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

A crystal of fluorite shows eight triangular faces meeting in an octahedron. Not seven, not nine — eight, and always eight, and if one of them is missing it is because the crystal grew against something.

The reason is not mysterious once it is said. Fluorite’s class is m3̅m; the class contains an operation carrying the face (111) onto (11̅1), and another carrying it onto (1̅11), and so on; and any face the class produces from a face that is present must itself be present, because the two are indistinguishable to the crystal. The eight faces are the orbit of one of them.

That word has been on this site from the beginning, where the orbit of a motif is the pattern. Nothing about it changes here except what is being acted on: face indices in place of points, and one of the thirty-two crystal classes in place of a plane group.

Multiplicity is orbit-stabiliser, and nothing else #

The class m3̅m has forty-eight operations and the octahedron has eight faces. The missing factor of six is the number of operations that leave (111) exactly where it is: the three-fold rotation about the body diagonal, its square, and the three mirrors containing that diagonal, with the identity making six.

That set is the face’s stabiliser, and the relation

$$|\text{orbit}| \times |\text{stabiliser}| = |G|$$

is the orbit-stabiliser theorem. It is asserted here on every figure that draws a form, by counting both sides rather than dividing one into the other — a division cannot fail, and a count can.

Every table of face multiplicities in every mineralogy textbook is that identity, tabulated by people who mostly did not have the group. The numbers are correct because the geometry forced them, and reading them as a consequence rather than as a list is what makes the pattern in them visible: the multiplicities of a class are exactly the divisors of its order that appear as indices of subgroups fixing a direction.

General and special, and how many kinds there are #

A face on no symmetry element has the identity for its stabiliser — the same condition a general direction has to satisfy for a stereogram to show the whole class — and gives the largest orbit the class allows: the general form. There is exactly one general form per class.

A face lying on a mirror, or along an axis, has a larger stabiliser and a smaller orbit: a special form. How many kinds of special form a class has is a question about how many conjugacy classes of stabiliser its faces can have, and that is a finite question decided in integers.

Running it over all thirty-two classes gives 113 kinds of form. The general one for each class, and eighty-one special ones between them. Nothing in that number was looked up; it comes from applying each class to every set of small indices, sorting the orbits by the symmetry of the face that produced them, and counting the sorts.

It is also a bounded search, and a bounded search that stops too early returns an answer that is correct as far as it went — which is a failure this site met in the space-group enumeration and again in the class enumeration. So the census is computed at three search widths and required to agree at the last two. It does not agree at the first: searching only to indices of two gives 112, one short, because the general form of the cubic classes needs three distinct non-zero indices and the smallest such triple is (123). The one that is missing is the largest form there is.

Whether a form closes, decided without a ruler #

Here is the question that makes this more than bookkeeping.

Some forms enclose a volume. A cube does; an octahedron does; a crystal can be a cube and nothing else. Other forms do not: a hexagonal prism is six faces around an axis and open at both ends, and no crystal is a prism and nothing else — it must have something on the ends, and that something is a second form.

The distinction is called closed and open, and the surprise is what decides it.

A set of faces fails to enclose anything exactly when they all lean the same way — when some direction exists with every face normal on the same side of it. So the form is closed if and only if no half-space contains all of its poles.

Now, a half-space is given by a direction r in the crystal, and a pole is a set of indices (hkl) in the reciprocal lattice, and the pairing between them is hu + kv + lw. That expression has no metric in it. The lengths of the cell edges, the angles between them, the axial ratio the whole of the previous rung was about — none of them appear, because the reciprocity relation cancels them exactly.

So whether a form is open or closed is decided by integer arithmetic, and the answer is the same for every substance in a class. That is why the mineralogists could tabulate open and closed forms in the eighteenth century, before anyone could measure a cell parameter: the question they were answering never needed one.

There are two ways a form can fail to close and the test tells them apart, which is worth knowing because they look different on a specimen. A pyramid has every pole strictly on one side of some plane, so the shape runs off in a single direction and a crystal closes it with faces at one end. A prism has its poles lying in a plane through the origin — every face parallel to one axis — so the shape is open at both ends and needs faces at each. The difference is the rank of the poles as integer vectors: three for the first, two for the second, and one for a pinacoid, which is two parallel faces and a slab.

The test needs care in one place, and the care is what makes it a decision rather than a sample. To ask whether some half-space contains every pole, it is not enough to try a few directions. The set of directions that would work is a cone, and a cone in three dimensions cut out by planes has its extreme rays on the intersections of pairs of those planes — so trying every cross product of two poles is trying every candidate that could possibly win. Nothing is sampled and nothing is approximated.

What the census says #

Of the 113 forms, 43 are closed and 70 are open. Open forms outnumber closed ones by nearly two to one, which is not the impression a case of museum specimens gives, and the reason is that the closed ones are the ones that make a crystal by themselves and so are the ones that get photographed.

Twelve of the thirty-two classes have no closed form at all. Every crystal in one of those classes is a combination — two forms at minimum, and the shape a mineral collection labels as a single habit is two orbits stacked.

The twelve are worth listing, because the pattern in them is immediate: 1, 1̅, 2, m, 3, 4, 6, 2/m, mm2, 3m, 4mm and 6mm. Every one of them is a class whose operations leave some direction alone, or leave a whole plane alone, so no orbit of a face can wrap all the way round.

There is a reading of that list which makes it more than a list. A form fails to close when all of its poles fit in a half-space, and an orbit’s poles fit in a half-space when the class leaves some direction alone — which is exactly the condition for the class to be polar. And indeed all ten polar classes are here. A crystal with a built-in electrical polarisation is a crystal with a top and a bottom, and a shape with a top and a bottom cannot be finished by one orbit of faces.

The other two entries, 1̅ and 2/m, are centrosymmetric and so not polar, and they fail for a weaker reason: their orbits are too small to surround anything. The general form of 1̅ is two parallel faces, which is a slab; the general form of 2/m is four faces around an axis, which is a prism. Neither leans all one way, and neither has enough faces to close, so the four-face bound rules them out before the half-space test is reached — which is why that second, entirely independent, route to the closure verdict was worth having.

What the round trip checked, and what it caught #

Two claims are asserted every time a form is drawn.

The first is orbit-stabiliser, counted both ways. The second is a second route to the closure verdict: a form of fewer than four faces cannot enclose a volume, because four planes is the least that bounds anything. That is a completely different argument from the half-space test — one counts faces, the other searches directions — and the two are required to agree.

There is a third check, and it is the one that stopped a bad picture. The angle-table figures draw a form in section, by intersecting the half-planes the faces define. If some direction in the plane of the page has no face across it, the section is unbounded, and joining the two ends of the gap draws a straight edge the crystal does not have. That is exactly the failure this site exists to prevent — a well-formed picture of a shape nobody named — so the section counts unbounded directions and refuses to draw when there are any.

The stretch the previous essay uses to change every angle in a drawing has a refusal of its own with the same flavour. Scaling c and leaving a and b alone is legitimate for a tetragonal or hexagonal class and destroys a cubic one, because a cubic class contains operations that exchange c with a. Ask for a stretched cubic cell and no figure appears at all, rather than a picture of a smaller group under a cubic caption.

Forty-seven, and why this site does not report it #

Every mineralogy text says there are 47 crystal forms. This one says 113, and the difference is not a disagreement — it is two different equivalences, and both are worth naming because the mismatch runs in both directions.

The classical list is coarser in one respect. It gives one name to congruent shapes wherever they occur. A cube is a cube: it appears as an orbit in m3̅m, in m3̅, in 432, in 4̅3m and in 23, and the classical count records it once. The count here records it five times, because in each of the five it is a different orbit of a different group.

The classical list is finer in another. A face can move along a mirror without its stabiliser changing, and still cross a direction at which the shape changes. The worked case is {hhl} in m3̅m. Two faces of the family have the same site symmetry and the same twenty-four-fold multiplicity, and when l is smaller than h the twenty-four faces make a trisoctahedron, while when l is larger they make a trapezohedron. Two names, one stabiliser, and the calculation here merges them.

So neither number is wrong and neither is a correction of the other. A count is only as meaningful as the equivalence attached to it, and this site has now said that three times about three different numbers: about two hundred and thirty against two hundred and nineteen, about the two-colour groups, and here.

Where this sits beside the special positions #

A reader who has been through this site’s space-group field will have noticed that this is Wyckoff positions again.

There, a point in the cell was classified by its stabiliser, its orbit size was the multiplicity, and the positions with larger stabilisers were the special ones. Here a face is classified by its stabiliser, its orbit size is the number of faces, and the faces with larger stabilisers are the special forms. The two are the same computation on two different objects — one on points in the cell, one on directions in the reciprocal lattice.

The difference is where the group comes from. Wyckoff positions are about a space group, so translations are in play and the answers are about a cell. Forms are about a point group, so nothing translates and the answers are about directions. That is why a form has a multiplicity and no coordinates, and why a Wyckoff position has both.

Naming, and who did it #

The names in this subject are Victorian and there are a lot of them: pinacoid, pedion, sphenoid, dome, prism, pyramid, dipyramid, trapezohedron, scalenohedron, rhombohedron, tetrahedron, hexahedron, octahedron, dodecahedron, tetartoid, gyroid, diploid, hexoctahedron. Each names a shape rather than an orbit, which is why the classical count is a count of shapes.

The system is largely Christian Samuel Weiss’s and Carl Friedrich Naumann’s, from the 1810s to the 1830s, and it was consolidated by Paul Groth later in the century. It was built to be spoken: a mineralogist at a specimen case needed a word for what was in front of them, and “hexoctahedron” is a better word for that purpose than “the general form of m3̅m”. Hermann–Mauguin went the other way and encodes the operations rather than the shape, which is why the two vocabularies do not translate term by term.

Reading it as orbits is a twentieth-century move and it buys two things the names do not. It says why the multiplicities are what they are, and it makes the open-and-closed distinction a theorem instead of a fact about each shape. What it costs is the vocabulary, and the vocabulary is worth keeping — a form has a name for the same reason a group has a symbol, and inventing a rival would be the worst thing this site could do.

The closure test, stated as a condition on the poles #

The test is described above as asking whether all the faces lean the same way. There is an equivalent statement that is easier to compute with and easier to be sure about, and it is worth writing down because it makes the decision a standard one.

A form is closed exactly when its poles positively span space — when every direction can be written as a non-negative combination of the face normals. Equivalently, and more usefully, the origin lies in the interior of the convex hull of the poles.

Those two statements are the same statement, and the second is a linear feasibility question with rational data: the poles are integer triples, the hull is a polyhedron with rational vertices, and asking whether a given point is inside it is exact arithmetic with no tolerance anywhere.

The direction the essay tests — is there a direction with every pole on one side? — is the negation, and it is the same question asked of the dual: a separating direction exists exactly when the origin is on the boundary of the hull or outside it. A form is open when its poles all lie in some closed half-space through the origin, and that is a condition an exact rational computation settles in one pass.

The care the essay mentions is about the boundary case. A pole lying exactly on the separating plane is neither inside nor outside, and a test using strict inequalities where it should use weak ones — or the reverse — gets the boundary forms wrong. Those are precisely the forms whose faces are parallel to some axis, which is to say most of the interesting ones.

What an open form has to be combined with #

An open form encloses nothing, so a crystal cannot be bounded by one alone — and the way real crystals resolve that is worth stating, because it explains what a habit actually is.

Open forms appear in combinations. A hexagonal prism is six faces parallel to one axis and bounds nothing; add a pinacoid — two faces perpendicular to that axis — and the pair encloses a volume. That is the shape of an ordinary quartz crystal’s prism and its terminations, and neither half closes on its own.

The condition on a combination is the same test applied to the union of the poles. A set of forms closes when their poles together positively span space, so the question of which combinations are possible is decided by the same integer arithmetic and, again, before any lengths are known.

That gives the classification a practical reading it otherwise lacks. A class with no closed form cannot grow a crystal showing a single form, so every specimen of every substance in one of the twelve shows a combination — and which combinations are available is a finite list computed from the class alone. The list is long and it is a list, which is what a mineralogist’s tables of habits are.

Where the ladder goes next #

The five cubic classes all grow cubes. That is a small observation with a large consequence, and it is the next rung: if several classes produce the same form, then the shape of a crystal does not determine its class, and a century of mineralogy spent working out how much it does determine.

The answer is that morphology narrows the class to a short list and cannot in general finish the job, and that the tie-breakers are not shapes at all — they are etch figures, optical activity, pyroelectric response and, eventually, diffraction. The classification the shapes suggested turned out to need instruments that measure something else entirely.