Cubical

Cubical is an adjective meaning related to, or shaped like, a cube. In geometry, a cube is a regular polyhedron with six square faces, twelve edges, and eight vertices. Objects described as cubical are often built from cubic elements or exhibit symmetry consistent with a cube.

In mathematics, the term is used in several contexts. A cubical complex is a combinatorial structure used in algebraic topology that is assembled from cubes of various dimensions by gluing their faces. It serves as an alternative to simplicial complexes for modeling spaces and defining (co)homology theories. The associated boolean hypercube or n-cube, denoted as {0,1}^n, is the n-dimensional cube; these cubical sets provide a convenient framework for discretized geometry and computation in some areas of topology and data analysis.

In type theory and related logic, "cubical" describes a family of ideas collectively known as cubical type theory and cubical sets. Here, a special interval object I with endpoints 0 and 1, together with face and degeneracy maps, enables constructive interpretations of paths and homotopies. This framework supports computational interpretations of univalence and higher inductive types, and has influenced contemporary implementations of homotopy type theory.

Outside pure mathematics, cubical concepts appear in computer graphics, digital imaging, and design wherever cube-like discretization or cubic tessellations are used, such as voxel-based representations and cubical tilings of space.