In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star regular 4-polytopes, giving a total of sixteen.
The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures.
Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F - E + V 2). That excludes cells and vertex figures such as the great dodecahedron and small stellated dodecahedron .
Edmund Hess (1843 - 1903) published the complete list in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder.
The existence of a regular 4-polytope
\{p,q,r\}
\{p,q\},\{q,r\}
| \sin | \pi | \sin |
| p |
| \pi | |
| r |
>\cos
| \pi | |
| q |
to ensure that the cells meet to form a closed 3-surface.
The six convex and ten star polytopes described are the only solutions to these constraints.
There are four nonconvex Schläfli symbols that have valid cells and vertex figures, and pass the dihedral test, but fail to produce finite figures:,,, .
The regular convex 4-polytopes are the four-dimensional analogues of the Platonic solids in three dimensions and the convex regular polygons in two dimensions.
Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and size. These are fitted together along their respective faces (face-to-face) in a regular fashion, forming the surface of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of the earth is a closed, curved 2-dimensional space).
Like their 3-dimensional analogues, the convex regular 4-polytopes can be naturally ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content within the same radius. The 4-simplex (5-cell) has the smallest content, and the 120-cell has the largest.
The following table lists some properties of the six convex regular 4-polytopes. The symmetry groups of these 4-polytopes are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group.
| Names | Image | Family | Dual | Symmetry group | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| n-simplex (An family) | 5 | 10 | self-dual | A4 [3,3,3] | 120 | ||||||
| n-orthoplex (Bn family) | 8 | 24 | 8-cell | B4 [4,3,3] | 384 | ||||||
| hypercube n-cube (Bn family) | 16 | 32 | 16-cell | ||||||||
| Fn family | 24 | 96 | self-dual | F4 [3,4,3] | 1152 | ||||||
| n-pentagonal polytope (Hn family) | 120 | 720 | 120-cell | H4 [5,3,3] | 14400 | ||||||
| n-pentagonal polytope (Hn family) | 600 | 1200 | 600-cell | ||||||||
John Conway advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD).
Norman Johnson advocated the names n-cell, or pentachoron, hexadecachoron, tesseract or octachoron, icositetrachoron, hexacosichoron, and hecatonicosachoron (or dodecacontachoron), coining the term polychoron being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the Greek roots poly ("many") and choros ("room" or "space").[1] [2]
The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analogue of Euler's polyhedral formula:
N0-N1+N2-N3=0
The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.[3]
A regular 4-polytope can be completely described as a configuration matrix containing counts of its component elements. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers (upper left to lower right) say how many of each element occur in the whole 4-polytope. The non-diagonal numbers say how many of the column's element occur in or at the row's element. For example, there are 2 vertices in each edge (each edge has 2 vertices), and 2 cells meet at each face (each face belongs to 2 cells), in any regular 4-polytope. The configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.[4]
The following table shows some 2-dimensional projections of these 4-polytopes. Various other visualizations can be found in the external links below. The Coxeter-Dynkin diagram graphs are also given below the Schläfli symbol.
The Schläfli - Hess 4-polytopes are the complete set of 10 regular self-intersecting star polychora (four-dimensional polytopes).[5]