Forms — the series · Crystallography & Symmetry

The angles between the faces of {102̅}. The form {102̅} of class 3̅m in section, with each face labelled by its indices. Its 6 faces make 15 pairs and only 3 distinct angles, the smallest being 76.43°. Every value is computed from the cell's metric — the one calculation in this family that is not integer arithmetic, because an angle is a real number and a lattice does not constrain it.

Why a crystal face carries small whole numbers

A crystal face is flat because it lies on a plane of lattice points, and its orientation is therefore named by three integers rather than by two angles. That the integers exist is a theorem about lattices; that they are usually smaller than four is a separate claim about growth, and the two are routinely run together.

part 1 · applied

Two habits of {101̅}, one set of angles. The same 12 faces of class 6/mmm, grown to different distances from the centre. The outline changes completely and not one interfacial angle moves, because a face's orientation is set by the lattice and its extent by how fast it grew. That is Steno's law, and it is the reason a goniometer measures something about the substance rather than about the specimen.

The angles belong to the substance, the shape to the specimen

Two crystals of the same mineral can look nothing like each other and still have exactly the same angles between corresponding faces. That is the oldest quantitative law in the subject, and what it measures turns out to be the shape of the unit cell — which means a brass instrument from 1809 was reading lattice parameters a century before anyone knew there were any.

part 2 · applied

{111} in class m3̅m. The form {111} of crystal class m3̅m: 8 faces, being the orbit of one face under the 48 operations of the class, with a stabiliser of order 6. 4 poles lie in the upper hemisphere or in the plane of the page and are drawn filled; the other 4 lie below and are drawn open at the same positions, which is the stereographic convention and the reason only the upper ones carry their indices. The form is closed: the faces enclose a volume, so a crystal can be bounded by this form alone.

A form is an orbit, and whether it closes is an integer question

Name one face of a crystal and its class names the rest. That set is a form, it is an orbit in exactly the sense this site has used since its first essay, and whether it encloses a volume — whether a crystal could be bounded by it alone — is decided without any lengths or angles entering the calculation anywhere.

part 2 · applied

{100} offered to 5 classes: one form between them — the shape names none of the 5. The same face, {100}, handed to 5 crystal classes — m3̅m, m3̅, 432, 4̅3m, 23 — with the orbit each one returns drawn as a stereogram. Filled marks are poles in the upper hemisphere and open ones their partners below. The face counts are 6, 6, 6, 6, 6, taking 1 distinct value; the sets of faces take 1, which is the number that matters, since two classes can return the same count and different faces. Here every class returns the identical set, so a crystal bounded by this form alone has said nothing about which of them grew it.

Five classes grow the same cube

A crystal's shape is the most obvious thing about it and the least informative. Five of the thirty-two classes produce an identical cube, diffraction cannot see an inversion centre and so collapses the thirty-two to eleven, and the measurements that finally separate them are etch pits, optical rotation and a heated crystal attracting ash.

part 3 · applied

[001]: 8 of 24 faces. The 24 faces of the form {210} of a cubic crystal, stereographically projected, with one zone marked. The great circle is the set of directions perpendicular to the axis [001]; the 8 faces on it are the zone, and each is on it because the integer sum hu + kv + lw is exactly zero. No length and no angle enters that test. Faces of the lower hemisphere are drawn faintly.

A zone is a vanishing dot product

Look at a crystal and the obvious thing about it is that the faces run in bands — sets whose edges are all parallel. A face belongs to such a band exactly when three integers multiplied by three others sum to zero. No length enters, no angle enters, and that is why every index on a nineteenth-century mineral specimen is still the index used now.

part 5 · applied

A hundred and thirteen orbit types, merged into shapes. The three counts, and what stands between them. 113 is the number of kinds of form this site publishes: one for every stabiliser a face can have, in every class. Allowing a stratum to change shape along its own family raises it to 164. Merging entries that are the same solid with the same symmetry, wherever they occur, brings it down to 48 — 30 that enclose a volume and 18 that do not, which is the count every mineralogy text prints, with the dome and the sphenoid kept apart rather than merged. The last line is the warning: throwing away the symmetry of the solid and keeping only its combinatorial type leaves 35, because a rhombic dipyramid, a tetragonal dipyramid and an octahedron are one and the same arrangement of eight triangles.

A hundred and thirteen orbits, and forty-eight shapes

This collection reports 113 kinds of crystal form and every mineralogy text reports 47. That difference was explained here in a paragraph and never computed, which means nobody had checked it. Computing it needs a definition of *shape* a program can decide, and the definition turns out to be the interesting part.

part 6 · applied