View source for Gyrate rhombicosidodecahedron

{{short description|72nd Johnson solid}}
{{Infobox polyhedron
 | image = Gyrate rhombicosidodecahedron.png
 | type = [[Canonical polyhedron]]<br>[[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]]s <br> 30 [[square (geometry)|square]]s <br> 12 [[pentagon]]s
 | edges = 120
 | vertices = 60
 | symmetry = <math> C_{5v} </math>
 | vertex_config = <math> \begin{align}
 &10 \times (3 \times 4^2 \times 5) + \\
 &4 \times 5 + 3 \times 10 \times (3 \times 4 \times 5 \times 4)
 \end{align} </math> 
 | properties = [[convex polytope|convex]],<br>[[Rupert property]]
 | net = Johnson solid 72 net.png
}}

In [[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}}

== Construction ==
The gyrate rhombicosidodecahedron can be constructed similarly as [[rhombicosidodecahedron]]: it is constructed from [[parabidiminished rhombicosidodecahedron]] by attaching two regular [[pentagonal cupola]]s onto its [[Decagon|decagonal]] faces. As a result, these pentagonal cupolas cover its dodecagonal faces, so the resulting polyhedron has 20 [[equilateral triangle]]s, 30 [[Square (geometry)|square]]s, and 10 [[regular pentagon]]s as its faces. The difference between those two polyhedrons is that one of two pentagonal cupolas from the gyrate rhombicosidodecahedron is rotated through 36°.{{r|berman}} A [[Convex set|convex]] polyhedron in which all faces are [[regular polygon]]s is called the [[Johnson solid]], and the gyrate rhombicosidodecahedron is among them, enumerated as the 72th Johnson solid <math> J_{72} </math>.{{r|francis}}
{{multiple image
 | image1 = Small rhombicosidodecahedron.png
 | image2 = Gyrate rhombicosidodecahedron color.png
 | footer = The difference between [[rhombicosidodecahedron]] (left) and gyrate rhombicosidodecahedron (right) by their construction
 | align = center
 | total_width = 400
}}

== Properties ==
Because the two aforementioned polyhedrons have similar construction, they have the same surface area and volume. A gyrate rhombicosidodecahedron with edge length has a surface area by adding all of the area of its faces:{{r|berman}}
<math display="block"> \left(30+5\sqrt{3}+3\sqrt{25+10\sqrt{5}}\right)a^2 \approx 59.306a^2. </math>
Its volume can be calculated by slicing it into two regular pentagonal cupolas and one parabigyrate rhombicosidodecahedron, and adding their volumes:{{r|berman}}
<math display="block"> \frac{60+29\sqrt{5}}{3}a^3 \approx 41.615a^3. </math>

The gyrate rhombicosidodecahedron is one of the five Johnson solids that do not have [[Rupert property]], meaning a polyhedron of the same or larger size and the same shape as it cannot pass through a hole in it. The other Johnson solids with no such property are [[parabigyrate rhombicosidodecahedron]], [[metabigyrate rhombicosidodecahedron]], [[trigyrate rhombicosidodecahedron]], and [[paragyrate diminished rhombicosidodecahedron]].{{r|fred}}

== See also ==
Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are: 
* The [[parabigyrate rhombicosidodecahedron]] <math> J_{73} </math> where two opposing cupolae are rotated;
* The [[metabigyrate rhombicosidodecahedron]] <math> J_{74} </math> where two non-opposing cupolae are rotated; 
* And the [[trigyrate rhombicosidodecahedron]] <math> J_{75} </math> where three cupolae are rotated.

== References ==
{{reflist|refs=

<ref name="berman">{{citation
 | last = Berman | first = Martin
 | doi = 10.1016/0016-0032(71)90071-8
 | journal = Journal of the Franklin Institute
 | mr = 290245
 | pages = 329–352
 | title = Regular-faced convex polyhedra
 | volume = 291
 | year = 1971| issue = 5
}}.</ref>

<ref name="francis">{{citation
 | last = Francis | first = Darryl
 | title = Johnson solids & their acronyms
 | journal = Word Ways
 | date = August 2013
 | volume = 46 | issue = 3 | page = 177
 | url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118
}}.</ref>

<ref name="fred">{{citation
 | last = Fredriksson | first = Albin
 | title = Optimizing for the Rupert property
 | journal = [[The American Mathematical Monthly]]
 | pages = 255–261
 | volume = 131
 | issue = 3
 | year = 2024
 | doi = 10.1080/00029890.2023.2285200
 | arxiv = 2210.00601
}}.</ref>

}}

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

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