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Pyritohedron

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Pyritohedron

Pyritohedron

A pyritohedron has 30 edges, divided into two lengths: 24 and 6 in each group.
Face polygon irregular pentagon
Coxeter diagrams
Faces 12
Edges 30 (6+24)
Vertices 20 (8+12)
Symmetry group Th, [4,3+], (3*2), order 24
Rotation group T, [3,3]+, (332), order 12
Dual polyhedron Pseudoicosahedron
Properties convex

In geometry, 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.

Crystal pyrite

Its name comes from one of the two common crystal forms of pyrite, the other one being cubical.


Cubic pyrite

Pyritohedral
quasicrystal

Cartesian coordinates

The coordinates of the 8 vertices:

(±1, ±1, ±1)

The coordinates of the 12 vertices are the permutations of:

(0, 1+h, 1−h2)

where h is the height of the wedge roof above the faces of the cube. When h=1, the 6 edges degenerate to points and rhombic dodecahedron is formed. For the regular dodecahedron, h=(√5−1)/2, the golden ratio.

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.

Special cases of the pyritohedron
1 : 1 1 : 1 2 : 1 1.3092... : 1 1 : 1 0 : 1
h=0 h=(√5−1)/2 h=1
pentagrams
Concave pyritohedral dodecahedron
cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions. Weaire–Phelan structure dodecahedron is an intermediate case with equal edge lengths. rhombic dodecahedron is the limiting case with the 6 crossedges reducing to length zero.

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".

Related polyhedra and tilings

The pyritohedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

It is also constructed in half-symmetry of the tetrahedron, as a dual of the snub tetrahedron.

See also

External links

  • MathWorld.
  • stellations.
  • Plato's Fourth Solid and the "Pyritohedron", by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–226 [1]
  • THE GREEK ELEMENTS
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