Small complex rhombicosidodecahedron

Small complex rhombicosidodecahedron
Small complex rhombicosidodecahedron
Small complex rhombicosidodecahedron
Type Uniform star polyhedron
Elements F = 62, E = 120 (60x2)
V = 20 (χ = -38)
Faces by sides 20{3}+12{5/2}+30{4}
Wythoff symbol 5/2 3 | 2
Symmetry group Ih, [5,3], *532
Index references U-, C-, W-
Small complex rhombicosidodecahedron
(3.4.5/2.4)3
(Vertex figure)		 120px
Small complex rhombicosidodecacron
(dual polyhedron)

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 triangles, 12 pentagrams and 30 squares), 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 (5/2.4.3.4)3, thus making it look identical to the cantellated great icosahedron.

Contents

As a compound

It can be seen as a compound of the small ditrigonal icosidodecahedron, U30, and the compound of five cubes. It is also a facetting of the dodecahedron.

Compound polyhedron
Small ditrigonal icosidodecahedron.png Compound of five cubes.png Cantellated great icosahedron.png
Small ditrigonal icosidodecahedron Compound of five cubes Compound

As a cantellation

It can also be seen as a cantellation of the great icosahedron (or, equivalently, of the great stellated dodecahedron).

(p q 2) Fund.
triangle		 Parent Truncated Rectified Bitruncated Birectified
(dual)		 Cantellated Omnitruncated
(Cantitruncated)		 Snub
Wythoff symbol q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
Schläfli symbol t0{p,q} t0,1{p,q} t1{p,q} t1,2{p,q} t2{p,q} t0,2{p,q} t0,1,2{p,q} s{p,q}
Coxeter–Dynkin diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Vertex figure pq (q.2p.2p) (p.q.p.q) (p. 2q.2q) qp (p. 4.q.4) (4.2p.2q) (3.3.p. 3.q)
Icosahedral
(5/2 3 2)		   Great icosahedron.png
{3,5/2}		 Great truncated icosahedron.png
(5/2.6.6)		 Great icosidodecahedron.png
(3.5/2)2		 Icosahedron.png
[3.10/2.10/2]		 Great stellated dodecahedron.png
{5/2,3}		 Cantellated great icosahedron.png
[3.4.5/2.4]		 Omnitruncated great icosahedron.png
[4.10/2.6]		 Great snub icosidodecahedron.png
(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 cantellating regular polyhedra.

See also

References

  • Richard Klitzing, 3D uniform polyhedra, sicdatrid
  • Richard Klitzing, 3D uniform polyhedra, cadditradid
  • Richard Klitzing, 3D uniform polyhedra, gicdatrid
Stub icon This geometry-related article is a stub. You can help Wikipedia by expanding it.