Pyritohedral crystal

The three images above show photographs of the mineral pyrite as a crystal having twelve faces. This specimen is not perfectly regular.

How to determine when such crystals are genuine habits rather than sliced or faceted is an interesting question. I must attempt to measure or calculate the dihedral angles, though this may require a special instrument to do accurately, as the spokes of most protractors are not continued to such a small scale size. In this sample, pairs of opposite faces appear to be parallel. Casually, not all dihedral angles seem equal. Some of the faces are smoother and better mirrors than others.

I read that pyritohedral crystals are hypothesized to be how the regular dodecahedron was first discovered.

Of course, the perfectly regular solid of twelve regular pentagonal faces cannot be the unit cell of a crystallographic lattice in three-dimensional Euclidean space.

References

• http://mathworld.wolfram.com/Pyritohedron.html
  
• https://www.minerals.net/mineral_glossary/pyritohedron.aspx
  
• https://dozenal.forumotion.com/t21-no-more-than-six-regular-4-dimensional-figures
  

The pyritohedral symmetry ( 2*3) is a subgroup of the icosahedral group (*2 3 5), by a factor 5.

The twelve faces are tangent to points on the edges of the octahedron. If the shape forms correctly, it corresponds to \2*\3, and will have three edges (at the octahedron-vertex) of equal length, and split by mirrors both ways, and the remaining edges, 3 per octant, form the supports to an inscribed cube.

The planes are indeed natural, rathered than cleven. It's a natural cut along the 1:1:0 face symmetry. The underlying lattice is the semicubic, or close-pack cubic, and there are several different planes (in crystalography) that identify it. Some planes are weaker-bonds than the others, which is why you get clean faces as in the photos.

It probably was how the dodecahedron was discovered.

The irregularity of the crystal appears to be in the distances of the faces from the centre of symmetry. The crystal does appear to be well formed in a particular orientation. Let a vertical co-ordinate axis be called the z-axis, and the horizontal axes be called the y-axis and x-axis.

Mirrors
The faces of the best mirrors are vertical faces. Their normals have no vertical component. They face more towards the x-axis than y-axis. The dullest faces are those at the top and bottom of the crystal and have the greatest horizontal aspect of the faces. Their normals are in the xz-plane and directed closer angularly to the z-axis. The remaining faces are intermediate for mirror and slope. The normals of these faces are in the yz-plane and closer to the y-axis.

Striations
The vertical faces have vertical striations. The other faces have horizontal striations. On the top and bottom faces, the striations are aligned with the y-axis. On the faces facing a direction in the yz-plane, the striations are aligned with the x-axis.

Edges
Between the vertical faces are vertical edges, aligned with the z-axis. Between the top and bottom faces are horizontal edges aligned with the y-axis. Between the remaining faces are horizontal edges aligned with the x-axis.

Angles
It was difficult to measure angles accurately without proper equipment.
Surface Face Angles: These were examined using a protractor. There are at least two sizes of angles between edges on the faces. Angles between edges neither of which is aligned with a co-ordinate axis are about zot four tau radians.  The obtuse angles having one arm aligned with a co-ordinate axis are smaller.
Dihedral Angles: These were inspected visually and are roughly a third of a turn. Attempt was made to calculate them using distances between vertices measured using the ruler of calipers and a compass fitted with two needles. The main error is from deciding where the vertices on the crystal are. However, a very small error, of say a few square dozenths of a centimetre, in the distances could lead to large errors, of a few per gross tau radians, in the angles calculated. Calculation of the dihedral angles statistically using many measurements may produce a reasonably good estimate, although this study would be time-consuming.

It would be interesting to compare the inclinations of the faces to those in the pyritohedron of the Weaire–Phelan structure.

Nomenclature
For a dozenist, writing the word dodecahedron feels strange. There has been a proposal for naming of polyhedrons dozenally.

References
https://en.wikipedia.org/wiki/Dodecahedron#Pyritohedron
https://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure
https://dozenal.forumotion.com/t20-dozenal-nomenclature-of-geometrical-figures

Assuming Miller indices (xyz) for the faces as permutations of plus or minus zero, one, and two in the places of the three axes, and writing equations of two planes Ax + By + Cz + D = 0 and Ex + Fy + Gz + H = 0, ignoring the translations D and H along the co-ordinate axes, and using the Miller indices of one face for A, B, C and of the other for E, F, G, then the dihedral angle, θ, between those faces may be calculated by $$\cos{θ} = \frac{AE + BF +GC}{\sqrt{A^{2} + B^{2} + C^{2}}\sqrt{E^{2} + F^{2} + G^{2}}}.$$ This provides for the larger dihedral angle a slope of four to minus three. The cosine of the lesser dihedral angle may be similarly obtained to be minus two fifths, if not mistaken.

References

• https://en.wikipedia.org/wiki/Miller_index
  
• https://study.com/academy/lesson/dihedral-angle-definition-calculation.html
  

Phaethon wrote:This provides for the larger dihedral angle a slope of four to minus three. The cosine of the lesser dihedral angle may be similarly obtained to be minus two fifths, if not mistaken.

The dihedral angles so obtained conform to the observations of being around a third of a turn and may be used to calculate the face angles.

Phaethon, Sun 27 Oct 2019 wrote:There are at least two sizes of angles between edges on the faces. Angles between edges neither of which is aligned with a co-ordinate axis are about zot four tau radians.  The obtuse angles having one arm aligned with a co-ordinate axis are smaller.

There is more than one size of angle between edges not aligned with the co-ordinate axes. One of these is at vertices where the three dihedral angles are the same and the lesser, from which the three equal face angles between edges at that vertex are calculated to have a cosine of minus two sevenths. This angle agrees with its approximate measurement by protractor.

The other of the two angles on a face between edges not aligned with co-ordinate axes is at a vertex of an edge that is aligned with a co-ordinate axis. This angle is calculated to have a cosine of minus 11/21. It is a bit more than zot four tau radians as approximately measured.

The angle on a face beside an edge as an arm or leg aligned with a co-ordinate axis is calculated to have a slope ratio of twice the square root of five to minus one. This is consistent with the observation.