# Pentagonal polytope

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.
Dodecahedral pentagonal polytopes :
| n | Coxeter group | Petrie polygon projection | Name Coxeter diagram Schläfli symbol | Facets | Elements | Elements | Elements | Elements | Elements |
| n | Coxeter group | Petrie polygon projection | Name Coxeter diagram Schläfli symbol | Facets | Vertices | Edges | Faces | Cells | 4-faces |
| 1 | $H_{1}$ [ ] (order 2) | | Line segment { } | 2 vertices | 2 | | | | |
| 2 | $H_{2}$ [5] (order 10) | | Pentagon {5} | 5 edges | 5 | 5 | | | |
| 3 | $H_{3}$ [5,3] (order 120) | | Dodecahedron {5, 3} | 12 pentagons | 20 | 30 | 12 | | |
| 4 | $H_{4}$ [5,3,3] (order 14400) | | 120-cell {5, 3, 3} | 120 dodecahedra | 600 | 1200 | 720 | 120 | |
| 5 | ${\bar {H}}_{4}$ (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.
Icosahedral pentagonal polytopes :
| n | Coxeter group | Petrie polygon projection | Name Coxeter diagram Schläfli symbol | Facets | Elements | Elements | Elements | Elements | Elements |
| n | Coxeter group | Petrie polygon projection | Name Coxeter diagram Schläfli symbol | Facets | Vertices | Edges | Faces | Cells | 4-faces |
| 1 | $H_{1}$ [ ] (order 2) | | Line segment { } | 2 vertices | 2 | | | | |
| 2 | $H_{2}$ [5] (order 10) | | Pentagon {5} | 5 Edges | 5 | 5 | | | |
| 3 | $H_{3}$ [5,3] (order 120) | | Icosahedron {3, 5} | 20 equilateral triangles | 12 | 30 | 20 | | |
| 4 | $H_{4}$ [5,3,3] (order 14400) | | 600-cell {3, 3, 5} | 600 tetrahedra | 120 | 720 | 1200 | 600 | |
| 5 | ${\bar {H}}_{4}$ (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.
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.