A filter of great precision and no predictive power

Thirty-two classes and six properties is 192 numbers. None of them is quoted, all of them are integers, and each is a character averaged over one group’s own operations.

That plate is the field’s conclusion, and reading it correctly takes a distinction that is easy to state and easy to lose.

The four headline counts #

Four numbers come out of the table that a reader is likely to meet elsewhere, and all four are computed here rather than quoted.

Eleven classes contain the inversion. They are the eleven Laue classes — what a diffraction pattern’s symmetry reports — and they forbid every odd-rank polar property outright, along with every even-rank axial one — which is the gyration tensor, and is not quite the same statement as “axial properties are forbidden”.

Ten classes are polar: they have a direction left alone by every operation, so they may have a spontaneous polarisation. This is where pyroelectricity and ferroelectricity must live.

Twenty classes permit piezoelectricity — every non-centrosymmetric class except 432, which fails through a cancellation of its whole group rather than through any single operation.

Fifteen permit optical activity, which is four more than the eleven enantiomorphic classes, because a crystal can have directions with a handedness without having one itself.

None of the four lists predicts any other. That is the observation the field is organised around, and it is why the table is computed rather than reasoned about.

What a symmetry argument establishes #

The strong direction is a prohibition, and prohibitions are the most reliable statements in the subject.

A zero in this table is exact and universal. It does not depend on the material, the temperature, the sample quality or the measurement. If the class forbids a component, no crystal in that class has ever had one, and a measurement reporting one is a measurement of something else — a different phase, a twin, a mis-assigned structure, or a higher-order effect masquerading as a linear one.

That is worth more than it sounds, because it turns a symmetry table into an error detector. A tetragonal crystal with a reported non-zero c₁₄; a centrosymmetric structure with a measured pyroelectric response; a cubic material claimed to be piezoelectric in class 432. Each of those is a contradiction rather than a curiosity, and the resolution is always that something in the experimental chain is wrong.

What it does not establish #

The weak direction is a permission, and permissions carry almost no information about a material.

A non-zero entry says only that nothing has been forbidden. It does not say the effect occurs, that it is measurable, or that it is useful. Class 1 heads three of the five columns — 21 elastic constants, 18 piezoelectric moduli, 6 dielectric constants — for the entirely uninformative reason that a class with one operation constrains nothing.

The mistake is easy to make because the numbers look like magnitudes. They are dimensions of a permitted space, and a large permitted space means a small group. Permitted is not present is the whole of it, and it is the sentence this field repeats most often because it is the one most often dropped in transmission.

The two things it is genuinely good for #

Designing a measurement. An experimentalist determining the elasticity of an orthorhombic crystal needs nine numbers, not twenty-one, and the table says which nine and in which orientations. Designing for twenty-one wastes effort; designing for six leaves the problem under-determined. This is the everyday use and it is worth more than the headline results.

Ruling out a structure. A measured effect that a proposed structure forbids is evidence the structure is wrong, and it is strong evidence — much stronger than the corresponding positive inference. The classic case is a structure refined in a centrosymmetric space group that turns out to be non-centrosymmetric, which happens often enough to be a standing caution in crystallography, and the symptom is usually a measured property the centrosymmetric assignment forbids.

Both uses run in the direction of the prohibition. Every safe use of this table does.

Reading the table by column #

Each column falls at its own rate, and putting the five rates beside each other says more than any one of them.

Elasticity falls from 21 to 3 — a factor of seven, and it never reaches zero, because a crystal always has an elasticity. It is even-rank and polar, so no operation can forbid it outright; the most a group can do is tie its components together.

The dielectric column falls from 6 to 1 and likewise never reaches zero. Same reason, one rank lower.

Piezoelectricity falls from 18 to 0 and hits zero twelve times. It is odd-rank and polar, so the inversion annihilates it, and one further class manages it by cancellation.

The polar vector falls from 3 to 0 and hits zero twenty-two times — the most of any column. It is the smallest tensor here and therefore the easiest to constrain to nothing.

Optical activity falls from 6 to 0 and hits zero seventeen times. Even-rank, so parity does not kill it, but axial, so every improper operation contributes with a reversed sign.

Rank and parity between them decide the shape of a column, and the group decides the entries. A reader who knows only whether a property is odd or even rank and polar or axial can predict which classes forbid it outright, without any calculation at all — and cannot predict a single one of the non-zero entries.

Where the field’s numbers came from #

Everything on this site’s version of the table is computed, and it is worth restating the chain, because five separate results have to be right before the last one means anything.

The thirty-two classes are enumerated by cyclic extension over the subgroups of two holohedries, quotiented by conjugacy, and merged — with every merge witnessed by an explicit matrix rather than by the fingerprint that proposed it. The operations are typed by determinant and trace, which is the only classifier that separates 3̅ from 6̅. The names are derived from each group’s own symmetry directions and required to be the Tables’ thirty-two. The property counts are character averages in the lattice basis, exact in integers with no Cartesian frame. And each count is checked against a second, independent calculation of the tensor’s shape in a Cartesian frame, which shares no arithmetic with the first.

Five results, five separate assertions, and the build stops if any of them fails. That is the site’s arrangement everywhere and this field is the third place it has been built: a plane pattern’s group rediscovered from its point set, a space group’s rediscovered in three dimensions, and now a class’s properties computed twice from the same group along routes with nothing in common.

What the phase found, and where it was wrong first #

Three findings from building this field, and all three are about the machinery rather than the crystallography.

An operation’s type must be read from its determinant and trace, and not from its order. 3̅ and 6̅ both have matrix order six, so an order-based classifier merges them, merges 3̅m with 6̅2m, and returns twenty-eight classes — every one of which is a real point group. The output is well-formed and short by four, which is the failure mode this site keeps meeting and named during the space-groups phase: a search that is correct as far as it went.

A fingerprint that returns the right answer is still a fingerprint. The merge between the two holohedries’ class lists was proposed by the element-type census and the census gives thirty-two. Checking it turned up nothing wrong with the classification and two things wrong with the checker: taking a single null-space vector as the conjugating matrix fails for 1̅, whose intertwiner space is all nine dimensions, and walking a curve through that space fails too, because the curve is rank one at every point. The degenerate case is the one where careless picking is most likely to pick badly, which is the opposite of the intuition.

And a procedure that considers one thing when it should consider two does not look broken from the inside. The symbol derivation turns a group inside its holohedry to find the standard setting, and for 3m the turning does nothing — both of its subgroups are normal in 6/mmm, so each orbit has one member. It derived 31m, which is a real symbol for a real setting and is not one of the thirty-two, and the assertion comparing the derived symbols with the Tables’ caught it by name.

Two smaller ones belong to the fleet rather than to this site and are recorded there: a font-size presentation attribute loses to a stylesheet rule, so every numeric size: at a figure call site is inert; and a colour role declared as a stroke puts a 2.2 px outline around every glyph of a label that uses it.

What is not here #

Three things this field deliberately does not do, each stated in the essays that come nearest to it.

It does not compute the 230. Getting from the thirty-two classes and the fourteen lattices to the space groups needs the translations and a second quotient by each point group’s normaliser in GL(3,ℤ), and that is the content of the classification rather than an application of it. 230 appears as a number from the literature wherever it is used, and every essay using it says so.

It does not compute the 73 arithmetic classes. Requiring the change of basis to carry one lattice onto another is a finer equivalence than the one used here, and under it 3m splits back into 3m1 and 31m. This site computes the geometric classification.

And it does not do representation theory. Character tables, irreducible representations, selection rules and the reduction of a reducible representation are the standard next step and are not taken. The characters used here are the characters of tensor representations, evaluated to get one number each, and nothing is decomposed.

Reading the table backwards #

The two uses named above both start from a known class. There is a third that starts from measurements and ends at a class, and it is the one a mineralogist or a materials chemist actually performs — the table read as a key rather than as a lookup.

Measure which effects a specimen shows and which it does not. Each answer is a filter: a measured effect eliminates every class whose entry for it is zero, and that is a hard elimination, since a zero is a prohibition. A measured absence eliminates nothing, for the reason the essay gives, so only the positives narrow the list.

Run it. Optical activity observed? Seventeen classes gone. Pyroelectricity observed? Twenty-two gone. Piezoelectricity observed? Twelve gone. Each is independent of the others in the sense that no one of them can be deduced from the rest, so the intersections are small.

Two positive results are usually enough to reach a handful of candidates, and three to reach one or two — which is exactly the procedure the classical mineralogists ran with etch figures, a polarising microscope and a warm crystal, before diffraction existed.

The limit is what makes it a key rather than a determination. Every filter runs one way, so a specimen showing nothing at all is consistent with every class that permits nothing — which is the eleven centrosymmetric ones and several others besides. A negative result narrows nothing, and a determination resting on negatives has established only that no measurement was sensitive enough.

How much the six columns distinguish #

That raises a countable question, and it is worth asking because it bounds what the table can ever do.

Take a class’s row — its five or six entries — as a signature, and ask how many of the thirty-two classes share one. The answer is that the table separates most of them and not all: classes of the same system with the same permissions differ only in entries the table does not carry, and several pairs agree in every column.

So the property table is a fingerprint of exactly the kind this collection distrusts: a set of invariants that separates when it separates and identifies nothing. Two classes with identical rows are not thereby the same class, and no measurement of any of these properties will ever distinguish them.

What distinguishes them is where the components sit, which is the tensor shape rather than the count — the orientation of the permitted components relative to the crystal’s axes, which a measurement made in a known orientation reports and a count does not. That is the honest ceiling of a table of integers, and it is the reason this field’s figures draw shapes as well as printing numbers.

One page a reader could keep #

If the field reduces to anything portable, it is four questions asked in order, and each is answered by looking rather than by computing.

Does the class contain the inversion? If so, every odd-rank polar property is zero and so is every even-rank axial one, and eleven of the thirty-two are disposed of in one step. This is the single most consequential fact about a crystal class.

Is there a direction every operation fixes? If so the class is polar, ten of the thirty-two are, and spontaneous polarisation, pyroelectricity and ferroelectricity become possible.

Are all the operations proper? If so the class is enantiomorphic, eleven are, and the material can hold a single-handed structure and its space group may come in a mirror-image pair.

And what rank and parity is the property in question? Replacing an operation M by −M multiplies a polar character of rank r by (−1)ʳ and an axial one by (−1)ʳ⁺¹, because the determinant changes sign as well. So even-rank polar properties are never killed by a centre, odd-rank polar ones always are, even-rank axial ones always are — and odd-rank axial ones, of which the magnetic moment is the one that matters, are not.

Four questions, and between them they predict every zero in the table. What they do not predict is a single one of the non-zero entries, and that asymmetry is the field in one sentence.

Where the exactness stops #

Every number is a count of permitted components. Magnitudes are measurements and none appears anywhere in this field.

Every count belongs to a class, and a real crystal’s class is an experimental inference with its own error rate — one that fails asymmetrically, since refining in too high a symmetry usually fits the data and then forbids things the material does.

And the intrinsic symmetry of each tensor is an input from physics, not a result: that stress and strain are symmetric, that elasticity exchanges its index pairs, that gyration is axial. Six characters were chosen and six were computed; a property with a different intrinsic symmetry needs a different sum and is not in this table.

What the field cost, in machinery #

A note on scale, because the ratio between the argument and the code is part of what makes this subject unusual.

The classification is about two hundred lines: cyclic extension, conjugacy, a signature, a character test, an averaging construction. The property counts are about forty — six character functions and one sum. The symbol derivation is about eighty, and most of that is the settings rule and its exception.

Everything else in the file is assertions and the reasons for them. Five counts on the enumeration, one per merge, the containment of all seven holohedries, the distinctness of the thirty-two symbols, their agreement with the Tables, the integrality of every character average, and the agreement between the two routes to every property count. Those outnumber the arithmetic they guard.

That ratio is the site’s habit made visible, and the phase justified it three times: an order-based classifier that returns twenty-eight real point groups, a conjugator construction that fails on the most symmetric input in the list, and a settings search that considers one subgroup where two exist. None of the three produced an error message on its own. Each produced a plausible, well-formed, wrong answer, and each was caught by an assertion comparing it with something computed another way.

The general form is the one worth carrying out of the field: the cheapest thing to compute is usually the answer, and the expensive thing is the second route to it. A subject where claims are decidable makes the second route available, and taking it is the whole of what “decidable” is worth.

Where this goes #

The field is complete as a classification and it is the beginning of an application. The obvious continuations — the tensor shapes class by class rather than only their dimensions, the magnetic point groups where time reversal is a symmetry operation and the thirty-two become ninety, the representation theory that turns a class into selection rules — are each larger than the field this page has assembled, and none of them is begun here.

What the site takes forward is smaller and is the habit rather than the content: a count is worth having when it comes with the equivalence it is a count of, and a permission is worth having when it is not mistaken for a prediction. Both sentences have been earned three times now, in the plane, in space, and here.