A hundred and thirteen orbits, and forty-eight shapes

A form is an orbit ends with a paragraph that has been waiting to be checked. It reports 113 kinds of form across the thirty-two crystal classes, notes that every mineralogy text reports 47, and explains the gap: the classical list is coarser in one respect, because it gives one name to congruent shapes wherever they occur, and finer in another, because a face can slide along a mirror without changing its stabiliser and still cross a direction at which the solid changes.

Both halves of that explanation are correct. Neither was computed. A difference stated in prose and never enumerated is a difference nobody has tested, and this collection’s own habit says what to do about it: give the second equivalence a definition a program can decide, run it, and see whether 113 becomes 47.

It becomes 48, and the two extra entries are worth more than the agreement.

What a shape has to mean before it can be counted #

The classical count is a count of shapes, so the whole computation rests on deciding when two forms are the same shape. There are two obvious answers and both are wrong, in opposite directions.

The first: the combinatorial type of the solid. Take the polyhedron a form bounds, forget its metric, and keep only which faces meet which vertices — its face lattice, up to isomorphism. It is exact, it needs no tolerance, and it is far too coarse. A rhombic dipyramid, a tetragonal dipyramid, a tetragonal scalenohedron and a regular octahedron are eight triangles meeting three to a vertex, in every one of the four cases. Nothing about the incidence separates them, and the classical list names all four.

The second: the group that produced the form. That is the 113, and it is too fine by construction, because five classes grow the same cube and a count keyed on the generating group records the cube five times.

The definition that works uses both, and the pairing is the content of this essay:

A shape is the combinatorial type of the solid, together with the symmetry group of the solid.

The second half is the one that does the work, and it needs saying carefully. A form has two groups and they are not the same group. One is the crystal class it is an orbit under. The other is the symmetry of the resulting solid taken by itself, as an object in space with no crystal around it — and that one is m3̅m for a cube whether the class that grew it was m3̅m of order forty-eight or 23 of order twelve. Keying on the first splits the cube into five; keying on the second merges it into one, and separates the four dipyramids above, because their solids have symmetries of order 8, 16, 8 and 48.

The group of a solid is not a group of the lattice #

Computing that second group is where the first version of this went wrong, and the mistake is instructive because everything it produced looked reasonable.

The obvious method is to ask which operations of the crystal’s own holohedry carry the set of face poles onto itself. That is a question about integer matrices, it takes a line to write, and it answers a subtly different question: not what symmetry does this solid have but what symmetry does this solid share with its lattice. Those differ whenever the solid sits at an angle to the crystal axes.

A hexagonal dipyramid whose faces straddle the mirror planes of its lattice comes back from the integer question with symmetry 6/m rather than 6/mmm — and it is the same solid as the one whose faces lie on the mirrors, turned about its own axis. Every “form of the second kind” and “of the third kind” in the older tables is exactly that distinction: a statement about orientation relative to the axes, recorded in the name as though it were a statement about the geometry. Counted the integer way, the closed shapes come to fifty-eight instead of thirty, and the excess is entirely made of solids counted twice for sitting at two angles.

So the group is computed as a group of space rather than of the lattice. Take three face normals that span; try every ordered triple of normals with the same three pairwise angles as their image; the linear map sending one triple to the other is orthogonal by construction, and it is a symmetry exactly when it carries every remaining normal to a normal. Nothing is searched that a matching set of angles has not already admitted, so the work is a small multiple of the face count rather than its cube.

Two details in that paragraph are load-bearing. The operations are typed by matching the cosine of their turn against the five values a lattice permits — 1, ½, 0, −½ and −1 — rather than by taking an arc cosine and dividing it into a full turn. Both routes agree where the arithmetic is exact and only the first is stable, because the arc cosine loses most of its precision near ±1 and a rounded quotient then calls a four-fold a three-fold in one figure and not in the next. That is not a hypothetical: it happened, it produced one group under two names, and it split hexagonal dipyramids in half. And a turn matching none of the five is refused rather than named, which puts the crystallographic restriction to work as a check on the geometry instead of a fact recalled about it.

The cell has to be generic, and the reason is this site’s own subject #

The shape of a form must not depend on the axial ratios. “The angles belong to the substance, the shape to the specimen” is an essay in this anchor, and it says the shape is decided by the class and the indices while the angles are decided by the cell. So any cell of the right system should do.

It does not, and the way it fails is the failure this whole site is built around.

The cell this collection uses for its angle tables is orthorhombic with axial ratios 3 : 4 : 6 — exactly rational, chosen so the tables read cleanly. In that cell the form {323} of class mmm has its poles at (1, ½, ½): two components equal, by arithmetic coincidence rather than by symmetry. The solid it bounds is therefore a tetragonal dipyramid rather than a rhombic one, its own symmetry comes back 4/mmm, and the merge files an orthorhombic form under a tetragonal shape without a murmur. The total was still 48. The class list under one entry was wrong.

That is the motif must be a comma arriving from a new direction. There the danger was a motif accidentally too symmetric for the group it was illustrating; here it is a cell accidentally too symmetric for the system it belongs to. In both the picture is impeccable and only a computation notices.

The census therefore runs in a cell with irrational ratios — √2 and √3 for the two axes of an orthorhombic cell, √3 for c/a in the others — where no small integer triple can make two pole components equal, since that would make a surd a fraction. And before a single pole is drawn, every operation of the class is required to preserve that cell, which is the check that would have caught the first cell if it had been asked of it. Nothing else changes: the operations are the same integer matrices they always were.

The eighteen that do not close #

The open forms come out first and they come out exactly.

A form that does not enclose a volume has no polyhedron to take a combinatorial type of, so it is classified by the arrangement of its normals, and five arrangements exhaust the cases. One face is a pedion. Two antiparallel faces are a pinacoid. Two faces that are not antiparallel are a dihedron, which the classical list splits in two according to what relates the pair — a mirror makes a dome, a two-fold alone makes a sphenoid — and that is the one open distinction the arrangement of normals cannot see, so it is the one place the generating class is consulted. Coplanar normals make a prism. Everything else is a pyramid, its normals on a cone about an axis.

Prisms and pyramids then separate further, and the separator is the part worth stating, because the first attempt at it measured the wrong thing. A prism’s shape is not its angles: six normals at equal spacing make a hexagonal prism whatever the spacing is, and six alternating wide and narrow make a ditrigonal prism whatever the two widths are. So what a shape depends on is which gaps equal which, and nothing else. Comparing the gaps as numbers instead — which is what a first version did — reported forty-six different four-faced prisms in class 2/m, one for every pair of indices the search happened to reach. That is a measurement of the search.

Seven prisms, seven pyramids, a pinacoid, a dome, a sphenoid and a pedion: eighteen, which is the classical open count exactly, entry for entry.

The thirty that do #

The closed forms need the solid built. Every face of one form is at the same distance from the centre — not a simplification but the only distance a single form can have, since its faces are images of one another — and the solid is the intersection of those half-spaces, found by solving every triple of planes and keeping the points that satisfy all the rest. It is the dullest method available and it is the one this site already uses for Wigner–Seitz cells, for the reason that matters: its failure mode is a missing vertex rather than a wrong one.

Two checks then run on every solid before its type is taken. Every face of the form must reach the surface — a face cut away entirely by its neighbours would mean the orbit and the solid disagree about how many faces there are — and the vertices, edges and faces must satisfy Euler’s relation. Neither has ever failed, which is what makes them worth keeping: they are the tests that would catch a half-space intersection quietly returning something that is not a polyhedron.

Thirty, of which fifteen occur only in cubic classes — which is the classical division of the list into fifteen isometric forms and the rest, arrived at by counting rather than by taking it as the division.

Eighteen and thirty make forty-eight. The classical forty-seven is this list with the dome and the sphenoid merged into a single dihedron, which some texts do and others do not; both counts are in the literature and the difference between them is one deliberate decision about whether a mirror and a two-fold produce the same shape. They produce the same two planes at the same angle, so as shapes they are one; they are different forms of different classes, so as forms they are two. This computation keeps them apart and says why, which is the only defensible thing to do with a distinction that two books disagree about.

Where the two counts pull apart #

The merge runs in both directions at once and it is worth seeing them separately.

The classical list is coarser, and by a lot. Twenty-eight of the forty-eight shapes occur in more than one class. The pinacoid occurs in twenty of the thirty-two; the hexagonal prism and the pedion in ten each. Every one of those occurrences is a distinct orbit of a distinct group, so a count of orbit types is right to record them separately — and a count of shapes is right not to.

And the classical list is finer, in thirty-five places. Thirty-five of the 113 strata change shape along their own length: the stabiliser stays the same, the multiplicity stays the same, and the solid does not. The worked case is the one the earlier essay named. In m3̅m the faces of the form {hhl} all have the same site symmetry and the same twenty-four-fold multiplicity, and the solid they bound has fourteen vertices when l is smaller than h and twenty-six when it is larger.

That direction is what turns 113 into 164, and it is the half of the mismatch that a table of site symmetries can never show, because the thing that changed is not the symmetry of the face.

What the merge could get wrong #

An invariant that merges has one dangerous direction and it is the merging one. Colour refinement on the vertex–face incidence graph — faces coloured by how many vertices they carry, vertices by how many faces meet there, then rounds of recolouring by the neighbours’ colours — is not a complete test for isomorphism. Two non-isomorphic solids can share a refinement invariant; two isomorphic ones cannot differ in it. So the count of distinct shapes is a lower bound, and if the refinement is fooled somewhere the true number is larger than forty-eight rather than smaller.

One subtle failure in that machinery is worth recording because it produced a wrong answer that looked entirely sane. The short colour labels have to be assigned in sorted order of the long strings they stand for, never in order of first encounter — encounter order is a property of how the vertices came out of the half-space intersection, so two isomorphic solids get the same colour classes under it and different names for them. The two hands of the pentagonal icositetrahedron came out as two different shapes on that account, one labelled six-and-thirty-two and the other thirty-two-and-six, and the closed count was fifty-two.

Where the exactness stops #

Computed here: for each of the thirty-two classes, every face with indices in a bounded box, sorted into strata by the conjugacy class of its stabiliser; the solid each face bounds, when it closes; the refinement invariant of that solid’s incidence graph; the group of orthogonal maps carrying its normals onto themselves; and the merge of all of it across the thirty-two.

The bound is checked rather than trusted. The search runs to indices of three and again to four, and both give 113 types and 48 shapes. A form whose smallest indices were four would be missing from the first search and the answer would be correct as far as it went — which is the failure this collection has met twice before, in the space-group and point-group phases, and now checks for by habit.

Nothing here is about real crystals. A form is a set of directions; a mineral is a solid with several forms on it at once, in proportions decided by growth rather than by symmetry, and which faces a crystal shows is a different question with a different answer. Forty-eight is the number of shapes symmetry permits a single form to have, not the number of shapes crystals have.

And the classical list is a historical object, not a theorem. It was assembled over a century by people naming what they saw, and the fact that a definition written down in a program reproduces it entry for entry is evidence that the definition is the one they were using — not proof, because they never wrote one down. The two extra entries relative to forty-seven are the dome and the sphenoid, and that is a disagreement about one merge rather than about the method.

Who counted them, and how #

The names are Victorian and there are a great many of them — pinacoid, pedion, sphenoid, dome, prism, pyramid, dipyramid, trapezohedron, scalenohedron, rhombohedron, tetartoid, gyroid, diploid, hexoctahedron — and each names a shape rather than an orbit, which is exactly why the classical count is a count of shapes. Groth’s tables of the 1870s are the standard consolidation; the list had settled by the end of that century, and the forty-seven has been copied from text to text ever since with the equivalence behind it left unstated.

The modern statement of the same thing is in the International Tables, where a form is given by its face poles and its site symmetry, and the tables carry both the orbit view and the shape names side by side without ever saying how one becomes the other. The arithmetic above is that missing step.

A count is only as good as its equivalence #

This collection has now said the same sentence about four different numbers, and the fourth is the tidiest instance of it.

Two hundred and thirty, or two hundred and nineteen is the same question about space groups: the eleven enantiomorphic pairs are one group up to an affine change of basis and two up to a proper one, and neither count is wrong. Seventy-four colourings, forty-six groups is the same question about two-colour patterns. Thirty-two classes, eighteen groups is the same question about abstract group type against geometric class.

Here the two equivalences are orbit and shape, and what the arithmetic adds is that the second is not one idea but two put together — a combinatorial type and a group — and that leaving either out gives a number nobody has ever printed. Combinatorial type alone gives thirty-five. Orbit alone gives a hundred and thirteen. The classical forty-seven is between them and it is not a compromise; it is a third equivalence, stated precisely for the first time by having to be programmed.

Where the ladder goes next #

Back, to the orbit count and the integer test that decides whether a form closes at all: a form is an orbit, which is where 113 comes from and where 43 of the 113 turn out to be closed.

Sideways, to what a real crystal does with these shapes: which faces a crystal shows ranks the forms by interplanar spacing and builds the habit, and the angles belong to the substance is the measurement that made forms a science before anybody knew what a lattice was.

And across, to the same accounting on points rather than directions: the points a group treats differently classifies positions in the cell by their stabilisers exactly as this classifies faces by theirs, and has its own version of the question of when two of them are the same thing.