4cube

4cube is the name commonly used in geometry to refer to the four-dimensional hypercube, also called the tesseract. It is the four-dimensional analogue of a cube and is the regular polytope with Schläfli symbol {4,3,3}. In four dimensions it has 16 vertices, 32 edges, 24 square faces, and 8 cubic cells.

One way to describe it is as a figure formed by connecting corresponding vertices of two congruent cubes in four-dimensional space, or equivalently as the set of points with coordinates (±1, ±1, ±1, ±1). These vertices are linked by edges that differ in exactly one coordinate sign, yielding 32 edges. The 24 square faces are formed by the 2D faces of the four-cubes, and the 8 cubic cells are the 3D “slices” that bound the figure.

In projections to three dimensions, a common visualization shows two nested cubes connected by 16 edges, producing a wireframe that resembles a cube within a cube.

Historically, the tesseract was described by Ludwig Schläfli in the 1850s and has since become a standard object in higher-dimensional geometry, popular in textbooks and computer graphics as a tool for illustrating four-dimensional structure.

Outside mathematics, the term 4cube is sometimes used as a brand name or project title for products, software, or design initiatives; such uses are context-dependent and not part of a single canonical definition.