Icosahedron (EntityTopic, 12)
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{{STS Shape | {{STS Shape | ||
| name=Icosahedron | | name=Icosahedron | ||
| - | | image=<[# | + | | image=<[#embed [hash 9F9FGF6HTQC8EZ51GC34JEKCKN] [width 150]]> |
| dim=3 | | dim=3 | ||
| - | | elements=20, 30, 12 | + | | elements=20 [[triangle]]s, 30 [[digon]]s, 12 [[point]]s |
| + | | sym=[[Rhodohedral symmetry|I<sub>h</sub>, H<sub>3</sub>, [5,3], (*532)]] | ||
| genus=0 | | genus=0 | ||
| ssc={G3<sup>5</sup>} | | ssc={G3<sup>5</sup>} | ||
| Line 13: | Line 14: | ||
| petrie=10,2 | | petrie=10,2 | ||
| dual=[[Dodecahedron]] | | dual=[[Dodecahedron]] | ||
| + | | bowers=Ike | ||
}}{{STS Uniform polytope | }}{{STS Uniform polytope | ||
| wythoff=<nowiki>5 | 2 3 </nowiki> | | wythoff=<nowiki>5 | 2 3 </nowiki> | ||
| schlaefli={[[Triangle|3,]]5} | | schlaefli={[[Triangle|3,]]5} | ||
| + | | dynkin=o5o3x, s3s3s | ||
| conway=s[[Tetrahedron|Y3]] | | conway=s[[Tetrahedron|Y3]] | ||
| vlayout=[[Triangle|3]]<sup>5</sup> | | vlayout=[[Triangle|3]]<sup>5</sup> | ||
| vfigure=Regular [[pentagon]], edge 1 | | vfigure=Regular [[pentagon]], edge 1 | ||
| - | |||
}}}} | }}}} | ||
| + | The icosahedron is one of the five Platonic solids. It contains 20 triangles joined five to a vertex. It can also be considered as a snubbed tetrahedron. | ||
| + | ==Coordinates== | ||
| + | The coordinates of an icosahedron with edge length 2 are: | ||
| + | <blockquote>(0, ±1, ±φ)<br>(±1, ±φ, 0)<br>(±φ, 0, ±1)</blockquote> | ||
== Equations == | == Equations == | ||
*The [[hypervolume]]s of an icosahedron with side length ''l'' are given by: | *The [[hypervolume]]s of an icosahedron with side length ''l'' are given by: | ||
| Line 27: | Line 33: | ||
surface area = 5√3 {{DotHV}}<br> | surface area = 5√3 {{DotHV}}<br> | ||
volume = {{Over|5(3+√5)|12}} {{DotHV|3}}</blockquote> | volume = {{Over|5(3+√5)|12}} {{DotHV|3}}</blockquote> | ||
| - | {{ | + | |
| + | == Relation to other polyhedra == | ||
| + | Non-convex polyhedra sharing vertex coordinates with the icosahedron can be morphed into J91 ([[bilbiro]]) and J92 ([[thawro]]) as seen in the following animations: | ||
| + | *[http://hddb.teamikaria.com/dl/VH65013KB0RBM2245X71MWVZY2.gif bilbiro] | ||
| + | *[http://hddb.teamikaria.com/dl/G6TYPETKGR822V4FPSQYPQ5PEF.gif thawro] | ||
| + | {{Selfref|The above animations are not embedded into the page due to a wiki bug. Please leave them as links and do not embed them.}} | ||
| + | |||
| + | <[#polytope [id 5]]> | ||
| + | |||
{{Trishapes}} | {{Trishapes}} | ||
Latest revision as of 14:50, 26 March 2017
The icosahedron is one of the five Platonic solids. It contains 20 triangles joined five to a vertex. It can also be considered as a snubbed tetrahedron.
Coordinates
The coordinates of an icosahedron with edge length 2 are:
(0, ±1, ±φ)
(±1, ±φ, 0)
(±φ, 0, ±1)
Equations
- The hypervolumes of an icosahedron with side length l are given by:
total edge length = 30l
surface area = 5√3 · l2
volume = 5(3+√5)∕12 · l3
Relation to other polyhedra
Non-convex polyhedra sharing vertex coordinates with the icosahedron can be morphed into J91 (bilbiro) and J92 (thawro) as seen in the following animations:
The above animations are not embedded into the page due to a wiki bug. Please leave them as links and do not embed them.
Incidence matrix
Dual: dodecahedron
| # | TXID | Va | Ea | 3a | Type | Name |
|---|---|---|---|---|---|---|
| 0 | Va | = point | ; | |||
| 1 | Ea | 2 | = digon | ; | ||
| 2 | 3a | 3 | 3 | = triangle | ; | |
| 3 | C1a | 12 | 30 | 20 | = icosahedron | ; |
Usage as facets
- 6× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron
| 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 |
