Related Polyhedra
The quotient of the cube by the antipodal map yields a projective polyhedron, the hemicube.
If the original cube has edge length 1, its dual polyhedron (an octahedron) has edge length .
The cube is a special case in various classes of general polyhedra:
| Name | Equal edge-lengths? | Equal angles? | Right angles? |
|---|---|---|---|
| Cube | Yes | Yes | Yes |
| Rhombohedron | Yes | Yes | No |
| Cuboid | No | Yes | Yes |
| Parallelepiped | No | Yes | No |
| quadrilaterally faced hexahedron | No | No | No |
The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron; more generally this is referred to as a demicube. These two together form a regular compound, the stella octangula. The intersection of the two forms a regular octahedron. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other.
One such regular tetrahedron has a volume of 1⁄2 of that of the cube. The remaining space consists of four equal irregular tetrahedra with a volume of 1⁄6 of that of the cube, each.
The rectified cube is the cuboctahedron. If smaller corners are cut off we get a polyhedron with six octagonal faces and eight triangular ones. In particular we can get regular octagons (truncated cube). The rhombicuboctahedron is obtained by cutting off both corners and edges to the correct amount.
A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes.
If two opposite corners of a cube are truncated at the depth of the three vertices directly connected to them, an irregular octahedron is obtained. Eight of these irregular octahedra can be attached to the triangular faces of a regular octahedron to obtain the cuboctahedron.
The cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.
| Symmetry: | + | ||||||||
| {4,3} | t0,1{4,3} | t1{4,3} | t1,2{4,3} | {3,4} | t0,2{4,3} | t0,1,2{4,3} | s{4,3} | h{4,3} | h1,2{4,3} |
|---|---|---|---|---|---|---|---|---|---|
| Duals to uniform polyhedra | |||||||||
All these figures have octahedral symmetry.
The cube is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.
| Spherical polyhedra | Euclidean tiling | Hyperbolic tiling | ||||
|---|---|---|---|---|---|---|
| Spherical/planar symmetry |
*332 Td |
*432 Oh |
*532 Ih |
*632 P6m |
*732 |
*832 |
| Rhombic figures |
Cube |
Rhombic dodecahedron |
Rhombic triacontahedron |
Rhombille |
||
| Face configuration | V3.3.3.3 | V3.4.3.4 | V3.5.3.5 | V3.6.3.6 | V3.7.3.7 | V3.8.3.8 |
| Coxeter diagram | ||||||
Compound of three cubes |
Compound of five cubes |
Read more about this topic: Cubes
Famous quotes containing the word related:
“No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.”
—Isaac Newton (1642–1727)