Permitted is not present

The table this field computes has 192 entries and every one of them is exact. It is also, more often than not, read as saying something it does not say.

A zero means the property is forbidden. That direction is airtight: if the symmetry requires a component to equal its own negative, the component is zero, and no material in that class has ever had one or will.

A non-zero number means nothing has been forbidden. It is the absence of a prohibition, and it is not a prediction, an estimate, or evidence that anybody will find the effect.

The asymmetry, stated plainly #

Neumann’s principle is an inclusion: the symmetry of the property must include the symmetry of the crystal. It is not an equality.

So a property may be more symmetric than the crystal, and frequently is. A triclinic crystal’s permittivity has six independent components, and nothing whatever stops five of them being indistinguishable from zero. The material would then behave, to any measurement, as though it were uniaxial — and it is triclinic, and the table is not wrong, and the six independent components are genuinely permitted.

This is the direction the principle does not constrain, and it is the direction most real materials live in. Most components that are permitted are small.

Four ways the reading goes wrong #

“This class is piezoelectric.” No class is piezoelectric. Twenty of the thirty-two permit piezoelectricity, and the strength of the effect in a given material spans many orders of magnitude — quartz is used because its coupling is usable and stable, not because class 32 is special among the twenty. Plenty of materials in permitting classes have piezoelectric constants too small to be worth measuring.

“Eighteen moduli means highly piezoelectric.” The count is the dimension of the permitted space, and a large space is a property of the class having few operations rather than of the material having a strong effect. Class 1 heads the piezoelectric column with eighteen because it forbids nothing, which is the least informative thing a class can do.

“This material is not piezoelectric, so its class is centrosymmetric.” A failed measurement is evidence and it is weak evidence. The effect may be present and below the noise, or the sample may be twinned so that opposite domains cancel, or the measurement may be along a direction where the permitted components happen to vanish. Only the forbidding direction is airtight.

“The crystal is nearly cubic, so use the cubic counts.” This is the most consequential of the four and it has its own essay: near-symmetry is not symmetry, and a crystal whose cell is cubic to within a measurement is in whatever class its structure has, not in the class its cell suggests. The property tensor obeys the real class exactly, and the extra components are small rather than absent.

A worked case: quartz, and what symmetry contributed #

Quartz is class 32, and it is the standard example of a piezoelectric material, so it is a good place to ask exactly which part of that sentence symmetry is responsible for.

Symmetry says 32 permits two independent piezoelectric moduli. It says which two — d₁₁ and d₁₄ in the usual notation — it says that thirteen of the eighteen entries are exactly zero, and it says that the remaining three are not free either: d₁₂ is minus d₁₁, d₂₅ is minus d₁₄ and d₂₆ is minus twice d₁₁. Five entries may be non-zero and two numbers decide all five. It also says the class is enantiomorphic, so quartz comes in left and right forms, and that it permits optical activity, so those forms rotate polarised light in opposite senses.

Symmetry does not say d₁₁ is 2.3 pC/N. That is a measurement. It does not say quartz’s coupling is stable over temperature, which is why quartz oscillators exist and most piezoelectric materials do not go into watches. It does not say the effect is large — it is not; the ferroelectric ceramics used in transducers are two orders of magnitude stronger, and they are in polar classes where the mechanism is different.

So the contribution is real and it is narrow: symmetry supplied the shape of the answer and the number of numbers, and the physics supplied all of the numbers. A reader who takes “quartz is piezoelectric because it is in class 32” as an explanation has been given a permission and mistaken it for a cause.

Where the useful information actually is #

None of this makes the table weak. It makes it a filter, and a filter is exactly what a structural or materials argument needs at the point where this table is consulted.

Ruling things out is the strong direction and it is used constantly. A material reported to show a spontaneous polarisation must be in one of the ten polar classes; a report of pyroelectricity in a centrosymmetric structure is a report that the structure is wrong, and that inference is sound. Symmetry-forbidden effects are the most reliable evidence in the subject.

Counting the measurements needed is the practical direction. Nine numbers for an orthorhombic crystal’s elasticity, not twenty-one, and the table says which nine and in which orientations. An experiment designed for twenty-one is wasted effort; one designed for six on an orthorhombic crystal is under-determined.

And checking a measurement against its own class is the direction that catches errors. A tetragonal crystal with a reported non-zero c₁₄ has been mis-oriented, mis-indexed, twinned or misidentified. The constraint is exact and the measurement violating it is wrong, which is a much more useful conclusion than “interesting”.

The site’s own version of the same caution #

This is not a new rule for this field. It is the standing rule about aperiodic patterns, applied to a different object.

The machinery here decides symmetry exactly, in integers, and its answers are about symmetry. The moment an essay lets a symmetry answer stand in for a physical one, the whole decidability argument becomes decoration — because what makes it worth having is precisely that it says a small, certain thing rather than a large, plausible one.

The site has three of these boundaries now and they have the same shape.

• A Penrose tiling’s tile ratio is measured, not decided, because the integer machinery needs a lattice and there is not one.
• This site does not enumerate the 230, because getting from six arithmetic classes to seventy-three needs each point group’s normaliser in GL(3,ℤ), and that is the content of the classification rather than an application of it.
• A property count is a permission, not a value. Same rule: state what the machinery decided, and separately state what it did not.

The one place the boundary is genuinely subtle #

Everything above is about size. There is a second, sharper failure mode that has nothing to do with magnitudes, and it is worth separating.

The crystal may not be in the class it is assigned to. Real crystals are twinned, disordered, strained, and occasionally in a structure that differs from the published one by a symmetry element nobody noticed. A structure refined in a centrosymmetric space group when the true structure is non-centrosymmetric is one of the classic errors in crystallography, and it produces a property table that forbids things the material does.

The direction of that error is the awkward one. The mistake makes symmetry look too high, because refining in a higher-symmetry group nearly always fits the data acceptably, and the extra symmetry then forbids effects the material actually has. A measured pyroelectric response in a “centrosymmetric” crystal is far more likely to mean the assignment is wrong than that Neumann’s principle failed.

So the airtight direction is airtight given the class, and the class is an experimental inference with its own error rate. That is not a hedge on the mathematics; it is where the mathematics hands over.

The asymmetry has a name in logic and it is worth using it #

The whole essay is one observation from elementary logic, and naming it makes the misreadings easier to spot in the wild.

Neumann’s principle has the form if the crystal has symmetry S, then the property satisfies constraint C. From that, two inferences are valid and two are not.

Valid: the crystal has S, therefore C holds — so a permitted component is genuinely permitted and a forbidden one is genuinely zero. Also valid: C fails, therefore the crystal does not have S — which is the contrapositive, and it is the inference that makes a measured pyroelectric response into evidence against a centrosymmetric assignment.

Invalid: C holds, therefore the crystal has S. A material whose permittivity looks uniaxial is not thereby tetragonal; it may be triclinic with small off-diagonal components. And invalid: the crystal lacks S, therefore C fails — which is the one this essay is mostly about, because “not forbidden” is being read as “occurs”.

Two of the four inferences are sound and both of them run from symmetry to a prohibition. Every use of the table that stays on that side is safe, and every use that crosses to the other side is an argument the table is not making.

Where the exactness stops #

Nothing in this field is a measurement, and no number in the table came from one. The counts are dimensions of invariant subspaces, computed exactly in integers, and the essays quote them as such.

The intrinsic symmetry of each tensor is an assumption, taken from physics rather than derived: that stress and strain are symmetric, that the elastic tensor exchanges its index pairs, that the gyration tensor is axial. A property with different intrinsic symmetry has a different count, and this site computes the six it names and no others.

And the classes are geometric crystal classes. Real materials sit in space groups, and a space group carries information — screws, glides, centring — that no point property can see. The table is the right table for point properties and is silent about everything else.

Why the machinery is built to state its own limits #

There is a design decision behind all of this that is worth surfacing, because it is why the field’s code says what it says rather than only computing what it computes.

Every property function here returns a count and every caption that prints one is required to say what kind of statement it is making. That is not editorial fussiness. A number with no account of what it is a number of is the most reusable kind of wrong thing, because it survives being quoted, propagates into a summary, and arrives somewhere with all its qualifications stripped off.

The site has watched this happen to its own numbers. The standard pass of 2026-08-09 found a generator asserting assertCount(rows.length, 7) over a hard-coded array of seven frieze groups — an assertion that restates its own literal and can reject nothing — sitting under three captions that described sixteen candidate combinations. Sixteen rows were asserted by three essays and drawn by none of them, and it passed every gate for months.

The remedy there was to enumerate rather than to quote. The remedy here is the same shape: the counts are computed, and the captions are required to say permitted rather than has.

The failure mode this essay is really about #

There is a specific way a symmetry table goes wrong in practice, and it is not that anybody misunderstands Neumann’s principle. It is that the qualification gets dropped in transmission.

A calculation produces “class 4mm permits three independent piezoelectric moduli”. A summary records “4mm is piezoelectric”. A later summary records “this material is 4mm, so it is piezoelectric”. Nothing in that chain is a mistake anybody would defend, and the end of it is a claim about a material that the calculation never made.

The remedy is not vigilance, which does not scale, but making the qualification part of the number. In this field every caption printing a count says permitted, every essay quoting one says what kind of statement it is, and the property table’s own description ends by saying that a zero and a non-zero entry have completely different standing. That is deliberate redundancy, and it is there because the alternative has been observed.

The site has its own record of the same failure with its own numbers. A generator once asserted assertCount(rows.length, 7) over a hard-coded array — an assertion restating its own literal, capable of rejecting nothing — under three captions describing sixteen candidate combinations. Three essays asserted a count that no figure drew, and it passed every gate for months, because a well-formed picture of the wrong thing overflows nothing and contrasts fine.

A number that has lost its qualification behaves exactly like that. It is well-formed, it is quotable, it survives every check that asks whether it is an integer, and it is about something other than what the reader thinks.

A last word on why this essay sits in the middle of the field rather than at the end of it.

Everything after it — the ten polar classes, the twenty piezoelectric, the fifteen optically active — is a list of permissions, and each of those essays would be easy to read as a list of materials. Putting the caution before them rather than after is deliberate: a qualification met first is a lens, and one met last is a footnote.

The counts are worth having precisely because they are narrow. A statement that holds for every material in a class, forever, without exception, is worth a great deal more than a statement about what is likely — and it is worth that only while it is being read as the statement it is.

What a zero guarantees, and inside which approximation #

The prohibition direction has been called airtight throughout, and it is — but it is airtight about a particular idealisation, and saying which one is the difference between a rule and a slogan.

Neumann’s principle constrains a tensor, and a tensor is a linear relation between two bulk quantities in an infinite, perfect, unstrained crystal. Every one of those words is doing work.

Bulk. A crystal has surfaces, and a surface has no inversion centre whatever the interior has — the two sides of it are different. So a centrosymmetric crystal produces a second-harmonic optical signal from its surface layers, which is forbidden in its bulk and is routinely measured; the effect is the basis of a whole family of surface probes precisely because the bulk contribution is zero and cannot swamp it.

Linear, and to lowest order. The forbidden term is the leading one. A centrosymmetric crystal in a strong static field acquires an effective non-centrosymmetric response, because the field itself breaks the symmetry and the next term in the expansion is not forbidden. Field-induced second-harmonic generation is that term, and it is not a violation of anything.

Uniform. A tensor relation assumes the applied quantity is the same everywhere. Where it is not — a gradient across the sample — the response can couple to the gradient rather than to the quantity, and a gradient is a different object with different transformation properties. Effects forbidden at dipole order reappear at quadrupole order for that reason, smaller by roughly the ratio of a lattice spacing to a wavelength and entirely real.

And perfect. A crystal that is nearly in a class rather than in it has a small permitted response where the ideal has none, and the size of it is set by how far the structure sits from the symmetric one rather than by anything symmetry can compute.

So the honest form of the prohibition is: a property forbidden by the class is exactly zero in the bulk, at leading order, in a uniform field, for the ideal structure. Every one of those qualifications has an experimental literature attached, and a measurement that finds a forbidden effect is much more likely to have found one of them than to have refuted the principle. That is a useful thing to know in both directions: it says the table is not fragile, and it says what a surprising measurement is evidence of.

Where this goes #

The most interesting entry in the whole table is a zero where nobody expects one. Twenty-one classes lack an inversion centre and twenty of them permit piezoelectricity; the twenty-first is 432, and the reason it fails is not that it has too much symmetry in any ordinary sense. Twenty of twenty-one is about that single exception and what it shows about how the character sum works.