Revision 471670918 of "Small complex rhombicosidodecahedron" on enwiki

{{Uniform polyhedra db|Uniform polyhedron stat table|Scrid}}
In geometry, the '''small complex rhombicosidodecahedron''' (also known as the '''small complex ditrigonal rhombicosidodecahedron''') is a degenerate [[uniform star polyhedron]]. It has 62 faces (20 [[triangle]]s, 12 [[pentagram]]s and 30 [[Square (geometry)|square]]s), 120 (doubled) edges and 20 vertices. All edges are doubled (making it degenerate), sharing 4 faces, but are considered as two overlapping edges as a topological polyhedron.

It can be constructed from the vertex figure (<sup>5</sup>/<sub>2</sub>.4.3.4)<sup>3</sup>, thus making it look identical to the [[cantellation (geometry)|cantellated]] [[great icosahedron]].

==As a compound==
It can be seen as a [[polyhedron compound|compound]] of the [[small ditrigonal icosidodecahedron]], U<sub>30</sub>, and the [[compound of five cubes]]. It is also a [[facetting]] of the dodecahedron.

{| class=wikitable width=300
|+ '''Compound polyhedron'''
|[[File:Small ditrigonal icosidodecahedron.png|100px]]
|[[File:Compound of five cubes.png|100px]]
|[[File:Cantellated great icosahedron.png|101px]]
|- align=center
|[[Small ditrigonal icosidodecahedron]]
|[[Compound of five cubes]]
|Compound
|}

==As a cantellation==
It can also be seen as a [[cantellation (geometry)|cantellation]] of the [[great icosahedron]] (or, equivalently, of the [[great stellated dodecahedron]]).

{| class="wikitable"
|-
!(p q 2)
!Fund.<BR>triangle
!Parent
!Truncated
!Rectified
!Bitruncated
!Birectified<BR>(dual)
!Cantellated
!Omnitruncated<BR>(<small>Cantitruncated</small>)
!Snub
|-
![[Wythoff construction|Wythoff symbol]]
!
! q &#124; p 2
! 2 q &#124; p
! 2 &#124; p q
! 2 p &#124; q
! p &#124; q 2
! p q &#124; 2
! p q 2 &#124;
! &#124; p q 2
|-
![[Schläfli symbol]]
!
!t<sub>0</sub>{p,q}
!t<sub>0,1</sub>{p,q}
!t<sub>1</sub>{p,q}
!t<sub>1,2</sub>{p,q}
!t<sub>2</sub>{p,q}
!t<sub>0,2</sub>{p,q}
!t<sub>0,1,2</sub>{p,q}
!s{p,q}
|-
![[Coxeter–Dynkin diagram]]
!
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node_1|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node_1}}
!{{CDD|node|p|node|q|node_1}}
!{{CDD|node_1|p|node|q|node_1}}
!{{CDD|node_1|p|node_1|q|node_1}}
!{{CDD|node_h|p|node_h|q|node_h}}
|-
![[Vertex configuration|Vertex figure]]
!
!p<sup>q</sup>
!(q.2p.2p)
!(p.q.p.q)
!(p.&nbsp;2q.2q)
!q<sup>p</sup>
!(p.&nbsp;4.q.4)
!(4.2p.2q)
!(3.3.p.&nbsp;3.q)
|-
|Icosahedral<BR>(5/2 3 2)
|&nbsp;
|[[Image:Great icosahedron.png|64px]]<BR>[[Great icosahedron|{3,5/2}]]
|[[Image:Great truncated icosahedron.png|64px]]<BR>[[Truncated great icosahedron|(5/2.6.6)]]
|[[Image:Great icosidodecahedron.png|64px]]<BR>[[Great icosidodecahedron|(3.5/2)<sup>2</sup>]]
|[[Image:Icosahedron.png|64px]]<BR>[[Small complex icosidodecahedron|[3.10/2.10/2]]]
|[[Image:Great stellated dodecahedron.png|64px]]<BR>[[Great stellated dodecahedron|{5/2,3}]]
|[[Image:Cantellated great icosahedron.png|64px]]<BR>[3.4.5/2.4]
|[[Image:Omnitruncated great icosahedron.png|64px]]<BR>[[Rhombicosahedron|[4.10/2.6]]]
|[[Image:Great snub icosidodecahedron.png|64px]]<BR>[[Great snub icosidodecahedron|(3.3.3.3.5/2)]]
|}

==Related degenerate uniform polyhedra==
Two other degenerate uniform polyhedra are also facettings of the dodecahedron. They are the '''complex rhombidodecadodecahedron''' (a compound of the [[ditrigonal dodecadodecahedron]] and the compound of five cubes) with vertex figure (5/3.4.5.4)/3 and the '''great complex rhombicosidodecahedron''' (a compound of the [[great ditrigonal icosidodecahedron]] and the compound of five cubes) with vertex figure (5/4.4.3/2.4)/3. All three degenerate uniform polyhedra have each vertex really being three coincident vertices and each edge really being two coincident edges.

They can all be constructed by [[cantellation (geometry)|cantellating]] regular polyhedra.

==See also==
*[[Small complex icosidodecahedron]]
*[[Great complex icosidodecahedron]]
*[[Complex rhombidodecadodecahedron]]
*[[Great complex rhombicosidodecahedron]]

==References==
* {{KlitzingPolytopes|polyhedra-neu.htm|3D uniform polyhedra|sicdatrid}}
* {{KlitzingPolytopes|polyhedra-neu.htm|3D uniform polyhedra|cadditradid}}
* {{KlitzingPolytopes|polyhedra-neu.htm|3D uniform polyhedra|gicdatrid}}

[[Category:Polyhedra]]

{{polyhedron-stub}}