Pentagonal polytope
In geometry, a pentagonal polytope[1] is a regular polytope in n dimensions constructed from the Hn Coxeter group. The family was named by George Olshevsky, because the two-dimensional pentagonal polytope is a pentagon. It can be named by its Schläfli symbol as {5, 3n - 1} (dodecahedral) or {3n - 1, 5} (icosahedral).
Such polytopes can always be stellated to form new star regular polytopes. In two dimensions, this forms the pentagram; in three dimensions, this forms the Kepler-Poinsot polyhedra; and in four dimensions, this forms the Schläfli-Hess polychora.
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.
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)
Elements
| n | Petrie polygon projection |
Name Coxeter-Dynkin diagram Schläfli symbol |
Facets | Elements | ||||
|---|---|---|---|---|---|---|---|---|
| Vertices | Edges | Faces | Cells | 4-faces | ||||
| 1 | 
|
Line segment {} |
2 points | 2 | ||||
| 2 | 
|
Pentagon  {5} |
5 line segments | 5 | 5 | |||
| 3 | 
|
Dodecahedron {5, 3} |
12 pentagons
|
20 | 30 | 12 | ||
| 4 | 
|
120-cell {5, 3, 3} |
120 dodecahedra
|
600 | 1200 | 720 | 120 | |
| 5 | Order-3 120-cell honeycomb {5, 3, 3, 3} |
∞ 120-cells
|
∞ | ∞ | ∞ | ∞ | ∞ | |
References
- ^ Olshevsky, George. "{{{title}}}". Glossary for Hyperspace. Archived from the original on 4 February 2007.