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.

- History
- Construction
- Regular convex 4-polytopes
- Properties
- As configurations
- Visualization
- Regular star (Schläfli–Hess) 4-polytopes
- Names
- Symmetry
- Properties 2
- See also
- References
- Citations
- Bibliography
- External links

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.^{ [1] } 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 {5,5/2} and small stellated dodecahedron { 5/2,5}.

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 is constrained by the existence of the regular polyhedra which form its cells and a dihedral angle constraint

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 {p,q,r} that have valid cells {p,q} and vertex figures {q,r}, and pass the dihedral test, but fail to produce finite figures: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}.

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 (face-to-face) in a regular fashion.

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^{ [2] } within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering.

Regular convex 4-polytopes | |||||||
---|---|---|---|---|---|---|---|

Symmetry group | A_{4} | B_{4} | F_{4} | H_{4} | |||

Name | 5-cell Hyper-tetrahedron | 16-cell Hyper-octahedron | 8-cell Hyper-cube | 24-cell
| 600-cell Hyper-icosahedron | 120-cell Hyper-dodecahedron | |

Schläfli symbol | {3, 3, 3} | {3, 3, 4} | {4, 3, 3} | {3, 4, 3} | {3, 3, 5} | {5, 3, 3} | |

Coxeter mirrors | |||||||

Mirror dihedrals | 𝝅/2𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2 | 𝝅/2𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2 | 𝝅/2𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2 | 𝝅/2𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2 | 𝝅/2𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2 | 𝝅/2𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2 | |

Graph | |||||||

Vertices | 5 | 8 | 16 | 24 | 120 | 600 | |

Edges | 10 | 24 | 32 | 96 | 720 | 1200 | |

Faces | 10 triangles | 32 triangles | 24 squares | 96 triangles | 1200 triangles | 720 pentagons | |

Cells | 5 tetrahedra | 16 tetrahedra | 8 cubes | 24 octahedra | 600 tetrahedra | 120 dodecahedra | |

Tori | 1 5-tetrahedron | 2 8-tetrahedron | 2 4-cube | 4 6-octahedron | 20 30-tetrahedron | 12 10-dodecahedron | |

Inscribed | 120 in 120-cell | 1 16-cell | 2 16-cells | 3 8-cells | 25 24-cells | 10 600-cells | |

Great polygons | 2 𝝅/2 squares x 3 | 4 𝝅/2 rectangles x 3 | 4 𝝅/3 hexagons x 4 | 12 𝝅/5 decagons x 6 | 50 𝝅/15 dodecagons x 4 | ||

Petrie polygons | 1 pentagon | 1 octagon | 2 octagons | 2 dodecagons | 4 30-gons | 20 30-gons | |

Isocline polygons | 1 {8/2}=2{4} x {8/2}=2{4} | 2 {8/2}=2{4} x {8/2}=2{4} | 2 {12/2}=2{6} x {12/6}=6{2} | 4 {30/2}=2{15} x 30{0} | 20 {30/2}=2{15} x 30{0} | ||

Long radius | |||||||

Edge length | |||||||

Short radius | |||||||

Area | |||||||

Volume | |||||||

4-Content |

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 | Schläfli Coxeter | V | E | F | C | Vert. fig. | Dual | Symmetry group | |
---|---|---|---|---|---|---|---|---|---|---|---|

5-cellpentachoron pentatope 4-simplex | n-simplex (A _{n} family) | {3,3,3} | 5 | 10 | 10 {3} | 5 {3,3} | {3,3} | self-dual | A_{4}[3,3,3] | 120 | |

16-cellhexadecachoron 4-orthoplex | n-orthoplex (B _{n} family) | {3,3,4} | 8 | 24 | 32 {3} | 16 {3,3} | {3,4} | 8-cell | B_{4}[4,3,3] | 384 | |

8-celloctachoron tesseract 4-cube | hypercuben-cube (B _{n} family) | {4,3,3} | 16 | 32 | 24 {4} | 8 {4,3} | {3,3} | 16-cell | |||

24-cellicositetrachoron octaplex polyoctahedron (pO) | F_{n} family | {3,4,3} | 24 | 96 | 96 {3} | 24 {3,4} | {4,3} | self-dual | F_{4}[3,4,3] | 1152 | |

600-cellhexacosichoron tetraplex polytetrahedron (pT) | n-pentagonal polytope (H _{n} family) | {3,3,5} | 120 | 720 | 1200 {3} | 600 {3,3} | {3,5} | 120-cell | H_{4}[5,3,3] | 14400 | |

120-cellhecatonicosachoron dodecacontachoron dodecaplex polydodecahedron (pD) | n-pentagonal polytope (H _{n} family) | {5,3,3} | 600 | 1200 | 720 {5} | 120 {5,3} | {3,3} | 600-cell |

John Conway advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD).^{ [3] }

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").^{ [4] }^{ [5] }

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

where *N*_{k} 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.^{ [6] }

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.^{ [7] }^{ [8] }

5-cell {3,3,3} | 16-cell {3,3,4} | tesseract {4,3,3} | 24-cell {3,4,3} | 600-cell {3,3,5} | 120-cell {5,3,3} |
---|---|---|---|---|---|

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.

A_{4} = [3,3,3] | B_{4} = [4,3,3] | F_{4} = [3,4,3] | H_{4} = [5,3,3] | ||
---|---|---|---|---|---|

5-cell | 16-cell | 8-cell | 24-cell | 600-cell | 120-cell |

{3,3,3} | {3,3,4} | {4,3,3} | {3,4,3} | {3,3,5} | {5,3,3} |

Solid 3D orthographic projections | |||||

Tetrahedral envelope (cell/vertex-centered) | Cubic envelope (cell-centered) | Cubic envelope (cell-centered) | Cuboctahedral envelope (cell-centered) | Pentakis icosidodecahedral envelope (vertex-centered) | Truncated rhombic triacontahedron envelope (cell-centered) |

Wireframe Schlegel diagrams (Perspective projection) | |||||

Cell-centered | Cell-centered | Cell-centered | Cell-centered | Vertex-centered | Cell-centered |

Wireframe stereographic projections (3-sphere) | |||||

The **Schläfli–Hess 4-polytopes** are the complete set of 10 regular self-intersecting **star polychora** (four-dimensional polytopes).^{ [10] } They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol {*p*,*q*,*r*} in which one of the numbers is 5/2. They are thus analogous to the regular nonconvex Kepler–Poinsot polyhedra, which are in turn analogous to the pentagram.

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. (Example: a pentagon stellates into a pentagram)**greatening**– replaces the faces by large ones in same planes. (Example: an icosahedron greatens into a great icosahedron)**aggrandizement**– replaces the cells by large ones in same 3-spaces. (Example: a 600-cell aggrandizes into a grand 600-cell)

John Conway names the 10 forms from 3 regular celled 4-polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral 600-cell), pI=polyicoshedron {3,5,5/2} (an icosahedral 120-cell), and pD=polydodecahedron {5,3,3} (a dodecahedral 120-cell), with prefix modifiers: *g*, *a*, and *s* for great, (ag)grand, and stellated. The final stellation, the *great grand stellated polydodecahedron* contains them all as *gaspD*.

All ten polychora have [3,3,5] (H_{4}) 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.

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*(face-colored) orthographic projections.

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

Name Conway (abbrev.) | Orthogonal projection | Schläfli Coxeter | C {p, q} | F {p} | E {r} | V {q, r} | Dens. | χ |
---|---|---|---|---|---|---|---|---|

Icosahedral 120-cell polyicosahedron (pI) | {3,5,5/2} | 120 {3,5} | 1200 {3} | 720 {5/2} | 120 {5,5/2} | 4 | 480 | |

Small stellated 120-cell stellated polydodecahedron (spD) | {5/2,5,3} | 120 {5/2,5} | 720 {5/2} | 1200 {3} | 120 {5,3} | 4 | −480 | |

Great 120-cell great polydodecahedron (gpD) | {5,5/2,5} | 120 {5,5/2} | 720 {5} | 720 {5} | 120 {5/2,5} | 6 | 0 | |

Grand 120-cell grand polydodecahedron (apD) | {5,3,5/2} | 120 {5,3} | 720 {5} | 720 {5/2} | 120 {3,5/2} | 20 | 0 | |

Great stellated 120-cell great stellated polydodecahedron (gspD) | {5/2,3,5} | 120 {5/2,3} | 720 {5/2} | 720 {5} | 120 {3,5} | 20 | 0 | |

Grand stellated 120-cell grand stellated polydodecahedron (aspD) | {5/2,5,5/2} | 120 {5/2,5} | 720 {5/2} | 720 {5/2} | 120 {5,5/2} | 66 | 0 | |

Great grand 120-cell great grand polydodecahedron (gapD) | {5,5/2,3} | 120 {5,5/2} | 720 {5} | 1200 {3} | 120 {5/2,3} | 76 | −480 | |

Great icosahedral 120-cell great polyicosahedron (gpI) | {3,5/2,5} | 120 {3,5/2} | 1200 {3} | 720 {5} | 120 {5/2,5} | 76 | 480 | |

Grand 600-cell grand polytetrahedron (apT) | {3,3,5/2} | 600 {3,3} | 1200 {3} | 720 {5/2} | 120 {3,5/2} | 191 | 0 | |

Great grand stellated 120-cell great grand stellated polydodecahedron (gaspD) | {5/2,3,3} | 120 {5/2,3} | 720 {5/2} | 1200 {3} | 600 {3,3} | 191 | 0 |

- Regular polytope
- List of regular polytopes
- Infinite regular 4-polytopes:
- One regular Euclidean honeycomb: {4,3,4}
- Four compact regular hyperbolic honeycombs: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}
- Eleven paracompact regular hyperbolic honeycombs: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.

- Abstract regular 4-polytopes:
- Uniform 4-polytope uniform 4-polytope families constructed from these 6 regular forms.
- Platonic solid
- Kepler-Poinsot polyhedra — regular star polyhedron
- Star polygon — regular star polygons
- 4-polytope
- 5-polytope
- 6-polytope

In geometry, a **Kepler–Poinsot polyhedron** is any of four regular star polyhedra.

In elementary geometry, a **polytope** is a geometric object with flat sides (*faces*). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or **n-polytope**. In this context, "flat sides" means that the sides of a (*k* + 1)-polytope consist of k-polytopes that may have (*k* – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

In geometry, a **4-polytope** 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.

In geometry, the **Schläfli symbol** is a notation of the form that defines regular polytopes and tessellations.

In mathematics, a **regular polytope** is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ *n*.

In geometry, a **vertex figure**, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

In geometry, a **uniform 4-polytope** is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In geometry of 4 dimensions or higher, a **duoprism** is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (*n*+*m*)-polytope, where n and m are dimensions of 2 (polygon) or higher.

In geometry, a **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, a **star polyhedron** is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.

In geometry, the **great grand stellated 120-cell** or **great grand stellated polydodecahedron** is a regular star 4-polytope with Schläfli symbol {5/2,3,3}, one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and has the same vertex arrangement as the regular convex 120-cell.

In geometry, the **grand 600-cell** or **grand polytetrahedron** is a regular star 4-polytope with Schläfli symbol {3,3,5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells.

In geometry, the **small stellated 120-cell** or **stellated polydodecahedron** is a regular star 4-polytope with Schläfli symbol {5/2,5,3}. It is one of 10 regular Schläfli-Hess polytopes.

In geometry, the **icosahedral 120-cell**, **polyicosahedron**, **faceted 600-cell** or **icosaplex** is a regular star 4-polytope with Schläfli symbol {3,5,5/2}. It is one of 10 regular Schläfli-Hess polytopes.

In geometry, the **great stellated 120-cell** or **great stellated polydodecahedron** is a regular star 4-polytope with Schläfli symbol {5/2,3,5}. It is one of 10 regular Schläfli-Hess polytopes.

In geometry, the **great 120-cell** or **great polydodecahedron** is a regular star 4-polytope with Schläfli symbol {5,5/2,5}. It is one of 10 regular Schläfli-Hess polytopes. It is one of the two such polytopes that is self-dual.

In geometry, the **grand stellated 120-cell** or **grand stellated polydodecahedron** is a regular star 4-polytope with Schläfli symbol {5/2,5,5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is also one of two such polytopes that is self-dual.

In geometry, a **pentagonal polytope** is a regular polytope in *n* dimensions constructed from the H_{n} Coxeter group. The family was named by H. S. M. Coxeter, because the two-dimensional pentagonal polytope is a pentagon. It can be named by its Schläfli symbol as {5, 3^{n − 2}} (dodecahedral) or {3^{n − 2}, 5} (icosahedral).

In geometry, a **regular skew apeirohedron** is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.

- ↑ Coxeter 1973, p. 141, §7-x. Historical remarks.
- ↑ Coxeter 1973, pp. 292–293, Table I(ii): The sixteen regular polytopes {
*p,q,r*} in four dimensions: [An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.] - ↑ Conway, Burgiel & Goodman-Strass 2008 , Ch. 26. Higher Still
- ↑ "Convex and abstract polytopes", Programme and abstracts, MIT, 2005
- ↑ Johnson, Norman W. (2018). "§ 11.5 Spherical Coxeter groups".
*Geometries and Transformations*. Cambridge University Press. pp. 246–. ISBN 978-1-107-10340-5. - ↑ Richeson, David S. (2012). "23. Henri Poincaré and the Ascendancy of Topology".
*Euler's Gem: The Polyhedron Formula and the Birth of Topology*. Princeton University Press. pp. 256–. ISBN 978-0-691-15457-2. - ↑ Coxeter 1973 , § 1.8 Configurations
- ↑ Coxeter, Complex Regular Polytopes, p.117
- ↑ Conway, Burgiel & Goodman-Strass 2008 , p. 406, Fig 26.2
- ↑ Coxeter,
*Star polytopes and the Schläfli function f{α,β,γ)*p. 122 2.*The Schläfli-Hess polytopes*

- Coxeter, H.S.M. (1973) [1948].
*Regular Polytopes*(3rd ed.). New York: Dover. - Coxeter, H.S.M. (1969).
*Introduction to Geometry*(2nd ed.). Wiley. ISBN 0-471-50458-0. - D.M.Y. Sommerville (2020) [1930]. "X. The Regular Polytopes".
*Introduction to the Geometry of*. Courier Dover. pp. 159–192. ISBN 978-0-486-84248-6.**n**Dimensions - Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008). "26. Regular Star-polytopes".
*The Symmetries of Things*. pp. 404–8. ISBN 978-1-56881-220-5. - Hess, Edmund (1883). "Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder".
- Hess, Edmund (1885). "Uber die regulären Polytope höherer Art".
*Sitzungsber Gesells Beförderung Gesammten Naturwiss Marburg*: 31–57. - Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995).
*Kaleidoscopes: Selected Writings of H.S.M. Coxeter*. Wiley. ISBN 978-0-471-01003-6.- (Paper 10) Coxeter, H.S.M. (1989). "Star Polytopes and the Schlafli Function f(α,β,γ)".
*Elemente der Mathematik*.**44**(2): 25–36.

- (Paper 10) Coxeter, H.S.M. (1989). "Star Polytopes and the Schlafli Function f(α,β,γ)".
- Coxeter, H.S.M. (1991).
*Regular Complex Polytopes*(2nd ed.). Cambridge University Press. ISBN 978-0-521-39490-1. - McMullen, Peter; Schulte, Egon (2002). "Abstract Regular Polytopes" (PDF).

- Weisstein, Eric W. "Regular polychoron".
*MathWorld*. - Jonathan Bowers, 16 regular 4-polytopes
- Regular 4D Polytope Foldouts
- Catalog of Polytope Images A collection of stereographic projections of 4-polytopes.
- A Catalog of Uniform Polytopes
- Dimensions 2 hour film about the fourth dimension (contains stereographic projections of all regular 4-polytopes)
- Reguläre Polytope
- The Regular Star Polychora
- Hypersolids

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