# pyritohedron

PYRITE and QUARTZ (Iron Sulfide and Silicon Dioxide) from Peru. Taken with a microscope at 40X magnification, the pyritohedral pyrite crystals are associated with minute double terminated quartz crystals that are less than 1 millimeter in length.

Pyritohedron progression from cube to rhombic dodecahedron.

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A pyritohedron is a 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 necessarily regular, so the structure normally 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.

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 equa - Image:

1. A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions (2 : 1) & h= 0.
2. Regular star, great stellated dodecahedron, with pentagonal distorted into regular pentagrams (1 : 1)
3. Concave pyritohedral dodecahedron (1 : 1).
4. The geometric proportions of the pyritohedron in the Weaire–Phelan structure (1.3092… : 1)
5. A regular dodecahedron is an intermediate case with equal edge lengths (1 : 1) & h=(√5−1)/2.
6. A rhombic dodecahedron is the limiting case with the 6 crossedges reducing to length zero (0 : 1) & h=0.

Cartesian coordinates: The coordinates of the 8 vertices of the original cube are: (±1, ±1, ±1)
The coordinates of the 12 vertices of the cross-edges are:
(0, ±(1+h), ±(1−h^2))
(±(1+h), ±(1−h^2), 0)
(±(1−h^2), 0, ±(1+h))
where h is the height of the wedge-shaped “roof” above the faces of the cube. When h=1, the six cross-edges degenerate to points and a rhombic dodecahedron is formed. When h=0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h=(√5−1)/2, the inverse of the golden ratio, the original edges of the cube are absorbed in the facets of the wedges, which become co-planar, resulting in a regular dodecahedron.

Image & the article : Shared at http://en.wikipedia.org/wiki/Dodecahedron & Dodécaèdre on Wikipedia.

PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

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PYRITE (Iron Sulfide) from logrono, Spain. These are microscope photos taken at 15X magnification showing the well developed striations on the pyritohedron crystals.

PYRITE (Iron Sulfide) from Mexico. Pyritohedron crystals taken with a microscope at 10X.

PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

PYRITE (Iron Sulfide) from Logrono, Spain. Photo, taken with a microscope at 10X, shows the faces of a perfectly formed pyritohedron crystals.

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PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

GALENA (Lead Sulfide) with quartz and pyrite from Madan District, Bulgaria. Well developed cubic and octahedral galena crystals sit on a matrix of euhedral quartz with inter-grown pyritohedrons.

PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

PYRITE (Iron Sulfide) from Peru. Pyritohedron crystal.

PYRITE (Iron Sulfide) from Peru. Three large pyritohedron crystals.

PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

PYRITE (Iron Sulfide) from Logrono, Spain showing clusters of pyritohedron crystals.

GALENA (Lead Sulfide) with quartz and pyrite from Madan District, Bulgaria. Well developed cubic and octahedral galena crystals sit on a matrix of euhedral quartz with inter-grown pyritohedrons.