Small complex rhombicosidodecahedron

Small complex rhombicosidodecahedron
Cantellated great icosahedron.png
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-
Dual polyhedron Small complex rhombicosidodecacron
Vertex figure Cantellated great icosahedron vf.png
3(3.4.5/2.4)
Bowers acronym Sicdatrid

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 3(5/2.4.3.4), thus making it also a cantellated great icosahedron. The "3" in front of this vertex figure indicates that each vertex in this degenerate polyhedron is in fact three coincident vertices. It may also be given the Schläfli symbol rr{​52,3} or t0,2{​52,3}.

As a compoundEdit

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 Compound of five cubes Compound

As a cantellationEdit

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                                                
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
(​52 3 2)
   
{3,​52}
 
52.6.6
 
(3.​52)2
 
3.​102.​102
 
{​52,3}
 
3.4.​52.4
 
4.​102.6
 
3.3.3.3.​52

Related degenerate uniform polyhedraEdit

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 (​53.4.5.4)/3 and the great complex rhombicosidodecahedron (a compound of the great ditrigonal icosidodecahedron and the compound of five cubes) with vertex figure (​54.4.​32.4)/3. All three degenerate uniform polyhedra have each vertex in fact being three coincident vertices and each edge in fact being two coincident edges.

They can all be constructed by cantellating regular polyhedra. The complex rhombidodecadodecahedron may be given the Schläfli symbol rr{​53,5} or t0,2{​53,5}, while the great complex rhombicosidodecahedron may be given the Schläfli symbol rr{​54,​32} or t0,2{​54,​32}.

Cantellated polyhedron  
Small complex rhombicosidodecahedron
 
Complex rhombidodecadodecahedron
 
Great complex rhombicosidodecahedron
Related polyhedron  
Great icosahedron
 
Great stellated dodecahedron
 
Great dodecahedron
 
Small stellated dodecahedron
 
Regular dodecahedron
 
Regular icosahedron

See alsoEdit

ReferencesEdit

  • Klitzing, Richard. "3D uniform polyhedra sicdatrid".
  • Klitzing, Richard. "3D uniform polyhedra cadditradid".
  • Klitzing, Richard. "3D uniform polyhedra gicdatrid".