# Toward a More Perfect Pyritohedron

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

## Pyritohedron

The pyritohedron, so called because it is a common crystal form in pyrite, looks like a pentagonal dodecahedron, but of course lacks the five-fold symmetry.

Mathematician John Baez has a page extolling the Platonic dodecahedron and icosahedron, and when I saw it at Cliff Pickover's Reality Carnival, I was pleased to see it linked some of my drawings. The drawings above are of {210} faces (see Miller indices), which, bluntly, were the only pyritohedra I'd ever heard of. Baez pointed out that a true dodecahedron would have Miller indices of ratio phi (1.618..), which is irrational, but that {320} and {530} would be better approximations.

So who am I to pass up a challenge?

### {320} Pyritohedron

Below is a {320} pyritohedron. The mirror planes are in light blue. Cube faces of different orientations are shaded. Because of the isometric projection used, cubes at different depths on the left sde line up and make it impossible to distinguish depth. The middle left face uses a different shade to distinguish it.

Below is the same solid with pentagonal faces outlined.

In practice, steps three cells wide and two cells high are unlikely to occur in nature, because there would be a strong charge concentration in the corner that would attract oppositely charged ions. It seems a good deal more likely that the faces would have intermediate steps, alternately one and two cells wide. Such a solid is shown below.

Below is a [530] pyritohedron. Mirror planes are light blue and cell faces are shaded. Again the isometric projection used causes faces in certain directions to align so there is no depth indication. The left middle  face is shaded lavender to distinguish it.

Below is the same solid with pentagonal faces outlined. The large steps create a blocky apperance and, frankly, a less regular appearance than {320}.

Below is a {530} solid with the large steps filled in by smaller ones. The appearance is a lot more regular.

Below is the same shape with pentagonal faces outlined. It is a clear improvement over {210} and {320}.