# Regular 4-polytope

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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.

## Contents

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. [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 (18431903) 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 ${\displaystyle \{p,q,r\}}$ is constrained by the existence of the regular polyhedra ${\displaystyle \{p,q\},\{q,r\}}$ which form its cells and a dihedral angle constraint

${\displaystyle \sin {\frac {\pi }{p}}\sin {\frac {\pi }{r}}<\cos {\frac {\pi }{q}}}$

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 {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}.

## Regular convex 4-polytopes

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.

### Properties

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 A4 B4 F4 H4
Name 5-cell

Hyper-tetrahedron
5-point

16-cell

Hyper-octahedron
8-point

8-cell

Hyper-cube
16-point

24-cell

24-point

600-cell

Hyper-icosahedron
120-point

120-cell

Hyper-dodecahedron
600-point

Schläfli symbol {3, 3, 3}{3, 3, 4}{4, 3, 3}{3, 4, 3}{3, 3, 5}{5, 3, 3}
Coxeter mirrors
Mirror dihedrals𝝅/2𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2𝝅/2𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2𝝅/2𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2
Graph
Vertices581624120600
Edges102432967201200
Faces10 triangles32 triangles24 squares96 triangles1200 triangles720 pentagons
Cells5 tetrahedra16 tetrahedra8 cubes24 octahedra600 tetrahedra120 dodecahedra
Tori 1 5-tetrahedron 2 8-tetrahedron 2 4-cube4 6-octahedron 20 30-tetrahedron 12 10-dodecahedron
Inscribed120 in 120-cell1 16-cell2 16-cells3 8-cells25 24-cells10 600-cells
Great polygons 2 𝝅/2 squares x 34 𝝅/2 rectangles x 34 𝝅/3 hexagons x 412 𝝅/5 decagons x 650 𝝅/15 dodecagons x 4
Petrie polygons 1 pentagon 1 octagon 2 octagons 2 dodecagons 4 30-gons 20 30-gons
Isocline polygons 1 2 2 4 20
Long radius${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$
Edge length${\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}$${\displaystyle {\sqrt {2}}\approx 1.414}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle {\tfrac {1}{\phi }}\approx 0.618}$${\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}$
Short radius${\displaystyle {\tfrac {1}{4}}}$${\displaystyle {\tfrac {1}{2}}}$${\displaystyle {\tfrac {1}{2}}}$${\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}$${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$
Area${\displaystyle 10\left({\sqrt {\tfrac {8}{9}}}\right)\approx 9.428}$${\displaystyle 32\left({\sqrt {\tfrac {3}{16}}}\right)\approx 13.856}$${\displaystyle 24}$${\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}$${\displaystyle 1200\left({\tfrac {\sqrt {3}}{8\phi ^{2}}}\right)\approx 99.238}$${\displaystyle 720\left({\tfrac {25+10{\sqrt {5}}}{8\phi ^{4}}}\right)\approx 621.9}$
Volume${\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}$${\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}$${\displaystyle 8}$${\displaystyle 24\left({\sqrt {\tfrac {2}{9}}}\right)\approx 11.314}$${\displaystyle 600\left({\tfrac {1}{3\phi ^{3}{\sqrt {8}}}}\right)\approx 16.693}$${\displaystyle 120\left({\tfrac {2+\phi }{2\phi ^{3}{\sqrt {8}}}}\right)\approx 18.118}$
4-Content${\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}$${\displaystyle {\tfrac {2}{3}}\approx 0.667}$${\displaystyle 1}$${\displaystyle 2}$${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}$${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}$

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.

NamesImageFamily Schläfli
Coxeter
V E F C Vert.
fig.
Dual Symmetry group
5-cell
pentachoron
pentatope
4-simplex
n-simplex
(An family)
{3,3,3}
51010
{3}
5
{3,3}
{3,3} self-dualA4
[3,3,3]
120
16-cell
4-orthoplex
n-orthoplex
(Bn family)
{3,3,4}
82432
{3}
16
{3,3}
{3,4} 8-cellB4
[4,3,3]
384
8-cell
octachoron
tesseract
4-cube
hypercube
n-cube
(Bn family)
{4,3,3}
163224
{4}
8
{4,3}
{3,3} 16-cell
24-cell
icositetrachoron
octaplex
polyoctahedron
(pO)
Fn family{3,4,3}
249696
{3}
24
{3,4}
{4,3} self-dualF4
[3,4,3]
1152
600-cell
hexacosichoron
tetraplex
polytetrahedron
(pT)
n-pentagonal
polytope

(Hn family)
{3,3,5}
1207201200
{3}
600
{3,3}
{3,5} 120-cellH4
[5,3,3]
14400
120-cell
hecatonicosachoron
dodecacontachoron
dodecaplex
polydodecahedron
(pD)
n-pentagonal
polytope

(Hn family)
{5,3,3}
6001200720
{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:

${\displaystyle N_{0}-N_{1}+N_{2}-N_{3}=0\,}$

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. [6]

### 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 (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}
${\displaystyle {\begin{bmatrix}{\begin{matrix}5&4&6&4\\2&10&3&3\\3&3&10&2\\4&6&4&5\end{matrix}}\end{bmatrix}}}$${\displaystyle {\begin{bmatrix}{\begin{matrix}8&6&12&8\\2&24&4&4\\3&3&32&2\\4&6&4&16\end{matrix}}\end{bmatrix}}}$${\displaystyle {\begin{bmatrix}{\begin{matrix}16&4&6&4\\2&32&3&3\\4&4&24&2\\8&12&6&8\end{matrix}}\end{bmatrix}}}$${\displaystyle {\begin{bmatrix}{\begin{matrix}24&8&12&6\\2&96&3&3\\3&3&96&2\\6&12&8&24\end{matrix}}\end{bmatrix}}}$${\displaystyle {\begin{bmatrix}{\begin{matrix}120&12&30&20\\2&720&5&5\\3&3&1200&2\\4&6&4&600\end{matrix}}\end{bmatrix}}}$${\displaystyle {\begin{bmatrix}{\begin{matrix}600&4&6&4\\2&1200&3&3\\5&5&720&2\\20&30&12&120\end{matrix}}\end{bmatrix}}}$

### 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.

A4 = [3,3,3]B4 = [4,3,3]F4 = [3,4,3]H4 = [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)

## Regular star (Schläfli–Hess) 4-polytopes

The SchläfliHess 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 KeplerPoinsot 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:

1. stellation – replaces edges by longer edges in same lines. (Example: a pentagon stellates into a pentagram)
2. greatening – replaces the faces by large ones in same planes. (Example: an icosahedron greatens into a great icosahedron)
3. 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.

### Symmetry

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.

### Properties

Note:

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}
4480
Small stellated 120-cell
stellated polydodecahedron (spD)
{5/2,5,3}
120
{5/2,5}
720
{5/2}
1200
{3}
120
{5,3}
4480
Great 120-cell
great polydodecahedron (gpD)
{5,5/2,5}
120
{5,5/2}
720
{5}
720
{5}
120
{5/2,5}
60
Grand 120-cell
grand polydodecahedron (apD)
{5,3,5/2}
120
{5,3}
720
{5}
720
{5/2}
120
{3,5/2}
200
Great stellated 120-cell
great stellated polydodecahedron (gspD)
{5/2,3,5}
120
{5/2,3}
720
{5/2}
720
{5}
120
{3,5}
200
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}
660
Great grand 120-cell
great grand polydodecahedron (gapD)
{5,5/2,3}
120
{5,5/2}
720
{5}
1200
{3}
120
{5/2,3}
76480
Great icosahedral 120-cell
great polyicosahedron (gpI)
{3,5/2,5}
120
{3,5/2}
1200
{3}
720
{5}
120
{5/2,5}
76480
Grand 600-cell
grand polytetrahedron (apT)
{3,3,5/2}
600
{3,3}
1200
{3}
720
{5/2}
120
{3,5/2}
1910
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}
1910

## Related Research Articles

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

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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.

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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.

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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.

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

## References

### Citations

1. Coxeter 1973, p. 141, §7-x. Historical remarks.
2. 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.]
3. Conway, Burgiel & Goodman-Strass 2008 , Ch. 26. Higher Still
4. Johnson, Norman W. (2018). "§ 11.5 Spherical Coxeter groups". Geometries and Transformations. Cambridge University Press. pp. 246–. ISBN   978-1-107-10340-5.
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.
6. Coxeter 1973 , § 1.8 Configurations
7. Coxeter, Complex Regular Polytopes, p.117
8. Conway, Burgiel & Goodman-Strass 2008 , p. 406, Fig 26.2
9. Coxeter, Star polytopes and the Schläfli function f{α,β,γ) p. 122 2. The Schläfli-Hess polytopes