Regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogs 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.
History
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. That excludes cells and vertex figures as great dodecahedron| and small stellated dodecahedron|.
Edmund Hess 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.
Construction
The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which form its cells and a dihedral angle constraintto 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:,,,.
Regular convex 4-polytopes
The regular convex 4-polytopes are the four-dimensional analogs of the Platonic solids in three dimensions and the convex regular polygons in two dimensions.Five of them may be thought of as close analogs of the Platonic solids. One additional figure, the 24-cell, has no close three-dimensional equivalent.
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.
Properties
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, dodecaplex or polydodecahedron, and tetraplex or polytetrahedron.
Norman Johnson advocated the names n-cell, or pentachoron, tesseract or octachoron, hexadecachoron, icositetrachoron, hecatonicosachoron, and hexacosichoron, coining the term polychoron being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the Greek roots poly and choros.
The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analog of Euler's polyhedral formula:
where Nk denotes the number of k-faces in the polytope.
The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.
As configurations
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 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, and 2 cells meet at each face, in any regular 4-polytope. Notice that the configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.| 5-cell | 16-cell | tesseract | 24-cell | 600-cell | 120-cell |
Visualization
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.Regular star (Schläfli–Hess) 4-polytopes
The Schläfli-Hess 4-polytopes are the complete set of 10 regular self-intersecting star polychora. They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol in which one of the numbers is pentagram|. They are thus analogous to the regular nonconvex Kepler-Poinsot polyhedra, which are in turn analogous to the pentagram.Names
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:- stellation – replaces edges by longer edges in same lines.
- greatening – replaces the faces by large ones in same planes.
- aggrandizement – replaces the cells by large ones in same 3-spaces.
Symmetry
All ten polychora have hexacosichoric symmetry. They are generated from 6 related Goursat tetrahedra rational-order symmetry groups: , , , , , and .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.
Properties
Note:- There are 2 unique vertex arrangements, matching those of the 120-cell and 600-cell.
- There are 4 unique edge arrangements, which are shown as wireframes orthographic projections.
- There are 7 unique face arrangements, shown as solids orthographic projections.
| Name Conway | Orthogonal projection | Schläfli Coxeter | C | F | E | V | Dens. | χ |
| Icosahedral 120-cell polyicosahedron | 120 regular icosahedron| | 1200 Triangle| | 720 Pentagram| | 120 Great dodecahedron| | 4 | 480 | ||
| Small stellated 120-cell stellated polydodecahedron | 120 Small stellated dodecahedron| | 720 Pentagram| | 1200 Triangle| | 120 Dodecahedron| | 4 | −480 | ||
| Great 120-cell great polydodecahedron | 120 Great dodecahedron| | 720 Pentagon| | 720 Pentagon| | 120 Small stellated dodecahedron| | 6 | 0 | ||
| Grand 120-cell grand polydodecahedron | 120 Dodecahedron| | 720 Pentagon| | 720 Pentagram| | 120 Great icosahedron| | 20 | 0 | ||
| Great stellated 120-cell great stellated polydodecahedron | 120 Great stellated dodecahedron| | 720 Pentagram| | 720 Pentagon| | 120 regular icosahedron| | 20 | 0 | ||
| Grand stellated 120-cell grand stellated polydodecahedron | 120 Small stellated dodecahedron| | 720 Pentagram| | 720 Pentagram| | 120 Great dodecahedron| | 66 | 0 | ||
| Great grand 120-cell great grand polydodecahedron | 120 Great dodecahedron| | 720 Pentagon| | 1200 Triangle| | 120 Great stellated dodecahedron| | 76 | −480 | ||
| Great icosahedral 120-cell great polyicosahedron | 120 Great icosahedron| | 1200 Triangle| | 720 Pentagon| | 120 Small stellated dodecahedron| | 76 | 480 | ||
| Grand 600-cell grand polytetrahedron | 600 Tetrahedron| | 1200 Triangle| | 720 Pentagram| | 120 Great icosahedron| | 191 | 0 | ||
| Great grand stellated 120-cell great grand stellated polydodecahedron | 120 Great stellated dodecahedron| | 720 Pentagram| | 1200 Triangle| | 600 Tetrahedron| | 191 | 0 |