4DPolytopes

4D polytopes, or polychora, are the four-dimensional analogues of polygons and polyhedra. They exist in four-dimensional Euclidean space and are bounded by three-dimensional cells. A polychoron has vertices, edges, faces (2D polygons), and cells (3D polyhedra); its facets are the 3D figures that bound it, and the whole solid is described by its vertex figure and symmetry.

The six convex regular 4-polytopes are: the 5-cell or 4-simplex with five tetrahedral cells; the 8-cell or tesseract with eight cubic cells; the 16-cell with sixteen tetrahedral cells; the 24-cell with twenty-four octahedral cells; the 120-cell with 120 dodecahedral cells; and the 600-cell with 600 tetrahedral cells. Schläfli symbols are {3,3,3}, {4,3,3}, {3,3,4}, {3,4,3}, {5,3,3}, and {3,3,5}, respectively. Duality pairs relate these polychora: the 8-cell and 16-cell are duals; the 120-cell and 600-cell are duals; the 5-cell and 24-cell are self-dual.

Beyond the regular ones, there exist uniform (Archimedean-like) 4-polytopes, formed by truncations and cantellations of regular polychora, with uniform vertex figures and regular facets. The study of polychora involves symmetry groups of higher dimensional space, and their projections and cross-sections yield familiar 3D polyhedra when viewed from within or projected to three dimensions.