Pentagon (EntityTopic, 12)

From Hi.gher. Space

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{{Shape|Pentagon|''No image''|2|5, 5|0|{5}|N/A|G5|N/A|[[Line segment|Line]], length ?|N/A|''Self-dual''|N/A|N/A|N/A|none|''Unknown''|''Unknown''|N/A|SSC}}
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<[#ontology [kind topic] [cats 2D Regular Polytope] [alt [[freebase:05zns]] [[wikipedia:Pentagon]]]]>
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{{STS Shape
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| name=Pentagon
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| dim=2
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| elements=5 [[digon]]s, 5 [[point]]s
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| genus=0
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| ssc=G5
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| ssc2=G5
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| extra={{STS Matrix|
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5 0
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1 1}}{{STS Polytope
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| bowers=Peg
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| dual=''Self-dual''}}{{STS Uniform polytope
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| schlaefli={5}
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| dynkin=x5o
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| vfigure=[[Digon]], length (1+√5)/2
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}}}}
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The '''pentagon''' can be seen as the two-dimensional analog to the [[dodecahedron]] in 3D and the [[cosmochoron]] in 4D. It is also the lowest-dimensional member of the [[ursatope]]s, with a (trivial) [[digon]]al base.
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== Use ==
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==coordinates==
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Pentagonal faces are found in these trishapes on FGwiki:
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The coordinates of a regular pentagon centered at the origi and having side length 2 are:
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*[[Dodecahedron]] (12×, 100%)
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<blockquote>(√((10+2√5)/5), 0)<br>(√((5-√5)/10), ±φ)<br>√((5+2√5)/5), ±1)</blockquote>
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*[[Icosidodecahedron]] (12×, 38%)
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Where φ is the golden ratio (1+√5)/2.
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*[[Icosahedral truncate]] (12×, 38%)
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== Equations ==
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*[[Dodecahedral snub]] (12×, 13%)
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*The [[hypervolume]]s of a pentagon with side length ''l'' are given by:
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<blockquote>total edge length = 5''l''<br>
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area = {{Over|1|4}} &middot; √(25+10√5) {{DotHV}}</blockquote>
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{{Dishapes}}
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<[#polytope [id -5]]>
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[[Category:Regular polygons]]
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{{Dishapes}}

Latest revision as of 14:25, 26 March 2017

The pentagon can be seen as the two-dimensional analog to the dodecahedron in 3D and the cosmochoron in 4D. It is also the lowest-dimensional member of the ursatopes, with a (trivial) digonal base.

coordinates

The coordinates of a regular pentagon centered at the origi and having side length 2 are:

(√((10+2√5)/5), 0)
(√((5-√5)/10), ±φ)
√((5+2√5)/5), ±1)

Where φ is the golden ratio (1+√5)/2.

Equations

  • The hypervolumes of a pentagon with side length l are given by:
total edge length = 5l
area = 1∕4 · √(25+10√5) · l2

Incidence matrix

Dual: Self-dual

#TXIDVaEaTypeName
0 Va= point ;
1 Ea2= digon ;
2 5a55= pentagon ;

Usage as facets


Notable Dishapes
Flat: triangle • square • pentagon • hexagon • octagon • decagon
Curved: circle
Retrieved from "http://hi.gher.space/wiki/Pentagon"