Dodecahedron (EntityTopic, 12)

(Difference between revisions)

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.

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.

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

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 4: / Line 4: @@
 | image=<[#embed [hash MTJRRXEJQQJVGFCEBYAN84F3NF] [width 150]]>
 | dim=3
-| elements=12, 30, 20
+| 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
@@ Line 17: / Line 17: @@
 | wythoff=<nowiki>3 | 2 5 </nowiki>
 | schlaefli={[[Pentagon|5,]]3}
+| dynkin=x5o3o
 | conway=d[[Icosahedron|s]][[Tetrahedron|Y3]]
 | vlayout=[[Pentagon|5]]<sup>3</sup>
@@ Line 22: / Line 23: @@
 | dual=[[Icosahedron]]
 }}}}
+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: