Geometric Freedom
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. |
An example concave pyritohedral dodecahedron |
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".
Read more about this topic: Pyritohedron
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