| Pyritohedron | |
|---|---|
A pyritohedron has 30 edges, divided into two lengths: 24 and 6 in each group. |
|
| Face polygon | irregular pentagon |
| Faces | 12 |
| Edges | 30 (6+24) |
| Vertices | 20 (8+12) |
| Symmetry group | Th, [4,3+], (3*2) |
| Dual polyhedron | Pseudoicosahedron |
| Properties | convex |
In geometry, a pyritohedron is an irregular dodecahedron with pyritohedral (Th) symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular, and the structure has no fivefold symmetry axes. Its 30 edges are divided into two sets - containing 24 and 6 edges of the same length.
Although regular dodecahedra do not exist in crystals, the distorted pyritohedron form occurs in the crystal pyrite, and it may be an inspiration for the discovery of the regular Platonic solid form.
Contents |
Its name comes from one of the two common crystal forms of pyrite, the other one being cubical.
Cubic pyrite |
Pyritohedral |
Ho-Mg-Zn quasicrystal |
The coordinates of the 8 vertices:
The coordinates of the 12 vertices are the permutations of:
where h is the height of the wedge roof above the faces of the cube. When h=1, the 6 edges degenerate to points and rhombic dodecahedron is formed. For the regular dodecahedron, h=(√5−1)/2, the golden ratio.
The pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of colinear edges, and a rhombic dodecahedron as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal.
| 2 : 1 | 1.3092... : 1 | 1 : 1 | 0 : 1 |
|---|---|---|---|
A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions. |
The geometric proportions of the pyritohedron in the Weaire–Phelan structure |
A regular dodecahedron is an intermediate case with equal edge lengths. |
A rhombic dodecahedron is the limiting case with the 6 crossedges reducing to length zero. |
A regular dodecahedron can be formed from a cube in the following way: The top square in the cube is replaced by a "roof" composed of two pentagons, joined along the top of the roof. The diagonals in the pentagons parallel to the top of the roof coincide with two opposite sides of the square. The other five squares are replaced by a pair of pentagons in a similar way. The pyritohedron is constructed by changing the slope of these "roofs".
An example concave pyritohedral dodecahedron