Dodecahedron (EntityTopic, 12)

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Latest revision as of 14:41, 26 March 2017

The dodecahedron is one of the five Platonic solids. It contains 12 pentagons joining three to a vertex.

Coordinates

The coordinates of a dodecahedron with side length 2/φ (where φ = (1+√5)/2) are:

(±1, ±1, ±1
(0, ±1/φ, ±φ)
(±1/φ, ±φ, 0)
(±φ, 0, ±1/φ)

The first set of coordinates shows that a cube can be inscribed into a dodecahedron.

Equations

The hypervolumes of a dodecahedron with side length l are given by:

total edge length = 30l
surface area = 3√(25 + 10√5) · l²
volume = ^{(15 + 7√5)}∕₄ · l³

Incidence matrix

Dual: icosahedron

#	TXID	Va	Ea	5a	Type	Name
0	Va				= point	;
1	Ea	2			= digon	;
2	5a	5	5		= pentagon	;
3	C1a	20	30	12	= dodecahedron	;

Usage as facets

120× 1-facets of a cosmochoron

Notable Trishapes
Regular:	tetrahedron • cube • octahedron • dodecahedron • icosahedron
Direct truncates:	tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate
Mesotruncates:	stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron
Snubs:	snub staurohedron • snub rhodohedron
Curved:	sphere • torus • cylinder • cone • frustum • crind

@@ Line 1: / Line 1: @@
-{{Shape|Dodecahedron|http://img129.imageshack.us/img129/2226/dodecahedron8do.png|3|12, 30, 20|0|{[[Pentagon|5,]]3}|<nowiki>3 | 2 5 </nowiki>|''Unknown''|N/A|Equilateral [[triangle]], edge ''tau''|Doe|[[Icosahedron]]}}
+<[#ontology [kind topic] [cats 3D Regular Polytope] [alt [[freebase:02bls]] [[wikipedia:Dodecahedron]]]]>
-== Geometry ==
+{{STS Shape
-=== Equations ===
+| name=Dodecahedron
-*Assumption: Dodecahedron is centered at the origin.
+| image=<[#embed [hash MTJRRXEJQQJVGFCEBYAN84F3NF] [width 150]]>
-*Variables:
+| dim=3
-<blockquote>''l'' ⇒ length of edges of the dodecahedron</blockquote>
+| elements=12 [[pentagon]]s, 30 [[digon]]s, 20 [[point]]s
+| sym=[[Rhodohedral symmetry|I<sub>h</sub>, H<sub>3</sub>, [5,3], (*532)]]
+| genus=0
+| ssc={G5<sup>3</sup>}
+| ssc2=Ki1
+| extra={{STS Polytope
+| flayout={{FLD|a5|er|e3}}
+| petrie=10,10,0
+| dual=[[Icosahedron]]
+| bowers=Doe
+}}{{STS Uniform polytope
+| wythoff=<nowiki>3 | 2 5 </nowiki>
+| schlaefli={[[Pentagon|5,]]3}
+| dynkin=x5o3o
+| conway=d[[Icosahedron|s]][[Tetrahedron|Y3]]
+| vlayout=[[Pentagon|5]]<sup>3</sup>
+| vfigure=Equilateral [[triangle]], edge ''tau''
+| dual=[[Icosahedron]]
+}}}}
-*The [[hypervolume]]s of a dodecahedron are given by:
+The dodecahedron is one of the five Platonic solids. It contains 12 pentagons joining three to a vertex.
+==Coordinates==
+The coordinates of a dodecahedron with side length 2/φ (where φ = (1+√5)/2) are:
+<blockquote>(±1, ±1, ±1<br>(0, ±1/φ, ±φ)<br>(±1/φ, ±φ, 0)<br>(±φ, 0, ±1/φ)</blockquote>
+The first set of coordinates shows that a [[cube]] can be inscribed into a dodecahedron.
+== Equations ==
+*The [[hypervolume]]s of a dodecahedron with side length ''l'' are given by:
 <blockquote>total edge length = 30''l''<br>
-surface area = 15''l''<sup>2</sup>tan(3π10<sup>-1</sup>)<br>
+surface area = 3√(25 + 10√5) {{DotHV}}<br>
-volume = 5''l''<sup>3</sup>(tan(3π10<sup>-1</sup>))<sup>2</sup>(tan(sin<sup>-1</sup>(2sin(π5<sup>-1</sup>))<sup>-1</sup>))2<sup>-1</sup></blockquote>
+volume = {{Over|(15 + 7√5)|4}} {{DotHV|3}}</blockquote>
+<[#polytope [id 4]]>
-*The [[planar]] [[cross-section]]s (''n'') of a dodecahedron are:
+{{Trishapes}}
-<blockquote>''Unknown''</blockquote>
-<br clear="all"><br>
-{{Polyhedra}}