Regular 4-polytope

In mathematics, a regular 4-polytope 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.

Regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, although the full set were not discovered until later.

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.

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 six convex and ten star polytopes described are the only solutions to these constraints.

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.

Five of the six are clearly analogues of the five corresponding Platonic solids. The sixth, the 24-cell, has no regular analogue in three dimensions. However, there exists a pair of irregular solids, the cuboctahedron and its dual the rhombic dodecahedron, which are partial analogues to the 24-cell (in complementary ways). Together they can be seen as the three-dimensional analogue of the 24-cell.

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 in a regular fashion.

The following tables 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.

John Conway advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD).[1]

Norman Johnson advocated the names n-cell, or pentachoron, tesseract or octachoron, hexadecachoron, icositetrachoron, hecatonicosachoron (or dodecacontachoron), and hexacosichoron, 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").[2][3]

The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analogue of Euler's polyhedral formula:

where Nk denotes the number of k-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).

The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.[4]

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. Notice that the configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.[5][6]

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.

Their names given here were given by John Conway, extending Cayley's names for the Kepler–Poinsot polyhedra: along with stellated and great, he adds a grand modifier. Conway offered these operational definitions:

All ten polychora have [3,3,5] (H4) hexacosichoric symmetry. They are generated from 6 related Goursat tetrahedra rational-order symmetry groups: [3,5,5/2], [5,5/2,5], [5,3,5/2], [5/2,5,5/2], [5,5/2,3], and [3,3,5/2].

Each group has 2 regular star-polychora, except for two groups which are self-dual, having only one. So there are 4 dual-pairs and 2 self-dual forms among the ten regular star polychora.

The cells (polyhedra), their faces (polygons), the polygonal edge figures and polyhedral vertex figures are identified by their Schläfli symbols.