The same pattern, described twice

Two crystallographers determine the same structure and publish two lists of coordinates. Every number in the second list differs from the corresponding number in the first. Neither is wrong.

Nothing here is a measurement problem. A structure is reported as a group together with positions read against an origin and a set of axes, and the group does not fix the origin — so moving it to another point of equal standing rewrites every coordinate and changes nothing whatever about the crystal. The cell is a choice makes that case for the axes. This is the same case for the origin, and it has a sharper answer, because the set of moves that leave a description intact is a group and can be computed.

What the operations have to satisfy #

Write n for a candidate move: a translation, or a rotation, or a reflection — some isometry of the plane. Applying it to a description means conjugating every operation of the group by it, and the description is unchanged exactly when

$$n G n^{-1} = G.$$

That is the definition of the normaliser of G, and it is the same arithmetic that conjugation and sameness uses to decide when two operations inside a group are the same symmetry, asked one level up about the group as a whole.

The condition is stronger than it first looks. It does not say that the group is mapped to a group of the same kind — every conjugate of G is a group of the same kind, since conjugation is an isomorphism. It says the conjugate is G itself, operation for operation, with the same translation parts. So a normalising move takes a coordinate list to another coordinate list that could have come out of the same table of positions, and no computation on the description could tell which of the two it was handed.

Every element of G normalises G, which is a check rather than a remark: a search that misses one has a bug in its conjugation and not an interesting finding. So the group always sits inside its own normaliser, and what matters is the index between them — the number of descriptions of one arrangement that the arithmetic calls equal.

Four origins for p2, and none for p6m #

For p2 the answer is four, and they are the four half-turn centres. This is worth pausing on, because it links the normaliser to something already computed: the Wyckoff positions of p2 are exactly those four points, and an origin shift to any of them leaves the operation (−I, 0) looking like (−I, 0) rather than (−I, t) for some t.

One half of that observation is a theorem and the other half is false, and the difference is worth having because the two are constantly run together. An equivalent origin always has the highest site symmetry a point of the cell can have: moving the origin there is precisely what leaves every operation with the translation part it had, and that is the same statement as “as much of the group fixes this point as fixes any point”. That inclusion holds on all seventeen. The converse — that a point of highest site symmetry may be taken as an origin — fails on three of them.

p4g is the case that shows it. Four of its points have site symmetry of order four — the two four-fold centres at (0, 0) and (½, ½), and the two points at (½, 0) and (0, ½) where a half turn meets two mirrors. All four are special positions of the same order, and only the first two are places the origin may go. Site symmetry has a type as well as an order, and conjugation preserves the type: no operation of the plane carries a four-fold rotation onto a pair of mirrors, so no move of the origin can turn one description into the other. pmg fails the converse for the same reason with a whole mirror line at the highest order, and pg fails it trivially, having no point of any site symmetry at all while still admitting a line of equivalent origins.

So a table of Wyckoff positions is nearly a table of the places it is sensible to put the origin, and reading it that way is right fourteen times in seventeen. The three exceptions are the ones where the order of a site symmetry has been read as though it were the whole of it.

At the other end of the seventeen sits p6m, whose normaliser is p6m itself. Index one: there is exactly one way to describe a p6m pattern, up to the lattice. Every other group has at least two, and the arithmetic says which.

Reading the table: which groups have the most descriptions #

The census repays reading across rather than down. Three patterns are in it.

The richest is p3, at index twelve. Its own three operations are normalised by all twelve linear parts the hexagonal lattice permits and by three translations — the origin may be moved to any of the three threefold centres, and the whole hexagonal holohedry may be applied on top. So a p3 pattern has twelve descriptions that no computation on the description could tell apart, which is the largest number among the discrete cases and a good deal more than most tables of the seventeen would lead anybody to expect.

The poorest is p6m, at index one. A group that already contains everything its lattice permits has nothing left to be normalised by. There is exactly one way to write a p6m pattern down, and it is the only one of the seventeen of which that is true.

In between, the index tracks how much of the lattice’s symmetry the group has left on the table. p4 sits inside the square holohedry with index two and gains a factor of four from its two equivalent origins; p31m and p6 are each normalised by all twelve hexagonal linear parts and have a single origin, so both come out at index two. The rule is not that low-symmetry groups have many descriptions and high-symmetry groups few — pm has a continuum and p3 has twelve — but that the index measures the gap between the group and the symmetry available to it, in both the operations and the origins.

That gap has a name in the subgroup language of the previous essay: the normaliser of G within the holohedry is where G sits in the descent from the lattice’s full symmetry, and the index is how far down.

When the answer is a continuum, and how an exact computation says so #

The search runs over a grid of twelfths, which raises an obvious objection: a count of origins found on a grid is a count of grid points, not a fact about the group. The objection is right, and the answer is to run the search twice.

A group whose normalising translations are discrete gives the same count at a grid and at twice that grid. A group normalised by a continuum gives a count that doubles for each free direction — one free direction doubles it, two quadruple it — so the dimension of the freedom is read off the ratio and never estimated. p1, which constrains nothing, has two free directions: every translation normalises it, because conjugating a translation by a translation gives the translation back. pm has one: sliding the origin along a mirror line leaves every operation exactly where it was, and sliding it across the mirror does not.

That is how a continuous answer shows up honestly inside an exact computation. The number 144 is never printed as though it meant something; what is printed is two free directions, which is the true statement.

The normaliser, rediscovered from the drawing #

Everything so far is arithmetic on operations, which on this site is only half of an argument. The other half is the round trip: forget that the group has operations, and ask the question about pictures.

Let P be the pattern grown from a motif m, and let n be any isometry the lattice permits. Growing a pattern from the moved motif n·m gives G·n·m; moving the whole of P by n gives n·G·m, which is (nGn⁻¹)·n·m. So the two point sets are equal exactly when G and nGn⁻¹ have the same orbit through n·m — and the round trip is what turns that into an equivalence rather than an implication. If the orbits agree while the groups differ, the point set is invariant under both, so its detected symmetry is strictly larger than G, and the drawing does not appear at all rather than appearing under a false label.

Which means the normaliser is visible in the drawing. Move the motif to an equivalent origin and the picture is the same picture, carried by n. Move it anywhere else and it is not.

Doing that search over every operation the lattice permits recovers the normaliser from point sets alone, having looked at no matrix at all. It is then required to equal the normaliser found by conjugation, in both directions — an operation the drawings accept and the algebra rejects is as much a failure as the reverse. Seventeen groups, two routes, one answer each.

Why the normalising operations form a group #

The set is closed, and the proof is two lines worth having because it is the reason the count is an index rather than a tally.

If m and n both normalise G then (mn)G(mn)⁻¹ = m(nGn⁻¹)m⁻¹ = mGm⁻¹ = G, so their composition normalises. If n normalises then conjugating the equation nGn⁻¹ = G by n⁻¹ gives G = n⁻¹Gn, so the inverse normalises. The identity normalises trivially. That is a group and not a list, by the same three checks every symmetry group on this site passes.

Two consequences follow at once. The first is that G ⊆ N, so |N| is a multiple of |G| and the ratio is a whole number — which is asserted while the census is computed, because a normaliser order that failed to divide the group order would mean the search had found something that is not a group. The second is that N is itself one of the seventeen, when it is discrete: the normaliser of p2 is p2mm on the same lattice with an origin at any half-turn centre, the normaliser of p3 is p6m on a cell of a third the area, and so on down the table. A normaliser is not an exotic object bolted on to a group; it is another plane group, and a reader who knows the seventeen already knows the answers.

The one place that fails is the continuous cases, and it fails for a reason worth naming: p1’s normaliser contains every translation of the plane and is therefore not discrete, so it is not a wallpaper group at all. A group whose normaliser is not of the same kind as itself is the arithmetic signature of a group that constrains too little to pin anything down.

What is not in the normaliser, and what it does instead #

An operation outside the normaliser is not useless; it produces a conjugate. Conjugating p2 by a translation of a fifth of a cell gives a perfectly good group of half-turns about points a fifth of a cell away from the original ones. It is the same abstract group, it describes a pattern congruent to the original, and it is a different set of operations.

That distinction is the whole reason the normaliser is the right object here. Two descriptions are the same description when the move between them normalises; they are descriptions of congruent-but-differently-placed things when it does not. A comparison of two published structures that ignores the difference will call two distinct arrangements identical, and a comparison that ignores the normaliser will call one arrangement two.

The count of conjugates is also what makes a subgroup census come out in two different right answers. The maximal subgroups of p6m include three copies of cmm, permuted by the sixfold rotation; a table prints one entry with a multiplicity of three because the three are conjugate. Conjugacy is the equivalence, and the normaliser of a subgroup inside its parent is what counts how many copies there are: the number of conjugates is the index of the normaliser.

Where this computation stops #

Three limits, stated because the index numbers above are meaningless without them.

This is the normaliser within the symmetry the lattice has. The search runs over the holohedry — the finite list of integer matrices preserving the lattice — and over translations on a grid. That is a genuine subgroup of the Euclidean normaliser rather than an approximation to it, and it leaves out the part that needs a different metric. p1 is the extreme case: its lattice may be any lattice at all, so it is normalised by isometries that are symmetries of no lattice, and those cannot be integer matrices in this basis by construction.

The International Tables answer a slightly larger question, and their entries can carry a changed cell: the Euclidean normaliser of p3 is p6m on a cell of a third the area, which is the three equivalent origins found here together with the extra rotations, described in the smaller cell rather than as an index. The numbers agree; the presentation differs, and a reader comparing the two should compare indices rather than symbols.

Nothing here is about a specialised metric. A structure whose lattice happens to be more symmetric than its group requires — a monoclinic crystal whose β is accidentally 90° — has a larger normaliser than its symbol suggests, and that is a fact about the specimen rather than about the group. Near-symmetry and the tolerance is where this site keeps that distinction, and it applies here in full.

What it is used for, and by whom #

The normaliser is not a curiosity of the tables. Three practical questions turn on it.

Comparing structures. Deciding whether two published structures are the same one requires putting both into a standard description, and the moves available for doing that are exactly the normaliser’s. A comparison program that tries every origin shift is doing this computation.

Counting the ways a structure could be wrong. A structure solved in the wrong origin is shifted by a vector that is not in the normaliser — if it were, the description would be identical and there would be nothing wrong. So the failure modes are indexed by the coset space, and the number of distinct wrong answers is finite and computable in advance.

Superstructures and twins. When a cell is doubled, the daughter’s normaliser is smaller than the parent’s, and the descriptions that were equivalent before the transition are no longer equivalent afterwards. The ones that come apart are precisely the domain states — which is how many domains a transition makes seen from the side of descriptions rather than of orientations.

Where the ladder goes next #

This rung establishes the object. The next question is the one the International Tables actually print, and it is sharper: for some groups the conventions that pick an origin give two different answers, and the group is published twice with every coordinate shifted between the two tables. That is two origins for one group, and it is the same arithmetic doing damage rather than bookkeeping.

Above that sits the specialised-metric half — what happens to the normaliser when a lattice is accidentally more symmetric than its group requires — which is where the tables become genuinely intricate and where this site has computed nothing yet.

Why a structure solution has to fix an origin by hand #

The freedom this essay measures is usually met for the first time in the middle of solving a structure, where it appears as an obstacle rather than as an ambiguity, and seeing it there explains why the count matters to somebody who never thinks about normalisers.

A diffraction experiment measures intensities and loses phases. Recovering the phases is the phase problem, and the methods that attack it work from relations among phases rather than from the phases themselves — relations that hold whatever origin the description uses.

They have to, because the data cannot know the origin. Move the origin by an equivalent shift and every phase changes, each by an amount depending on its own indices, while every intensity is unaltered. So no function of the intensities can determine a phase: the intensities are the same for every one of the descriptions this essay counts, and the phases are not.

The fix is to choose. A direct-methods program assigns phases to a small number of reflections arbitrarily, and this is not a guess to be checked later — it is a declaration of which of the equivalent origins the solution will be expressed in. Everything the program computes afterwards is relative to that declaration.

How many reflections it takes is decided by the normaliser. The origin shifts that leave the group unchanged form a group of their own, and enough reflections must be fixed to pin down every one of them. In a centrosymmetric structure in the lowest symmetry there are eight equivalent origins, three independent binary choices, and three reflections whose signs are set by hand — one for each choice.

Choose badly and the program fails. Fixing reflections whose indices do not distinguish the candidate origins leaves some of them still available, and the phase relations then have several mutually inconsistent solutions rather than one. The rule for choosing them is a rule about which index parities are represented, which is the normaliser’s arithmetic in the form a programmer meets it.

The same freedom, in a database #

The other place this count is applied is in deciding whether two published structures are one structure, and there the freedom has to be removed rather than fixed arbitrarily.

Two descriptions of one crystal will not compare numerically. Their coordinate lists differ by an element of the normaliser, so a comparison of the numbers reports a disagreement that is not there — and a comparison across a database of hundreds of thousands of entries would report a great many.

So descriptions are standardised before they are compared. The procedure applies every element of the normaliser to the coordinate list, generating all the equivalent descriptions, and keeps whichever one is first under a stated ordering. Two structures are then the same when their standardised lists agree, and the ordering is arbitrary in exactly the way a normal form always is: what it must do is choose the same representative from every description of one structure.

The Wyckoff labels move too. A normaliser element that carries one half-turn centre to another exchanges the two Wyckoff positions attached to them, so which centre is called a and which is called b is a consequence of the description rather than a property of the crystal. Two reports of the same structure can therefore disagree about which position an atom occupies, both correctly.

That is the practical content of the whole essay. The number in the table is the number of ways a correct description can be written, and every operation that compares two descriptions — a database lookup, a difference map, an automated check for duplication — has either to divide by it or to be wrong by exactly that factor.