What a trace decides

A symmetry operation of a crystal, written in the lattice’s own basis, is a 3 × 3 matrix of integers. There are ten kinds it can be, and the whole of the identification is two numbers.

The determinant says whether handedness is preserved. A rotation has determinant +1 and a reflection or rotoinversion has −1, and nothing else is possible for a matrix that preserves lengths.

The trace says how far it turns. A rotation through angle θ about any axis has trace 1 + 2cos θ, because the matrix is conjugate to the one with the rotation in a coordinate plane and the axis untouched, and a trace is unchanged by conjugation.

The ten values #

Put the two together. The crystallographic restriction says the trace of an integer matrix is an integer, so 1 + 2cos θ is an integer, so cos θ is a half-integer, so θ is one of 0°, 60°, 90°, 120° or 180° — the rotation orders 1, 2, 3, 4 and 6 and no others.

That gives five proper operations with traces 3, 2, 1, 0 and −1. A rotoinversion n̅ is −1 times a rotation, so its trace is the negative of a rotation’s, giving −3, −2, −1, 0 and 1.

Ten pairs, and every pair is distinct. Reading the table by column, the determinant −1 row runs 1̅, m, 3̅, 4̅, 6̅ — the inversion, the mirror, and the three genuine rotoinversions. So the type of an operation is a lookup on two integers, computed without a trigonometric function, an eigenvector or a tolerance.

That is a small fact with a large consequence, and the consequence is the reason this essay exists rather than being a paragraph inside another one.

Both numbers survive a change of basis #

The matrices here are written in the lattice basis, which is the basis that makes them integers. A physicist’s basis is orthonormal, and in it the same operations are matrices of cosines.

Those two descriptions are related by conjugation: ρ_cart(g) = B ρ_lat(g) B⁻¹, where B is the matrix of basis vectors. And the trace and the determinant of a matrix are unchanged by conjugation — the trace because tr(XY) = tr(YX), the determinant because det is multiplicative and det(B)det(B⁻¹) = 1.

So the classification can be done entirely in integers, in a basis nobody would choose for physics, and the answers describe the operations in space exactly. Nothing is approximated and nothing is converted. That property is what carries the whole of Neumann’s principle into integer arithmetic later in this field, and it starts here.

The rotoinversions are the half of the table nobody draws #

Five of the ten types are improper, and three of those five are genuinely unfamiliar, because two dimensions has no room for them.

In the plane there are exactly four motions — translation, rotation, reflection, glide — and the point operations among them are rotations and mirrors. Nothing else. In space the determinant −1 operations are not exhausted by mirrors, because a reflection composed with a rotation about the mirror’s own normal is neither a rotation nor a reflection. It is a rotoinversion: turn through 2π/n, then invert through the centre.

Three of them are new objects rather than familiar ones with a new name.

4̅ is the one that catches people. It has order four, it contains a two-fold rotation as its square, and it contains neither a four-fold rotation nor a mirror nor the inversion. A tetrahedron has three of them, along the axes through opposite edge-midpoints, and a reader who expects the tetrahedron’s symmetry to be built from rotations and mirrors will be four operations short.

6̅ is 3/m in disguise, and both spellings are in use. Its cube is the mirror perpendicular to its axis, so a group containing 6̅ contains that mirror whether or not the symbol mentions it — which is why the symbol is written 6̅ and never 6̅/m. Writing the slash would name the same mirror twice.

3̅ is the inversion’s companion. Its cube is 1̅, so any group with a 3̅ axis is centrosymmetric, and there is no way to have three-fold symmetry and a centre without it.

The two integers separate all five without any of this being reasoned about. What the paragraphs above are for is that the reader needs them, and the classifier does not.

Matrix order is the plausible alternative, and it is wrong #

The obvious way to identify a rotation is by its order: the smallest n with Mⁿ = I. “Four-fold” surely means order four.

It gives twenty-eight crystal classes rather than thirty-two.

The reason is a fact about rotoinversions that has no analogue in the plane. For n̅ = −R where R is a rotation of order n,

$$(-R)^k = (-1)^k R^k$$

which is the identity exactly when k is even and n divides k. So the order of n̅ is n when n is even and 2n when n is odd. The inversion 1̅ has order 2, the mirror 2̅ = m has order 2, 4̅ has order 4 — and 3̅ and 6̅ both have order six.

An order-based classifier therefore cannot tell 3̅ from 6̅. It merges them, and it merges the classes built on them: 3̅m with 6̅2m, and two more pairs besides. Four merges, thirty-two down to twenty-eight.

Why the wrong answer is so hard to see #

Twenty-eight is not a list with a mistake in it. Every one of the twenty-eight is a real crystallographic point group. No impostor appears; nothing is malformed; the symbols all name things. Four classes are simply missing, and each of the four that swallowed them looks exactly as it should.

This is the shape of error the site keeps meeting, and it is worth naming precisely: a search that is correct as far as it went. The space-group enumeration turned it up four times over — a symmorphic test that asked whether any operation had an intrinsic translation, an origin-shift grid at sixths where ninths were needed, a catalogue that reported the strongest extinction rule it could express rather than the one the group obeys. In every case the output was well-formed and short.

And the consequence in this case is not cosmetic. 3̅m is centrosymmetric and 6̅2m is not; the eleven Laue classes are exactly the centrosymmetric classes, so an enumeration that has confused the two has confused the answer to what a diffraction experiment can see. The error propagates into every property count downstream.

The same two integers count the classes’ contents #

Once the type of a single operation is a lookup, the census of a whole group is a loop, and the census turns out to answer several questions the field asks.

Whether a class is centrosymmetric is whether its census contains a 1̅ — determinant −1 and trace −3, one operation, no search. Whether a class is enantiomorphic — able to hold a single-handed molecule without its mirror image — is whether the census contains any determinant −1 operation at all, which is eleven of the thirty-two. Which crystal system a class belongs to is read off the counts of 3, 4 and 6, with the cubic case identified by holding eight three-folds rather than by holding the highest-order axis.

None of that requires knowing where anything is. It is the reason the census is such a tempting invariant, and the reason it had to be checked rather than trusted when it was used to merge two lists of subgroups into one classification.

Where the two-integer rule stops #

It identifies the type of an operation and not its position. Two mirrors with different normals have the same determinant and the same trace, and they are different operations of the group. A classification that used only the type census is exactly the fingerprint the previous essay declines to trust for that reason: it knows what operations are present and nothing about where they sit or how they compose.

It is a statement about crystallographic operations, not about isometries. A rotation through 40° has determinant 1 and trace 1 + 2cos40° ≈ 2.53, and the table above has no row for it — correctly, because no lattice admits one. The ten-value table is a consequence of the restriction and is not independent of it.

It says nothing about screws and glides. In a space group an operation carries a translation as well as a linear part, and the determinant and trace see only the linear part. A two-fold rotation and a two-fold screw have identical matrices; what separates them is the intrinsic part of the translation, which no amount of looking at the matrix will produce.

What the machinery does with it #

Every function in this field’s library begins by calling the same three-line classifier, and it throws rather than guesses when handed anything else:

not a crystallographic operation: determinant −1, trace 4

That message has never appeared in a build, which is the point of having it. Its job is to fail loudly if a matrix arrives from somewhere it should not — an arithmetic slip in a change of basis, a metric that was not positive definite, a subgroup enumeration that wandered outside its holohedry. Ten values are legal and everything else is a bug, so the type lookup is also the field’s cheapest integrity check.

Who worked this out, and when #

The trace argument is the modern form of a result that predates matrices being used for it. The restriction itself — that a periodic pattern admits only two-, three-, four- and six-fold rotations — was known to Haüy’s successors as an empirical fact about crystal faces and given proofs through the nineteenth century, most memorably by the shrinking-vector construction that needs no arithmetic at all.

What matrices added was not a better proof but a decidable one. The geometric argument is a picture and has to be believed; the trace argument is an integer and can be checked by a machine that has no idea what a crystal is. Hessel’s 1830 enumeration of the thirty-two classes and Gadolin’s independent one in 1867 were both done by hand, by reasoning about which combinations of axes can coexist — which is a correct method and one whose completeness is very hard to be sure of, since the failure mode is precisely a case nobody thought of.

Frankenheim’s 1842 enumeration of the Bravais lattices is the cautionary case: he found fifteen, and Bravais corrected it to fourteen in 1850 by showing two of them were the same lattice in different cells. The error was not in the reasoning about symmetry. It was in the quotient at the end — deciding when two answers are the same answer — and that is exactly the step this site’s own enumerations count separately and assert separately, for the same reason.

The classifier as the field’s cheapest integrity check #

Because ten pairs are legal and everything else is impossible, the type lookup doubles as a validity test on any matrix that reaches it, and that is worth more than the classification it was written for.

A matrix arriving with determinant −1 and trace 4 is not an exotic operation; it is a bug. It means a change of basis was applied the wrong way round, or a metric was not positive definite, or a subgroup enumeration wandered outside its holohedry, or two matrices were multiplied in the wrong order. Every one of those failures produces a matrix that looks like a matrix, and none of them produces a matrix whose determinant and trace land on one of the ten legal pairs except by coincidence.

So the classifier throws rather than guesses, with a message naming both numbers. It has never fired in a build, which is the point of having it: the cost is two integer comparisons on every operation the field touches, and the return is that an arithmetic slip anywhere upstream stops the build at the first operation it corrupts rather than at whatever count comes out wrong later.

This is the cheap version of the site’s standing arrangement. The expensive version is the round trip — generate a pattern from a group, forget the group, rediscover it, and require the two to agree — which catches errors the type lookup never could, and costs a detection run per figure. The cheap version costs nothing and catches a smaller class of error at the earliest possible moment. Both are worth having and they are not substitutes.

The same two integers count what an operation leaves alone #

There is a third thing the determinant and the trace decide together, and it is the one that connects this page to several essays elsewhere in the collection.

Ask how many points of the cell an operation leaves where they are — points fixed modulo a lattice translation, which is the right question in a crystal, since a point carried to itself plus a lattice vector is the same point of the structure. The answer is |det(I − M)| when that determinant is not zero, and for an operation preserving lengths the determinant collapses to something written in the two integers:

$$\det(I - M) = (1 - \det M)(1 - \operatorname{tr} M).$$

Every entry of the ten-row table can be evaluated with it. The inversion has determinant −1 and trace −3, giving 2 × 4 = 8: a cell contains eight inversion centres, which is exactly the count the equivalent origins of P1̄ come to. A mirror has determinant −1 and trace 1, giving zero — which is the degenerate case and says the fixed set is not isolated but a plane. A 4̄ gives 2 × 2 = 4.

And every proper operation gives zero, because the first factor vanishes when the determinant is +1. That is the statement that a rotation fixes a whole line and never a finite set of points, arriving as an arithmetic identity rather than as a picture.

Where else that determinant has appeared #

det(I − M) is not a new quantity in this collection; it is the same one three other essays turn on, and seeing that they are one quantity is worth the paragraph.

Whether an operation confines its vectors to a plane. A Harker section exists exactly when I − M is singular, which is exactly when the formula above gives zero — so the operations producing Harker sections are precisely the ones with no isolated fixed point, and the inversion’s failure to produce one is the same fact as its having eight of them.

Whether an operation can act freely. The thirteen groups that fold space flat are the ones with no operation holding any point still, and the first cut in that search removes every class containing an operation whose I − M is invertible — because such an operation has a fixed point for every possible translation part.

Where a symmetry element sits. Locating an axis or a plane means solving (I − M)p = −t, and whether that has a unique solution, a line of them, or none is the rank of the same matrix.

One matrix, four questions. That is the strongest argument for working in the lattice basis with integers: a quantity computed once decides a real-space geometry, a reciprocal-space locus, a global classification and a fixed-point count, and it does so without anything ever being measured.

Where this goes #

Two integers name an operation. The next question is what a group of them is called, and that turns out to need something the type census cannot supply: not what operations are present, but which direction each one is about.

4̅2m is the case that shows what is missing. It holds one 4̅ axis, two two-folds, two mirrors and the identity — and so does 4̅m2, which is the same group turned an eighth of a turn about its own axis and is a real symbol printed in the Tables. The census of the two is identical entry for entry, because turning a group does not change what operations are in it; what it changes is which of the system’s directions each one sits on. So the two integers name every operation of the group and cannot name the group’s setting, and a notation that reports position is a notation the census cannot produce.

Reading a class off its own axes is the derivation of the Hermann–Mauguin symbol, and it has one honest convention in it and one exception, and the exception is orthorhombic.

The general lesson is about where to do arithmetic. Written in a Cartesian frame, an operation is a matrix of cosines and its determinant and trace are floating-point numbers that have to be rounded before they can be compared. Written in the lattice basis, both are integers, and every one of the four questions above becomes an exact test. The basis nobody would choose for describing a rotation is the basis in which every statement about it is decidable.