Pentagonal polytope: Difference between revisions
Content deleted Content added
added Category:Polytopes using HotCat |
No edit summary |
||
| Line 1: | Line 1: | ||
In [[geometry]], a '''pentagonal polytope'''<ref>{{GlossaryForHyperspace|blah}}</ref> is a [[ |
In [[geometry]], a '''pentagonal polytope'''<ref>{{GlossaryForHyperspace|blah}}</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). |
||
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]]. |
||
==Family members== |
==Family members== |
||
The family starts as [[ |
The family starts as [[polytope|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 |
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 [[pentagon]]al faces) |
|||
# [[120-cell]], {5, 3, 3} (120 [[dodecahedron|dodecahedral]] cells) |
|||
# [[Order-3 120-cell honeycomb]], {5, 3, 3, 3} (tessellates hyperbolic 4-space (∞ [[120-cell]] facets) |
|||
===Elements=== |
|||
{| class="wikitable" |
|||
|+ |
|||
Dodecahedral pentagonal polytopes |
|||
|- |
|||
!rowspan=2|n |
|||
!rowspan=2|[[Petrie polygon]] projection |
|||
!rowspan=2|Name<br>[[Coxeter-Dynkin diagram]]<br>[[Schläfli symbol]] |
|||
!rowspan=2|[[Facet (mathematics)|Facets]] |
|||
!colspan=5|Elements |
|||
|- |
|||
![[Vertex (geometry)|Vertices]] |
|||
![[Edge (geometry)|Edges]] |
|||
![[Face (geometry)|Faces]] |
|||
![[Cell (geometry)|Cells]] |
|||
!''4''-faces |
|||
|- |
|||
|1 |
|||
|[[File:Cross graph 1.svg|80px]] |
|||
|[[Line segment]]<br>{{CDD|node_1}}<br>{} |
|||
|2 [[point (geometry)|points]] |
|||
|2 |
|||
| |
|||
| |
|||
| |
|||
| |
|||
|- |
|||
|2 |
|||
|[[File:Regular polygon 5.svg|80px]] |
|||
|[[Pentagon]]<br>{{CDD|node_1|5|node}}<br>{5} |
|||
|5 [[line segment]]s |
|||
|5 |
|||
|5 |
|||
| |
|||
| |
|||
| |
|||
|- |
|||
|3 |
|||
|[[File:Dodecahedron t0 H3.png|80px]] |
|||
|[[Dodecahedron]]<br>{{CDD|node_1|5|node|3|node}}<br>{5, 3} |
|||
|12 [[pentagon]]s<br>[[File:Regular polygon 5.svg|80px]] |
|||
|20 |
|||
|30 |
|||
|12 |
|||
| |
|||
| |
|||
|- |
|||
|4 |
|||
|[[File:120-cell graph H4.svg|80px]] |
|||
|[[120-cell]]<br>{{CDD|node_1|5|node|3|node|3|node}}<br>{5, 3, 3} |
|||
|120 [[dodecahedron|dodecahedra]]<br>[[File:Dodecahedron t0 H3.png|80px]] |
|||
|600 |
|||
|1200 |
|||
|720 |
|||
|120 |
|||
| |
|||
|- |
|||
|5 |
|||
| |
|||
|[[Order-3 120-cell honeycomb]]<br>{CDD|node_1|5|node|3|node|3|node|3|node}}<br>{5, 3, 3, 3} |
|||
|∞ [[120-cell]]s<br>[[File:120-cell graph H4.svg|80px]] |
|||
|∞ |
|||
|∞ |
|||
|∞ |
|||
|∞ |
|||
|∞ |
|||
|} |
|||
==References== |
==References== |
||
{{reflist}} |
{{reflist}} |
||
{{ |
{{polytopes}} |
||
[[Category:Polytopes]] |
|||
Revision as of 09:50, 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 {CDD|node_1|5|node|3|node|3|node|3|node}} {5, 3, 3, 3} |
∞ 120-cells
|
∞ | ∞ | ∞ | ∞ | ∞ | |
References
- ^ Olshevsky, George. "{{{title}}}". Glossary for Hyperspace. Archived from the original on 4 February 2007.