4DHypercube

The 4DHypercube, also called the tesseract, is the four-dimensional analogue of the cube. In four-dimensional Euclidean space it is a regular polytope with Schläfli symbol {4,3,3}. It consists of eight cubic cells, 24 square faces, 32 edges, and 16 vertices.

A convenient coordinate model places the vertices at all points with coordinates (±1, ±1, ±1, ±1). Two vertices are joined by an edge if and only if their coordinates differ in exactly one sign. This yields 32 edges, 24 square faces, and 8 cube cells, arranged as two parallel cubes connected by 16 edges.

Construction and visualization often involve two congruent cubes in parallel 3D hyperplanes, with corresponding vertices connected. In projections to 3D or 2D, the 4DHypercube is typically drawn as a larger outer cube enclosing a smaller inner cube, with lines connecting corresponding vertices.

Symmetry and dual: The 4DHypercube has the full symmetry group of order 384 (the four-dimensional hyperoctahedral group); the orientation-preserving rotational subgroup has order 192. Its dual polytope is the 16-cell.

Applications and context: As a fundamental example of a 4D regular polytope, the 4DHypercube appears in mathematics, computer graphics, and education to illustrate higher-dimensional geometry. It serves as a mental model for four-dimensional space and is often used in discussions of polytope theory and visualization.

History: Regular four-polytopes were studied in the 19th century by mathematicians such as Ludwig Schläfli; the tesseract represents the four-dimensional analogue of the cube and is one of the six convex regular 4-polytopes.