Pyritohedron

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A pyritohedron is an irregular dodecahedron. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 corners. However, the pentagons are not regular, and the structure has no five-fold symmetry axes; instead, it has a tetrahedral symmetry. It is one of the two common crystal forms of pyrite, the other one being cubical.

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 to 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".