Cubic symmetry
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Cubic symmetry | |
---|---|
Rank | 3 |
Space | Spherical |
Order | 48 |
Info | |
Coxeter diagram | |
Elements | |
Axes | 3 × BC_{2}×A_{1}, 4 × (G_{2}×A_{1})/2, 6 × K_{3} |
Related polytopes | |
Omnitruncate | Great rhombicuboctahedron |
Cubic symmetry, also known as octahedral symmetry and notated B_{3} or BC_{3}, is a 3D spherical Coxeter group. It is the symmetry group of the cube and octahedron.
Subgroups[edit | edit source]
- B_{3}/2 (maximal)
- B_{3}+ (maximal)
- A_{3} (maximal)
- A_{3}+
- (G_{2}×A_{1})/2 (maximal)
- (G_{2}+×A_{1})/2
- B_{2}×A_{1} (maximal)
- (B_{2}×A_{1})/2
- (B_{2}×A_{1})+
- B_{2}×I
- B_{2}+×A_{1}
- (B_{2}+×A_{1})/2
- B_{2}+×I
- (A_{2}×A_{1})+
- A_{2}×I
- A_{2}+×I
- K_{3}
- K_{3}+
- K_{2}×I
- K_{2}+×A_{1}
- K_{2}+×I
- ±(I×I×I)
- A_{1}×I×I
- I×I×I
Convex polytopes with B_{3} symmetry[edit | edit source]
- Cube (regular)/Octahedron (regular)
- Cuboctahedron (isogonal)/Rhombic dodecahedron (isotopic)
- Truncated cube (isogonal)/Triakis octahedron (isotopic)
- Truncated octahedron (isogonal)/Tetrakis hexahedron (isotopic)
- Small rhombicuboctahedron (isogonal)/Deltoidal icositetrahedron (isotopic)
- Great rhombicuboctahedron (isogonal)/Disdyakis dodecahedron (isotopic)
Wythoffians with B_{3} symmetry[edit | edit source]
Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|
Cube | cube | {4,3} | x4o3o | |
Truncated cube | tic | t{4,3} | x4x3o | |
Cuboctahedron | co | r{4,3} | o4x3o | |
Truncated octahedron | toe | t{3,4} | o4x3x | |
Octahedron | oct | {3,4} | o4o3x | |
Small rhombicuboctahedron | sirco | rr{4,3} | x4o3x | |
Great rhombicuboctahedron | girco | tr{4,3} | x4x3x | |
Snub cube | snic | sr{4,3} | s4s3s |
Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|
Cube | cube | {4/3,3} | x4/3o3o () | |
Quasitruncated hexahedron | quith | t{4/3,3} | x4/3x3o () | |
Cuboctahedron | co | r{3,4/3} | o4/3x3o () | |
Truncated octahedron | toe | t{3,4/3} | o4/3x3x () | |
Octahedron | oct | {3,4/3} | o4/3o3x () | |
Quasirhombicuboctahedron | querco | rr{3,4/3} | x4/3o3x () | |
Quasitruncated cuboctahedron | quitco | tr{3,4/3} | x4/3x3x () | |
(degenerate, oct+6(4)) | o4/3o3ß () | |||
Icosahedron | ike | s{3,4/3} | o4/3s3s () |
Name | OBSA | CD diagram | Picture |
---|---|---|---|
(degenerate, double cover of cube) | x4/3o3o4*a () | ||
Great cubicuboctahedron | gocco | x4/3x3o4*a () | |
(degenerate, oct+6(4)) | o4/3x3o4*a () | ||
(degenerate, double cover of cho) | o4/3x3x4*a () | ||
(degenerate, oct+6(4)) | o4/3o3x4*a () | ||
Small cubicuboctahedron | socco | x4/3o3x4*a () | |
Cuboctatruncated cuboctahedron | cotco | x4/3x3x4*a () |