Dodecahedron (EntityTopic, 12)
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| image=<[#embed [hash MTJRRXEJQQJVGFCEBYAN84F3NF] [width 150]]> | | image=<[#embed [hash MTJRRXEJQQJVGFCEBYAN84F3NF] [width 150]]> | ||
| dim=3 | | 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 | | genus=0 | ||
| ssc={G5<sup>3</sup>} | | ssc={G5<sup>3</sup>} | ||
| Line 12: | Line 13: | ||
| petrie=10,10,0 | | petrie=10,10,0 | ||
| dual=[[Icosahedron]] | | dual=[[Icosahedron]] | ||
| + | | bowers=Doe | ||
}}{{STS Uniform polytope | }}{{STS Uniform polytope | ||
| wythoff=<nowiki>3 | 2 5 </nowiki> | | wythoff=<nowiki>3 | 2 5 </nowiki> | ||
| schlaefli={[[Pentagon|5,]]3} | | schlaefli={[[Pentagon|5,]]3} | ||
| + | | dynkin=x5o3o | ||
| conway=d[[Icosahedron|s]][[Tetrahedron|Y3]] | | conway=d[[Icosahedron|s]][[Tetrahedron|Y3]] | ||
| vlayout=[[Pentagon|5]]<sup>3</sup> | | vlayout=[[Pentagon|5]]<sup>3</sup> | ||
| vfigure=Equilateral [[triangle]], edge ''tau'' | | vfigure=Equilateral [[triangle]], edge ''tau'' | ||
| - | |||
| dual=[[Icosahedron]] | | 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 == | == Equations == | ||
*The [[hypervolume]]s of a dodecahedron with side length ''l'' are given by: | *The [[hypervolume]]s of a dodecahedron with side length ''l'' are given by: | ||
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 |
