Icosahedron (EntityTopic, 12)
From Hi.gher. Space
(Difference between revisions)
(add symmetry group) |
Username5243 (Talk | contribs) |
||
(2 intermediate revisions not shown) | |||
Line 4: | Line 4: | ||
| image=<[#embed [hash 9F9FGF6HTQC8EZ51GC34JEKCKN] [width 150]]> | | 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)]] | | sym=[[Rhodohedral symmetry|I<sub>h</sub>, H<sub>3</sub>, [5,3], (*532)]] | ||
| genus=0 | | genus=0 | ||
Line 18: | Line 18: | ||
| 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]]> | <[#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 |