Icosahedron (EntityTopic, 12)
From Hi.gher. Space
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(add morphing animations for bilbiro and thawro) |
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| 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 | ||
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| 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. | ||
+ | |||
== 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: |
Revision as of 02:18, 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.
Equations
- The hypervolumes of an icosahedron with side length l are given by:
total edge length = 30l
surface area = 5√3 · l^{2}
volume = ^{5(3+√5)}∕_{12} · l^{3}
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 |