Pyritohedron Dodecahedron

1 pyritohedron

1.1 crystal pyrite
1.2 cartesian coordinates
1.3 geometric freedom

pyritohedron

a pyritohedron dodecahedron pyritohedral (th) symmetry. regular dodecahedron, has twelve identical pentagonal faces, 3 meeting in each of 20 vertices. however, pentagons not constrained regular, , underlying atomic arrangement has no true fivefold symmetry axes. 30 edges divided 2 sets – containing 24 , 6 edges of same length. axes of rotational symmetry 3 mutually perpendicular twofold axes , 4 threefold axes.

although regular dodecahedra not exist in crystals, pyritohedron form occurs in crystals of mineral pyrite, , may inspiration discovery of regular platonic solid form. note true regular dodecahedron can occur shape quasicrystals icosahedral symmetry, includes true fivefold rotation axes.

crystal pyrite

its name comes 1 of 2 common crystal habits shown pyrite, other 1 being cube.

cartesian coordinates

the coordinates of 8 vertices of original cube are:

(±1, ±1, ±1)

the coordinates of 12 vertices of cross-edges are:

(0, ±(1 + h), ±(1 − h))
(±(1 + h), ±(1 − h), 0)
(±(1 − h), 0, ±(1 + h))

where h height of wedge-shaped roof above faces of cube. when h = 1, 6 cross-edges degenerate points , rhombic dodecahedron formed. when h = 0, cross-edges absorbed in facets of cube, , pyritohedron reduces cube. when h = √5 − 1/2, inverse of golden ratio, result regular dodecahedron.

pyritohedra in dual positions

a reflected pyritohedron made swapping nonzero coordinates above. 2 pyritohedra can superimposed give compound of 2 dodecahedra seen in image here.

geometric freedom

the pyritohedron has geometric degree of freedom limiting cases of cubic convex hull @ 1 limit of colinear edges, , rhombic dodecahedron other limit 6 edges degenerated length zero. regular dodecahedron represents special intermediate case edges , angles equal.