Icosahedron (EntityTopic, 12)
From Hi.gher. Space
(Difference between revisions)
m (ad) |
Username5243 (Talk | contribs) |
||
(22 intermediate revisions not shown) | |||
Line 1: | Line 1: | ||
- | {{Shape|Icosahedron| | + | <[#ontology [kind topic] [cats 3D Regular Polytope] [alt [[freebase:03vyw]] [[wikipedia:Icosahedron]]]]> |
- | = | + | {{STS Shape |
- | + | | name=Icosahedron | |
- | + | | image=<[#embed [hash 9F9FGF6HTQC8EZ51GC34JEKCKN] [width 150]]> | |
- | + | | dim=3 | |
- | + | | 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 | ||
+ | | ssc={G3<sup>5</sup>} | ||
+ | | ssc2=Ki4 | ||
+ | | pv_circle=~0.6055 | ||
+ | | extra={{STS Polytope | ||
+ | | flayout={{FLD|a3|er|e5}} | ||
+ | | petrie=10,2 | ||
+ | | dual=[[Dodecahedron]] | ||
+ | | bowers=Ike | ||
+ | }}{{STS Uniform polytope | ||
+ | | wythoff=<nowiki>5 | 2 3 </nowiki> | ||
+ | | schlaefli={[[Triangle|3,]]5} | ||
+ | | dynkin=o5o3x, s3s3s | ||
+ | | conway=s[[Tetrahedron|Y3]] | ||
+ | | vlayout=[[Triangle|3]]<sup>5</sup> | ||
+ | | 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. | ||
- | *The [[hypervolume]]s of an icosahedron are given by: | + | ==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 == | ||
+ | *The [[hypervolume]]s of an icosahedron with side length ''l'' are given by: | ||
<blockquote>total edge length = 30''l''<br> | <blockquote>total edge length = 30''l''<br> | ||
- | surface area = | + | surface area = 5√3 {{DotHV}}<br> |
- | volume = 5 | + | 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}} |
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 |