Dodecahedral symmetry
(Redirected from H3)
Dodecahedral symmetry | |
---|---|
Rank | 3 |
Space | Spherical |
Order | 120 |
Info | |
Coxeter diagram | |
Centrally symmetric | Yes |
Elements | |
Axes | 6 × (I2(10)×A1)/2, 10 × (G2×A1)/2, 15 × K3 |
Related polytopes | |
Omnitruncate | Great rhombicosidodecahedron |
Dodecahedral symmetry, also known as icosahedral symmetry, doic symmetry, and notated as H3, is a 3D spherical Coxeter group. It is the symmetry group of the dodecahedron and icosahedron.
Subgroups[edit | edit source]
- Chiral dodecahedral symmetry (maximal)
- Pyritohedral symmetry (maximal)
- Chiral tetrahedral symmetry
- Pentagonal antiprismatic symmetry (maximal)
- Propentagonal antiprismatic symmetry
- Chiral pentagonal prismatic symmetry
- Pentagonal pyramidal symmetry
- Chiral pentagonal pyramidal symmetry
- Triangular antiprismatic symmetry (maximal)
- Protriangular antiprismatic symmetry
- Chiral triangular prismatic symmetry
- Triangular pyramidal symmetry
- Chiral triangular pyramidal symmetry
- Digonal prismatic symmetry
- Chiral digonal prismatic symmetry
- Prodigonal prismatic symmetry
- Rectangular pyramidal symmetry
- Chiral digonal pyramidal symmetry
- Inversion symmetry)
- Reflection symmetry
- Identity symmetry
Convex polytopes with H3 symmetry[edit | edit source]
- Dodecahedron (regular)/Icosahedron (regular)
- Icosidodecahedron (isogonal)/Rhombic triacontahedron (isotopic)
- Truncated dodecahedron (isogonal)/Triakis icosahedron (isotopic)
- Truncated icosahedron (isogonal)/Pentakis dodecahedron (isotopic)
- Small rhombicosidodecahedron (isogonal)/Deltoidal hexecontahedron (isotopic)
- Great rhombicosidodecahedron (isogonal)/Disdyakis triacontahedron (isotopic)
Wythoffians with H3 symmetry[edit | edit source]
Name | OBSA | CD diagram | Picture |
---|---|---|---|
Small ditrigonary icosidodecahedron | sidtid | x5/2o3o3*a () | |
(degenerate, double cover of id) | x5/2x3o3*a () | ||
(degenerate, double cover of ike) | o5/2o3x3*a () | ||
Small icosicosidodecahedron | siid | x5/2o3x3*a () | |
(degenerate, double cover of ti) | x5/2x3x3*a () | ||
Small snub icosicosidodecahedron | seside | s5/2s3s3*a () |
Name | OBSA | CD diagram | Picture |
---|---|---|---|
Ditrigonary dodecadodecahedron | ditdid | ||
Small complex icosidodecahedron (degenerate, ike+gad) | cid | ||
Great complex icosidodecahedron (degenerate, sissid+gike) | gacid | ||
Icosidodecadodecahedron | ided | ||
Small ditrigonal dodecicosidodecahedron | sidditdid | ||
Great ditrigonal dodecicosidodecahedron | gidditdid | ||
Icosidodecatruncated icosidodecahedron | idtid | ||
Snub icosidodecadodecahedron | sided |
Name | OBSA | CD diagram | Picture |
---|---|---|---|
Great complex icosidodecahedron (degenerate, sissid+gike) | gacid | x5/3o5/2o3*a () | |
Great dodecicosidodecahedron | gaddid | x5/3x5/2o3*a () | |
(degenerate, double cover of gissid) | o5/3x5/2o3*a () | ||
(degenerate, ditdid+gidtid) | o5/3x5/2x3*a () | ||
Great complex icosidodecahedron (degenerate, sissid+gike) | gacid | o5/3o5/2x3*a () | |
(degenerate, double cover of sidhei) | x5/3o5/2x3*a () | ||
(degenerate, giddy+12(10/2)) | x5/3x5/2x3*a () | ||
Great snub dodecicosidodecahedron | gisdid | s5/3s5/2s2*a () |