Pentagonal polytope: Difference between revisions
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In [[geometry]], a '''pentagonal polytope'''<ref>{{GlossaryForHyperspace| |
In [[geometry]], a '''pentagonal polytope'''<ref>{{GlossaryForHyperspace|title=}}</ref> is a [[regular polytope]] in ''n'' dimensions constructed from the [[Coxeter group|H<sub>''n'' 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, 3<sup>''n'' - 1</sup>} (dodecahedral) or {3<sup>''n'' - 1</sup>, 5} (icosahedral). |
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Such polytopes can always be [[stellation|stellated]] to form new [[star polytope|star regular polytopes]]. In two dimensions, this forms the [[pentagram]]; in three dimensions, this forms the [[Kepler-Poinsot polyhedron|Kepler-Poinsot polyhedra]]; and in four dimensions, this forms the [[Schläfli-Hess polychoron|Schläfli-Hess polychora]]. |
Such polytopes can always be [[stellation|stellated]] to form new [[star polytope|star regular polytopes]]. In two dimensions, this forms the [[pentagram]]; in three dimensions, this forms the [[Kepler-Poinsot polyhedron|Kepler-Poinsot polyhedra]]; and in four dimensions, this forms the [[Schläfli-Hess polychoron|Schläfli-Hess polychora]]. |
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# [[Order-3 120-cell honeycomb]], {5, 3, 3, 3} (tessellates hyperbolic 4-space (∞ [[120-cell]] facets) |
# [[Order-3 120-cell honeycomb]], {5, 3, 3, 3} (tessellates hyperbolic 4-space (∞ [[120-cell]] facets) |
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===Elements=== |
====Elements==== |
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===Icosahedral=== |
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==References== |
==References== |
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Revision as of 09:54, 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
|
∞ | ∞ | ∞ | ∞ | ∞ | |
Icosahedral
The template {{Expand}} has been deprecated since 26 December 2010, and is retained only for old revisions. If this page is a current revision, please remove the template.
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.
{{cite web}}: Missing or empty|title=(help)