View source for Gyrate rhombicosidodecahedron

{{short description|72nd Johnson solid}}
{{Infobox polyhedron
|image=gyrate_rhombicosidodecahedron.png
|type=[[Johnson solid|Johnson]]<br>{{math|[[Triaugmented truncated dodecahedron|''J''<sub>71</sub>]] – '''''J''<sub>72</sub>''' – [[Parabigyrate rhombicosidodecahedron|''J''<sub>73</sub>]]}}
|faces=20 [[Triangle|triangle]]s <br> 30 squares <br> 12 [[Pentagon|pentagon]]s
|edges=120
|vertices=60
|symmetry={{math|''C''<sub>5v</sub>}}
|vertex_config={{math|10(3.4<sup>2</sup>.5)<br>4x5+3x10(3.4.5.4)}}
|dual=-
|properties=[[Convex polytope|convex]], [[Midsphere#Canonical polyhedron|canonical]]
|net=Johnson solid 72 net.png
}}

In [[Geometry|geometry]], the '''gyrate rhombicosidodecahedron''' is one of the [[Johnson solid]]s ({{math|''J''<sub>72</sub>}}). It is also a [[Midsphere#Canonical polyhedron|canonical polyhedron]].

{{Johnson solid}}

== Related polyhedron ==

It can be constructed as a [[Rhombicosidodecahedron|rhombicosidodecahedron]] with one [[Pentagonal cupola|pentagonal cupola]] rotated through 36 degrees. They have the same faces around each vertex, but [[Vertex configuration|vertex configuration]]s along the rotation become a different order, {{math|3.4.4.5}}.
{| class=wikitable
|[[File:Small rhombicosidodecahedron.png|160px]]<br>[[Rhombicosidodecahedron]]
|[[File:Gyrate rhombicosidodecahedron color.png|160px]]<br>'''Gyrate rhombicosidodecahedron'''
|}

Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are: 
* The [[Parabigyrate rhombicosidodecahedron|parabigyrate rhombicosidodecahedron]] ({{math|''J''{{sub|73}}}}) where two opposing cupolae are rotated;
* The [[Metabigyrate rhombicosidodecahedron|metabigyrate rhombicosidodecahedron]] ({{math|''J''{{sub|74}}}}) where two non-opposing cupolae are rotated; 
* And the [[Trigyrate rhombicosidodecahedron|trigyrate rhombicosidodecahedron]] ({{math|''J''{{sub|75}}}}) where three cupolae are rotated.

==External links==
* {{Mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid | urlname =GyrateRhombicosidodecahedron| title =   Gyrate rhombicosidodecahedron}}

[[Category:Johnson solids]]
{{Johnson solids navigator}}

{{Sourceattribution|Gyrate rhombicosidodecahedron}}