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 dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... Look up pentagon in Wiktionary, the free dictionary. ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... This article is about the mineral Pyrite or Fools Gold. ...
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".
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