Pentagon (EntityTopic, 12)

From Hi.gher. Space

(Difference between revisions)
m (add equations)
 
(6 intermediate revisions not shown)
Line 1: Line 1:
-
<[#ontology [kind topic] [cats 2D Regular Flat Shape] [alt [[freebase:05zns]] [[wikipedia:Pentagon]]]]>
+
<[#ontology [kind topic] [cats 2D Regular Polytope] [alt [[freebase:05zns]] [[wikipedia:Pentagon]]]]>
{{STS Shape
{{STS Shape
| name=Pentagon
| name=Pentagon
| dim=2
| dim=2
-
| elements=5, 5
+
| elements=5 [[digon]]s, 5 [[point]]s
| genus=0
| genus=0
| ssc=G5
| ssc=G5
Line 9: Line 9:
| extra={{STS Matrix|
| extra={{STS Matrix|
  5 0
  5 0
-
  1 1}}{{STS Uniform polytope
+
  1 1}}{{STS Polytope
 +
| bowers=Peg
 +
| dual=''Self-dual''}}{{STS Uniform polytope
| schlaefli={5}
| schlaefli={5}
-
| vfigure=[[Digon]], length ?
+
| dynkin=x5o
-
| dual=''Self-dual''
+
| vfigure=[[Digon]], length (1+√5)/2
}}}}
}}}}
 +
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.
 +
==coordinates==
 +
The coordinates of a regular pentagon centered at the origi and having side length 2 are:
 +
<blockquote>(√((10+2√5)/5), 0)<br>(√((5-√5)/10), ±φ)<br>√((5+2√5)/5), ±1)</blockquote>
 +
Where φ is the golden ratio (1+√5)/2.
== Equations ==
== Equations ==
*The [[hypervolume]]s of a pentagon with side length ''l'' are given by:
*The [[hypervolume]]s of a pentagon with side length ''l'' are given by:
<blockquote>total edge length = 5''l''<br>
<blockquote>total edge length = 5''l''<br>
-
surface area = {{Over|1|4}} &middot; √(25+10√5) {{DotHV}}</blockquote>
+
area = {{Over|1|4}} &middot; √(25+10√5) {{DotHV}}</blockquote>
-
== Use ==
+
<[#polytope [id -5]]>
-
Pentagonal faces are found in these trishapes on FGwiki:
+
-
*[[Dodecahedron]] (12×, 100%)
+
-
*[[Icosidodecahedron]] (12×, 38%)
+
-
*[[Icosahedral truncate]] (12×, 38%)
+
-
*[[Dodecahedral snub]] (12×, 13%)
+
{{Dishapes}}
{{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"