Pentagonal polytope: Difference between revisions
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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). |
In [[geometry]], a '''pentagonal polytope'''<ref>{{GlossaryForHyperspace|title=}}</ref> is a [[regular polytope]] in ''n'' dimensions constructed from the [[Coxeter group|H<sub>''n''</sub> 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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The facets of each dodecahedral pentagonal polytope are the dodecahedral pentagonal polytopes of one less dimension. |
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====Elements==== |
====Elements==== |
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{| class="wikitable" |
{| class="wikitable" |
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===Icosahedral=== |
===Icosahedral=== |
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The complete family of icosahedral pentagonal polytopes are: |
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{{Empty section|date=September 2011}} |
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# [[Line segment]], {} |
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# [[Pentagon]], {5} |
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# [[Icosahedron]], {3, 5} (20 [[equilateral triangle|triangular]] faces) |
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# [[600-cell]], {3, 3, 5} (120 [[tetrahedron]] cells) |
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# [[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 (geometry)|simplices]] of one less dimension. |
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====Elements==== |
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{| class="wikitable" |
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|+ |
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Icosahedral pentagonal polytopes |
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!rowspan=2|n |
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!rowspan=2|[[Petrie polygon]]<br>projection |
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!rowspan=2|Name<br>[[Coxeter-Dynkin diagram]]<br>[[Schläfli symbol]] |
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!rowspan=2|[[Facet (mathematics)|Facets]] |
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!colspan=5|Elements |
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![[Vertex (geometry)|Vertices]] |
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![[Edge (geometry)|Edges]] |
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![[Face (geometry)|Faces]] |
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![[Cell (geometry)|Cells]] |
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!''4''-faces |
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|1 |
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|[[File:Cross graph 1.svg|80px]] |
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|[[Line segment]]<br>{{CDD|node_1}}<br>{} |
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|2 [[point (geometry)|points]] |
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|2 |
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|2 |
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|[[File:Regular polygon 5.svg|80px]] |
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|[[Pentagon]]<br>{{CDD|node_1|5|node}}<br>{5} |
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|5 [[line segment]]s |
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|5 |
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|5 |
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|3 |
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|[[File:Icosahedron t0 H3.png|80px]] |
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|[[Icosahedron]]<br>{{CDD|node_1|3|node|5|node}}<br>{3, 5} |
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|20 [[equilateral triangle]]s<br>[[File:Regular polygon 3.svg|80px]] |
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|12 |
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|30 |
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|20 |
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|4 |
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|[[File:600-cell graph H4.svg|80px]] |
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|[[600-cell]]<br>{{CDD|node_1|3|node|3|node|5|node}}<br>{3, 3, 5} |
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|600 [[tetrahedron|tetrahedra]]<br>[[File:3-simplex t0.svg|80px]] |
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|120 |
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|720 |
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|1200 |
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|600 |
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|5 |
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|[[Order-5 5-cell honeycomb]]<br>{{CDD|node_1|3|node|3|node|3|node|5|node}}<br>{3, 3, 3, 5} |
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|∞ [[5-cell]]s<br>[[File:4-simplex t0.svg|80px]] |
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|∞ |
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|∞ |
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|∞ |
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|∞ |
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|∞ |
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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.
{{cite web}}: Missing or empty|title=(help)