Pentagonal polytope: Difference between revisions
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# [[Order-5 5-cell honeycomb]], {3, 3, 3, 5} (tessellates hyperbolic 4-space (∞ [[5-cell]] facets) |
# [[Order-5 5-cell honeycomb]], {3, 3, 3, 5} (tessellates hyperbolic 4-space (∞ [[5-cell]] facets) |
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The facets of each icosahedral pentagonal polytope are the [[simplex |
The facets of each icosahedral pentagonal polytope are the [[simplex|simplices]] of one less dimension. |
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====Elements==== |
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==References== |
==References== |
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Revision as of 11:19, 8 September 2011
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. 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.
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
|
∞ | ∞ | ∞ | ∞ | ∞ | |
Icosahedral
The complete family of icosahedral pentagonal polytopes are:
- Line segment, {}
- Pentagon, {5}
- Icosahedron, {3, 5} (20 triangular faces)
- 600-cell, {3, 3, 5} (120 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.
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 | 
|
Icosahedron {3, 5} |
20 equilateral triangles
|
12 | 30 | 20 | ||
| 4 | 
|
600-cell {3, 3, 5} |
600 tetrahedra
|
120 | 720 | 1200 | 600 | |
| 5 | Order-5 5-cell honeycomb {3, 3, 3, 5} |
∞ 5-cells
|
∞ | ∞ | ∞ | ∞ | ∞ | |
References
- ^ Olshevsky, George. Glossary for Hyperspace https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#{{{anchor}}}. Archived from the original on 4 February 2007.
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