Nature never grows a perfect dodecahedron, but pyrite comes close
Plato linked four of the perfect solids to earth, air, fire and water, and gave the twelve-sided dodecahedron to the cosmos itself. Ordinary crystals can never form its exact shape. Fool's gold manages a near miss, with twelve slightly lopsided pentagons, and that crystal may have inspired the geometric ideal in the first place.
Any solid with twelve flat faces is a dodecahedron, but the famous one is regular: twelve identical regular pentagons, three meeting at each corner, which works out to 20 corners joined by 30 edges. It belongs to the family of Platonic solids, and its dual, made by swapping faces for corners, is the twenty-sided icosahedron. It also hides a curious property: from one corner you can draw infinitely many straight paths across its surface that return to the starting point without passing through any other corner.
Pyrite often grows as a pyritohedron, which has the same arrangement of 12 faces and 20 vertices but pentagons that are not regular. Its edges split into a set of 24 and a set of 6, and the underlying atoms lack any true fivefold symmetry, which ordinary crystals cannot have. The mineral cobaltite can take a related form called the tetartoid, from the Greek for one quarter, because it keeps a quarter of full octahedral symmetry. A true regular dodecahedron does appear in quasicrystals, such as a holmium, magnesium and zinc alloy, which have genuine fivefold axes.
Squash the pyritohedron's six special edges down to nothing and the pentagons collapse into rhombuses, giving the rhombic dodecahedron. That shape, like the elongated dodecahedron, can pack together to fill space with no gaps. A golden-ratio relative described by Bilinski in 1960 is also a space-filler.
Stretch the regular form outward and you get three star-shaped versions, the small stellated, great and great stellated dodecahedra, which make up three of the four Kepler and Poinsot polyhedra. Counting every possible arrangement, there are 6,384,634 topologically different convex dodecahedra, not counting mirror images, with anywhere from 8 to 20 corners.
Source: Dodecahedron
