The normaliser is not a function of the group

The same pattern, described twice computes the normaliser of every plane group by conjugation, and one crystal and sixteen coordinate lists does the same in space and gets a number a database has to divide out. Both end with a table: a group’s name and, beside it, how many ways there are of writing one arrangement down.

The first of them names what it did not do. “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.”

The reason it matters is that the table’s shape is misleading. A number printed beside a group’s name looks like a property of the group. It is not. It is a property of the group and its cell, the tabulated value is the one for a cell as general as the group permits, and it is the smallest the number can be.

Where the metric enters #

The normaliser is searched over the holohedry of the lattice — the largest point group the cell’s metric admits — and the reason is not a convenience. A motion that does not map the lattice onto itself is not a symmetry of anything the group describes, so it cannot possibly normalise the group; the holohedry is the ceiling on the search, and nothing outside it needs trying.

A group constrains its cell and usually does not determine it. p2 needs a lattice and a half turn, and every lattice has a half turn, so p2 places no condition on the cell whatever. Its lattice is called oblique because that is the general case, not because the two lengths are forbidden to be equal.

So a p2 pattern may be drawn on a square cell. Nothing about it becomes a different group — the operations are the same two, the pattern is the same pattern — but the lattice now has eight symmetries instead of two, and eight candidates get tried where two did before.

The ladder for one group #

Running the same conjugation search over each holohedry in turn, for a group that permits all five, gives a ladder rather than a number.

Four descriptions on an oblique cell, twenty-four on a hexagonal one. The same pattern, the same group, the same arrangement of the same motif; six times as many coordinate lists that are all of it.

That number is not an abstraction. A structure determined in p2 on a cell that happens to be hexagonal has twenty-four descriptions, and a comparison that divides out four of them will report the same structure as different in five cases out of six. The tabulated value is not wrong — it answers the question for a general cell — but it is not the number that structure needs.

And a cell that buys nothing #

The obvious guess after that is that a larger holohedry always buys more, and it is wrong, which is why the search is a search.

pmg has mirrors in one direction and glides in the other, and the operations a square cell adds are the four-fold rotations and the diagonal mirrors. A four-fold rotation would have to carry the mirror direction onto the glide direction, and those are different kinds of operation, so conjugating by it produces a different group rather than the same one. The candidate fails, and it fails for a reason that is about the group’s own structure rather than about the lattice.

So specialising a metric enlarges the set of candidates and may or may not enlarge the answer. Which happens is decided by conjugation, group by group, and there is no reading it off the lattice type. Of the seventeen, four gain something and one has a specialisation available that gains nothing.

Where the factor of six lives #

The index is a product of two counts and the jump is entirely in one of them, which is worth separating because it says what a specialised cell actually gives a reader.

A normalising motion is a linear part together with a translation, so the normaliser’s order is the number of distinct linear parts that normalise, times the number of origins that do. For p2 on its own cell that is two linear parts and four origins, an order of eight, and an index of four against a group of order two. On a hexagonal cell it is twelve linear parts and four origins — an order of forty-eight and an index of twenty-four.

The origins did not move. All four of p2’s equivalent origins are its four half-turn centres, and those are the four half-turn centres whatever the shape of the cell; nothing about making the cell hexagonal creates a new place to put the origin. Every one of the extra twenty descriptions is the same origin with the axes turned or reflected.

That is the right thing to know before deciding whether the extra descriptions matter. A reader comparing two coordinate lists that were written on the same axes has not been affected at all — the origin freedom is what it always was. A reader comparing lists written by two people who chose their axes independently, on a cell symmetric enough to make several choices look natural, is facing six times as many candidates as the table led them to expect. The failure mode is a comparison across settings, not within one.

Which four, and why not the rest #

Nine of the seventeen groups have somewhere to go: p1 and p2 place no condition on the cell and may sit on any of the five lattices; the five rectangular groups may have their two lengths equalised; cm and cmm may become square or hexagonal. The other eight already occupy the most symmetric lattice they can use, and a group on a square or hexagonal cell has no further specialisation available — there is nothing more to make equal.

Of the nine, p1, pm, pg and cm are the starred ones, whose normalisers are continuous and whose counts are not indices. That leaves five with a real number and a real choice, and four of them gain: p2 by six, and pmm, pgg and cmm by two. pmg is the one that does not.

The pattern in that list is the amount of the lattice the group is already using. p2 uses a half turn, which every lattice has, so it constrains nothing and every extra symmetry of the cell is available to normalise it. pmm and pgg use the rectangular lattice’s mirrors, so equalising the two lengths adds the diagonal directions and the four-fold turn, and half of that is usable. pmg uses the two rectangular directions differently from each other — mirrors along one, glides along the other — so any operation exchanging them changes the group rather than preserving it, and the new symmetry is entirely wasted.

So the quantity that decides is not how symmetric the cell became but whether the group treats the directions the new operations exchange as interchangeable. That is a statement about the group and it is why the answer has to be computed by conjugating rather than deduced from two holohedry orders.

The number that is not an index at all #

Four rows of the census are starred, and the star is doing real work.

p1, pm, pg and cm are normalised by a continuum of translations: sliding the origin along a mirror line, or in any direction at all for p1, leaves every operation exactly where it was. There is no finite count of such origins, so there is no index — and what a search over a grid returns for them is the number of grid points, which doubles when the grid does.

That is why those rows cannot be compared across metrics. p1’s apparent rise from 288 to 1728 is arithmetic about a grid of twelfths and about the size of the hexagonal holohedry, not a statement that a p1 structure on a hexagonal cell has six times as many descriptions as one on an oblique cell. Both have infinitely many, and the honest report is that the question has no numerical answer for those four.

The distinction is detected rather than declared. A group’s origin count is computed at one grid and again at twice that grid: a discrete normaliser gives the same number twice, and a free direction doubles it once per dimension. That test is the previous rung’s and it carries over unchanged, which is what makes the starred rows separable from the four real ones.

What this does to a comparison #

The practical consequence runs through everything the normaliser is used for, and it runs the dangerous way.

A comparison that divides out too little produces false negatives. Two determinations of one structure, reduced by the four cosets the table prints, will disagree if the extra symmetry of the cell has put them in different cosets of the twenty-four. The reduction is incomplete, the canonical forms differ, and the database reports two compounds where there is one — which is the exact failure the space-group rung exists to prevent, arriving through the metric rather than through a forgotten step.

And it cannot be fixed by dividing out more. Applying the specialised normaliser to a structure whose cell is not specialised would identify arrangements that are genuinely different, which is a false positive and is worse. So the reduction has to be chosen from the cell as measured, not from the group as assigned — and that means a comparison needs the metric and not only the symbol.

The awkward case is the one in between, and it is common. A cell with a = 5.001 and b = 4.999 is not square, and is not meaningfully oblique either. The exact computation has nothing to say about it: the holohedry of that lattice is the oblique one, exactly, and the four extra motions are not symmetries of it by any amount. Whether to treat the structure as though they were is a decision about tolerance, and it is precisely the decision this site declines to make — because a threshold there would decide the answer, and there is no principled place to put it.

What can be said is which side of the question a mistake falls on. Treating a nearly-square cell as square divides out motions that are not quite symmetries, so the structures identified are nearly the same rather than the same. That is a defensible thing to do knowingly and an indefensible thing to do by accident, and knowing that the number depends on the metric is what makes the difference.

A third freedom, beside the two already counted #

A description of a structure has been shown to have two kinds of freedom in this collection, and this rung adds a condition on both rather than a third of its own.

The origin is what the normaliser’s translations count, and it is the freedom the International Tables acknowledge by printing twenty-four space groups twice. The axes are what a setting is: relabelling a, b and c gives another description of the same crystal, and the group of relabellings is what the affine normaliser adds to the Euclidean one.

The cell itself is not a third freedom — it is a fact about the crystal — and that is exactly why it is dangerous. An origin and a setting are choices somebody made and can be recorded; a metric is measured, it comes out of the diffraction, and nobody chose it. So the extra descriptions a special metric creates arrive without anyone having decided anything, and there is no field in a structure report that records them.

That makes the effect the quiet kind. Conjugation is the test of sameness throughout this collection, and the set of things to conjugate by is read off the lattice — so a lattice that is more symmetric than expected silently enlarges that set. Nothing warns, no count comes out fractional, and the arithmetic is correct at every step for a question nobody asked.

The reduction procedures are where it should be caught. A reduction with one rule produces a canonical cell from a lattice by its metric alone, and a cell that reduces to a more symmetric type than the group requires is exactly the flag this rung is about. A comparison that reduces the cell first and reads the normaliser from the reduced type rather than from the group’s symbol gets the right number automatically — which is a small change of order in a procedure, and the whole of the practical content here.

What was computed, and how it was checked #

Computed here: for each of the seventeen plane groups and each lattice type whose holohedry contains all of the group’s own operations, the motions that normalise the group — by conjugating over that holohedry and over every translation on a grid of twelfths, and asking whether the operation set comes back identical. Then the index, the origin count, the number of distinct linear parts, and the ratio between the general metric’s index and the best one’s.

The first check is that the general case reproduces what already existed. Running the new search over each group’s own lattice must give the number the previous rung computed, for all seventeen, and it does. That is the test that says the holohedry override is the only thing that changed.

A specialisation has to be legal and the legality is tested. A group whose operations are not all in the target lattice’s holohedry does not sit on that lattice at all — p4 on a rectangular cell is not a group with a longer ladder, it is a contradiction — and the search refuses it rather than returning a number for it.

And the exactness is not incidental. The translations are exact rationals, and the comparison of operation sets is a comparison of exact keys. The first run of this computation used floating-point translations and returned an index of zero for every group on every metric, because a third written as 0.3333333333333333 is not the key an exact third produces. Every index zero is a loud failure; a tolerance would have made it a quiet one.

Where the ladder goes next is one dimension up #

Nothing above is two-dimensional in substance, and in space the effect is larger for a reason worth stating.

There are more ways for a cell to be accidental. In the plane a metric is two lengths and an angle, and there are five lattice types. In space it is three lengths and three angles, fourteen Bravais types, and a great many more coincidences available — a monoclinic cell with β accidentally ninety degrees, an orthorhombic one with two lengths accidentally equal, a rhombohedral one with an angle at ninety.

And the tables acknowledge it. The International Tables print Euclidean normalisers with entries for specialised metrics alongside the general ones, which is what makes those pages intricate; the general entry is one line and the specialisations are a list beneath it. What this rung supplies is not that list, which is long, but the reason it exists and the arithmetic that generates it — and the demonstration that the effect is not marginal, since a factor of six turns up in the plane, where there is much less room for it.

The direction this leaves open is the one the two-dimensional case cannot reach. Every metric here is a lattice type, so every specialisation is a jump between five discrete possibilities. In space there are metrics that are special without being a different Bravais type at all — a cell whose a/c ratio takes one particular value, so that an extra motion maps the lattice onto itself without the lattice being of a more symmetric kind. Those are where the tabulated normalisers become genuinely long, they are not reachable by substituting one holohedry for another, and computing them needs the lattice’s automorphisms found from the metric rather than looked up from its name.