Ingeometry, a pentagonal polytope is a regular polytopeinn dimensions constructed from the Hn 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, 3n − 2} (dodecahedral) or {3n − 2, 5} (icosahedral).
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.
The complete family of dodecahedral pentagonal polytopes are:
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 | [ ] (order 2) |
Line segment { } |
2vertices | 2 | |||||
2 | [5] (order 10) |
Pentagon {5} |
5edges | 5 | 5 | ||||
3 | [5,3] (order 120) |
Dodecahedron {5, 3} |
12pentagons |
20 | 30 | 12 | |||
4 | [5,3,3] (order 14400) |
120-cell {5, 3, 3} |
120 dodecahedra |
600 | 1200 | 720 | 120 | ||
5 | [5,3,3,3] (order ∞) |
120-cell honeycomb {5, 3, 3, 3} |
∞ 120-cells |
∞ | ∞ | ∞ | ∞ | ∞ |
The complete family of icosahedral pentagonal polytopes are:
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 | [ ] (order 2) |
Line segment { } |
2vertices | 2 | |||||
2 | [5] (order 10) |
Pentagon {5} |
5Edges | 5 | 5 | ||||
3 | [5,3] (order 120) |
Icosahedron {3, 5} |
20equilateral triangles |
12 | 30 | 20 | |||
4 | [5,3,3] (order 14400) |
600-cell {3, 3, 5} |
600 tetrahedra |
120 | 720 | 1200 | 600 | ||
5 | [5,3,3,3] (order ∞) |
Order-5 5-cell honeycomb {3, 3, 3, 5} |
∞ 5-cells |
∞ | ∞ | ∞ | ∞ | ∞ |
The pentagonal polytopes can be stellated to form new star regular polytopes:
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;
Star polytopes can also be combined.
Fundamental convex regular and uniform polytopes in dimensions 2–10
| ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform polychoron | Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |