Pentagonal polytope: Difference between revisions
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!rowspan=2|[[Petrie polygon]] |
!rowspan=2|[[Petrie polygon]]<br>projection |
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!rowspan=2|Name<br>[[Coxeter-Dynkin diagram]]<br>[[Schläfli symbol]] |
!rowspan=2|Name<br>[[Coxeter-Dynkin diagram]]<br>[[Schläfli symbol]] |
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==References== |
==References== |
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{{reflist}} |
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Revision as of 09:51, 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.
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.