Multidimensional

Multidimensional refers to something that involves more than one dimension. In everyday usage, it can describe complex systems with multiple, interrelated aspects. In mathematics, a space is called multidimensional if it has more than three dimensions. The common case is n-dimensional Euclidean space R^n, with n independent coordinate axes. Distances and angles are defined by a metric, and geometric objects generalize to higher dimensions, such as hypercubes and hyperplanes. Multidimensional analysis frequently uses linear algebra and vector representations.

In statistics and data analysis, data are often described by many attributes, forming points in a high-dimensional feature space. High dimensionality can complicate visualization and modeling and leads to the curse of dimensionality, where sampling requirements grow exponentially. Dimensionality reduction techniques, including principal component analysis, t-distributed stochastic neighbor embedding, and multidimensional scaling, aim to project data into lower dimensions while preserving structure. Multidimensional arrays and tensors are standard data structures in computing, enabling representation of images, videos, and scientific data.

In physics, spacetime is four-dimensional, with three spatial and one temporal dimension. Some theoretical models posit additional spatial dimensions that are compactified at small scales, as in certain string theory frameworks. Multidimensional concepts also appear in psychology and social sciences, where constructs are modeled as having multiple interacting dimensions.

Visualization of multidimensional spaces is inherently challenging beyond three dimensions, often relying on projections, slices, or color and motion cues to convey structure.