Dimension

Dimension is a measure of the number of independent coordinates or parameters required to specify a point within a space or object. In mathematics, the dimension of a space is the minimum number of coordinates needed to uniquely determine any point in that space. For example, a line is one-dimensional, a plane two-dimensional, and ordinary three-dimensional space is three-dimensional. More generally, curves and surfaces embedded in higher-dimensional spaces have dimensions equal to their degrees of freedom.

Beyond ordinary Euclidean space, dimensions can be defined for abstract spaces such as manifolds, vector spaces, or topological spaces. A manifold's dimension is the dimension of its tangent space at each point, typically an integer; however, the concept can be local rather than global.

Several notions of dimension exist. Topological dimension (covering dimension) is one way to assign a nonnegative integer to a space; Hausdorff and box-counting (fractal) dimensions may be non-integer and describe scaling behavior of sets. Fractals often exhibit non-integer dimensions, reflecting intricate structure at small scales.

In physics, spacetime combines three spatial dimensions with one temporal dimension, though theories such as relativity treat time as a dimension on equal footing with space, and some theories propose additional spatial dimensions beyond the familiar three, typically compactified at small scales.

In data analysis and machine learning, dimension refers to the number of features or variables in a dataset. High-dimensional spaces pose challenges such as the curse of dimensionality, influencing modeling and visualization strategies.