page_title: Dodecahedron
section_title: Cartesian coordinates
hierarchical_section_title: Dodecahedron / Other pentagonal dodecahedra / Pyritohedron / Cartesian coordinates
caption_attribution_description: English: Compound pyritohedron and dual position
context_page_description: In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
  The pyritohedron, a common crystal form in pyrite, is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron, seen as a limiting case of the pyritohedron, has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There are numerous other dodecahedra.
context_section_description: If the eight vertices of a cube have coordinates of:
  (±1, ±1, ±1)
  Then a pyritohedron has 12 additional vertices:
  (0, ±(1 + h), ±(1 − h²))
  (±(1 + h), ±(1 − h²), 0)
  (±(1 − h²), 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 the pyritohedron reduces to a rhombic dodecahedron. When h = 0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = , the multiplicative inverse of the golden ratio, the result is a regular dodecahedron. When h = , the conjugate of this value, the result is a regular great stellated dodecahedron. For natural pyrite, h = .
  A reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra. The image to the left shows the case where the pyritohedra are convex regular dodecahedra.
caption: Pyritohedra in dual positions
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