4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.
| {3,3,3} | {3,3,4} | {4,3,3} |
|---|---|---|
5-cell Pentatope 4-simplex |
16-cell Orthoplex 4-orthoplex |
8-cell Tesseract 4-cube |
| {3,4,3} | {3,3,5} | {5,3,3} |
24-cell Octaplex |
600-cell Tetraplex |
120-cell Dodecaplex |
The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron.
Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space.