4polytope

A four-dimensional polytope, or 4-polytope, is a geometric object in four-dimensional Euclidean space bounded by three-dimensional polyhedral cells. Its boundary consists of 3D cells joined along shared polygonal faces, with lower-dimensional faces corresponding to vertices, edges, and 2D faces. Like their 2D and 3D counterparts, 4-polytopes can be convex or non-convex, and their combinatorial structure is studied through counts of k-faces for k = 0 to 3.

Regular 4-polytopes are the most symmetric four-dimensional polytopes. In Euclidean 4-space there are six convex regular 4-polytopes, described by Schläfli symbols {p,q,r}. Each cell is a regular {p,q} polyhedron, and the vertex figure is {q,r}. The six convex regular 4-polytopes are the 5-cell {3,3,3}; the tesseract or hypercube {4,3,3}; the 16-cell {3,3,4}; the 24-cell {3,4,3}; the 120-cell {5,3,3}; and the 600-cell {3,3,5}. The 5-cell and 16-cell form a dual pair, as do the tesseract and the 24-cell, while the 120-cell and 600-cell are dual to each other.

Typical combinatorial data for the regular 4-polytopes (vertices V, edges E, 2D faces F, and 3D cells C) include: 5-cell has V=5, E=10, F=10, C=5; tesseract has V=16, E=32, F=24, C=8; 16-cell has V=8, E=24, F=32, C=16; 24-cell has V=24, E=96, F=96, C=24; 120-cell has V=600, E=1200, F=720, C=120; 600-cell has V=120, E=720, F=1200, C=600. These figures illustrate the richness of 4-polytopes and their dual relationships.