Icosahedron (EntityTopic, 12)

From Hi.gher. Space

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| image=<[#embed [hash 9F9FGF6HTQC8EZ51GC34JEKCKN] [width 150]]>
| image=<[#embed [hash 9F9FGF6HTQC8EZ51GC34JEKCKN] [width 150]]>
| dim=3
| dim=3
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| elements=20, 30, 12
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| 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}
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| 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
}}}}
}}}}
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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.
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==Coordinates==
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The coordinates of an icosahedron with edge length 2 are:
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<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:

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

#TXIDVaEa3aTypeName
0 Va = point ;
1 Ea 2 = digon ;
2 3a 33 = triangle ;
3 C1a 123020 = icosahedron ;


Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind