objects have higher-dimensional analogues: lines, planes, and spheres generalize to five dimensions, and the unit 5-ball consists of all points with the sum of squares less than or equal to one.

In geometry and topology, five-dimensional spaces illustrate how concepts such as convexity, projection, and cross-sections extend beyond three dimensions. The study of five-dimensional manifolds and their curvature leads to insights applicable to higher-dimensional spaces in general.

In physics, five-dimensional spacetime has been explored as a simple extension of four-dimensional relativity. The original Kaluza-Klein theory adds a compact fifth dimension, often modeled as a small circle, so that the five-dimensional metric contains both gravitational and electromagnetic fields when viewed from four dimensions. Quantum considerations motivated the idea that extra dimensions might be hidden at small scales. More broadly, higher-dimensional frameworks appear in string theory and related models, where additional dimensions may be compactified, warped, or otherwise rendered unobservable at ordinary energies.

Applications of five-dimensional concepts include serving as a tractable setting for testing mathematical techniques, exploring dimensional dependence of physical laws, and providing a stepping stone toward understanding more complex higher-dimensional theories. See also higher-dimensional spaces, Kaluza-Klein theory, and braneworld models.