# Pentagonal polytope

__: Regular polytope whose 2D form is a pentagon__

**Short description**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).

## Family members

The family starts as 1-polytopes and ends with *n* = 5 as infinite tessellations of 4-dimensional hyperbolic space.

There are two types of pentagonal polytopes; they may be termed the *dodecahedral* and *icosahedral* types, by their three-dimensional members. The two types are duals of each other.

### Dodecahedral

The complete family of dodecahedral pentagonal polytopes are:

- Line segment, { }
- Pentagon, {5}
- Dodecahedron, {5, 3} (12 pentagonal faces)
- 120-cell, {5, 3, 3} (120 dodecahedral cells)
- Order-3 120-cell honeycomb, {5, 3, 3, 3} (tessellates hyperbolic 4-space (∞ 120-cell facets)

The facets of each dodecahedral pentagonal polytope are the dodecahedral pentagonal polytopes of one less dimension. Their vertex figures are the simplices of one less dimension.

n | Coxeter group | Petrie polygon projection |
Name Coxeter diagram Schläfli symbol |
Facets | Elements | ||||
---|---|---|---|---|---|---|---|---|---|

Vertices | Edges | Faces | Cells | 4-faces
| |||||

1 | [math]\displaystyle{ H_1 }[/math] [ ] (order 2) |
Line segment { } |
2 vertices | 2 | |||||

2 | [math]\displaystyle{ H_2 }[/math] [5] (order 10) |
Pentagon {5} |
5 edges | 5 | 5 | ||||

3 | [math]\displaystyle{ H_3 }[/math] [5,3] (order 120) |
Dodecahedron {5, 3} |
12 pentagons |
20 | 30 | 12 | |||

4 | [math]\displaystyle{ H_4 }[/math] [5,3,3] (order 14400) |
120-cell {5, 3, 3} |
120 dodecahedra |
600 | 1200 | 720 | 120 | ||

5 | [math]\displaystyle{ {\bar{H}}_4 }[/math] [5,3,3,3] (order ∞) |
120-cell honeycomb {5, 3, 3, 3} |
∞ 120-cells |
∞ | ∞ | ∞ | ∞ | ∞ |

### Icosahedral

The complete family of icosahedral pentagonal polytopes are:

- Line segment, { }
- Pentagon, {5}
- Icosahedron, {3, 5} (20 triangular faces)
- 600-cell, {3, 3, 5} (600 tetrahedron cells)
- Order-5 5-cell honeycomb, {3, 3, 3, 5} (tessellates hyperbolic 4-space (∞ 5-cell facets)

The facets of each icosahedral pentagonal polytope are the simplices of one less dimension. Their vertex figures are icosahedral pentagonal polytopes of one less dimension.

n | Coxeter group | Petrie polygon projection |
Name Coxeter diagram Schläfli symbol |
Facets | Elements | ||||
---|---|---|---|---|---|---|---|---|---|

Vertices | Edges | Faces | Cells | 4-faces
| |||||

1 | [math]\displaystyle{ H_1 }[/math] [ ] (order 2) |
Line segment { } |
2 vertices | 2 | |||||

2 | [math]\displaystyle{ H_2 }[/math] [5] (order 10) |
Pentagon {5} |
5 Edges | 5 | 5 | ||||

3 | [math]\displaystyle{ H_3 }[/math] [5,3] (order 120) |
Icosahedron {3, 5} |
20 equilateral triangles |
12 | 30 | 20 | |||

4 | [math]\displaystyle{ H_4 }[/math] [5,3,3] (order 14400) |
600-cell {3, 3, 5} |
600 tetrahedra |
120 | 720 | 1200 | 600 | ||

5 | [math]\displaystyle{ {\bar{H}}_4 }[/math] [5,3,3,3] (order ∞) |
Order-5 5-cell honeycomb {3, 3, 3, 5} |
∞ 5-cells |
∞ | ∞ | ∞ | ∞ | ∞ |

## Related star polytopes and honeycombs

The pentagonal polytopes can be stellated to form new star regular polytopes:

- In two dimensions, we obtain the pentagram {5/2},
- In three dimensions, this forms the four Kepler–Poinsot polyhedra, {3,5/2}, {5/2,3}, {5,5/2}, and {5/2,5}.
- In four dimensions, this forms the ten Schläfli–Hess polychora: {3,5,5/2}, {5/2,5,3}, {5,5/2,5}, {5,3,5/2}, {5/2,3,5}, {5/2,5,5/2}, {5,5/2,3}, {3,5/2,5}, {3,3,5/2}, and {5/2,3,3}.
- In four-dimensional hyperbolic space there are four regular star-honeycombs: {5/2,5,3,3}, {3,3,5,5/2}, {3,5,5/2,5}, and {5,5/2,5,3}.

In some cases, the star pentagonal polytopes are themselves counted among the pentagonal polytopes.^{[1]}

Like other polytopes, regular stars can be combined with their duals to form compounds;

- In two dimensions, a decagrammic star figure {10/2} is formed,
- In three dimensions, we obtain the compound of dodecahedron and icosahedron,
- In four dimensions, we obtain the compound of 120-cell and 600-cell.

Star polytopes can also be combined.

## Notes

- ↑ Coxeter, H. S. M.: Regular Polytopes (third edition), p. 107, p. 266

## References

**Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]- (Paper 10) H.S.M. Coxeter,
*Star Polytopes and the Schlafli Function f(α,β,γ)*[Elemente der Mathematik 44 (2) (1989) 25–36]

- (Paper 10) H.S.M. Coxeter,
- Coxeter,
*Regular Polytopes*, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Table I(ii): 16 regular polytopes {p, q,r} in four dimensions, pp. 292–293)

Original source: https://en.wikipedia.org/wiki/Pentagonal polytope.
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