The same symmetry, somewhere else

A pattern in p4m has mirror lines running four ways: two along the cell axes and two along the diagonals. A reader asked whether the horizontal mirrors and the diagonal mirrors are the same symmetry has to guess, because the question has not been given a meaning. Giving it one is the whole of this essay, and the answer for that pattern turns out to be no.

The operation that settles it is conjugation: given a symmetry $h$ and another symmetry $g$, form $g h g^{-1}$. That composite is what $h$ looks like after $g$ has moved everything, and two operations count as the same symmetry exactly when the group contains a $g$ carrying one to the other.

What the composite does #

Reading $g h g^{-1}$ from the inside out makes it obvious rather than formal.

Apply $g^{-1}$: undo whatever $g$ did. Apply $h$: perform the symmetry in its original position. Apply $g$: redo $g$. The net effect is $h$ performed in the place $g$ moved things to.

If $h$ is a reflection in a vertical line and $g$ is a translation to the right by half a cell, then $g h g^{-1}$ is a reflection in a vertical line half a cell to the right. If $g$ is a quarter turn, $g h g^{-1}$ is a reflection in the line that the quarter turn takes the original mirror to. The conjugate is always the same kind of thing, relocated.

That last sentence is a theorem and not an observation, and it is worth seeing why. The linear part of $g h g^{-1}$ is $M_g M_h M_g^{-1}$, which is a similarity transformation, and a similarity preserves determinant and trace. The determinant separates the direct motions from the reversing ones, and the trace fixes the rotation order. So a conjugate of a threefold rotation is a threefold rotation, a conjugate of a mirror is a mirror or a glide, and nothing can turn into anything else. That is a stronger statement than it looks, because it means the inventory of a pattern’s symmetries — how many of each kind — is fixed before any question of arrangement arises.

The classes, and what they are for #

Grouping the operations of a group by conjugacy gives its conjugacy classes, and the classes are the honest answer to “how many different symmetries does this pattern have”.

For p4m the answer is not eight, which is the number of operations per cell, and it is not two, which is the number of kinds. Running the classification on the group gives five classes, and their sizes are the interesting part.

The identity is always alone in its class, since $g \cdot 1 \cdot g^{-1} = 1$ for every $g$, which is the first thing the group axioms give back. The half turn at the centre of a fourfold pattern is alone as well. The two quarter turns pair up, because a reflection sends a quarter turn clockwise to a quarter turn anticlockwise. And the four mirrors split into two classes of two rather than falling into one class of four — which is the fact the two figures above exhibit, and the reason the answer to the opening question is no.

Nothing about a picture of p4m suggests that split. Both families are solid lines drawn the same way, and both are mirrors of a pattern whose whole point is its high symmetry. Only the composition tells them apart.

Why the split happens #

The axis-parallel mirrors and the diagonal mirrors of p4m are distinguished twice over — by the directions they run in, and by what sits on them — and the two accounts are the same fact in different clothes.

The direct argument is about directions. Conjugating an operation by $g$ turns its linear part into $M_g M_h M_g^{-1}$, so the direction a mirror runs in is carried by $M_g$ — and every linear part in p4m is built from a quarter turn and an axis reflection. Those map the two axis directions to each other and the two diagonal directions to each other, and never mix the families, because the smallest rotation available is $90°$ and exchanging the families would need $45°$. No amount of composing gets from one family to the other, and the computation is the exhaustive version of that sentence.

The geometric shadow of the same fact is about what sits on each line. The axis-parallel mirrors of p4m run through the fourfold centres and through the twofold centres at the cell-edge midpoints. The diagonal mirrors run through fourfold centres only. Since conjugation moves the whole configuration — mirror and everything on it together — an operation carrying an axis mirror to a diagonal one would have to carry a twofold centre onto something that is not one. The classes are separated by site symmetry, which is a property nothing can change.

This is the same argument that separates p3m1 from p31m, run inside a single group rather than between two. The recurring lesson is that placement relative to the rest of the group is part of what an operation is, and every question that seems to be about an operation alone turns out to be about its relationship to the others.

What the computation checks #

The classes on this site are computed rather than tabulated, and the computation carries two assertions that would fire if the group were wrong.

The first is closure under conjugation. Every conjugate of a group element by a group element must itself be in the group; if it is not, the operation list was not a group in the first place. The check runs over every ordered pair, which for a group of order twelve is a hundred and forty-four compositions, and it fails on the first conjugate it cannot find.

The second is preservation of kind and order. The theorem above says a conjugate of an $n$-fold rotation is an $n$-fold rotation, so the classifier is run on both and the results compared. That assertion has no business ever failing, which is exactly why it is worth having: it tests the classifier and the composition code against one another, and an error in either shows up as a mirror that conjugates into a glide.

Both checks pass on all seventeen groups, and the classes they produce are the ones the International Tables list, which is a useful external agreement rather than a proof of anything.

The smaller groups, where the counting is easy to follow #

The seventeen span the whole range of behaviour, and looking at the two extremes makes the general shape clear.

At one end sits pmm, whose four operations per cell — the identity, a half turn and two perpendicular mirrors — fall into four classes of one. Every operation commutes with every other, so conjugation moves nothing, and the group’s own arithmetic says that its two mirrors are different symmetries. They are: one runs horizontally and one vertically, and nothing in pmm turns the pattern by a quarter turn to exchange them. A reader who thinks of pmm as “the group with mirrors both ways” is describing two symmetries and calling them one.

At the other end sits pgg, whose four operations are the identity, a half turn and two glides, again in four classes of one — and here the arithmetic is saying something less obvious. The two glides run in perpendicular directions and neither is conjugate to the other, even though the pattern looks as though its two glide directions ought to be interchangeable. They are not, because the group has no operation that exchanges the axes. Adding one gives p4g, a different group, and in that group the two glides do become conjugate.

The general rule is short enough to carry: an operation of the group can only be moved by the group. Whether a pattern looks as if two of its features ought to be interchangeable is a question about the reader; whether they are is a question about a composition table.

The convention this depends on #

Conjugacy is a property of a group, and a group here means the operations modulo the lattice translations.

That qualification is doing real work and it is easy to lose. A wallpaper group is infinite: it contains every translation by every lattice vector, and every mirror in the infinite family of parallel mirrors a cell’s worth apart. Working with the infinite object would put every one of those mirrors in its own place and make the classes infinite too.

Reducing modulo translations collapses each infinite family to one representative and makes the group finite — order eight for p4m, twelve for p6m — which is what makes the classes computable and small. The price is that the classes describe families of elements rather than individual mirror lines, and a statement like “these two mirrors are conjugate” means “these two families are related by an operation of the group”.

Stated that way it stays true and stops being ambiguous. The alternative convention — working with the infinite group — gives the same answer to the question this essay asks, by a longer route.

Where the exactness stops #

The comparison is exact, and what it is exact about is worth pinning down.

Two operations are conjugate or they are not, and the test is a search over a finite list with an exact equality at the end of it. There is no tolerance. But the question the test answers is about a given group, and which group a real pattern has is where the softness lives: a pattern whose symmetry has been assigned by eye, or a structure whose group came from a refinement, brings its uncertainty in with it, and the class computation inherits whatever that was.

There is a second limit, sharper and easier to miss. Conjugacy inside a group is not the same as conjugacy inside the normaliser — the larger group of motions that map the pattern’s group onto itself without necessarily preserving the pattern. Two operations can fail to be conjugate in the group and be conjugate in its normaliser, which is the formal statement of “there is a symmetry of the description relating them, but not a symmetry of the pattern”. Crystallography uses that distinction constantly when deciding whether two structure determinations are the same answer in different settings.

So “the same symmetry” has at least two useful meanings and this essay computes the narrower one. Naming which is meant costs a clause and saves an argument.

Where else the classes show up #

Three consequences, each of which is the reason somebody would want this computed.

Equivalent positions. The International Tables list, for every group, the positions an atom may occupy and the site symmetry of each. Positions related by the group are one entry; positions with the same site symmetry that the group does not relate are two entries. That table is a conjugacy computation, run on the stabilisers rather than on the operations — and the stabiliser of a point is exactly what makes a motif at a careless position illustrate the wrong group.

Spectroscopy. The number of vibrational modes of each symmetry type is determined by the character table of the point group, and a character table has exactly one column per conjugacy class. So the count of classes fixes the shape of every selection-rule argument in infrared and Raman spectroscopy, and getting the classes wrong changes how many peaks a calculation predicts.

Twinning. When a crystal grows in two orientations related by an operation that is not one of its symmetries, the result is a twin, and the twin law is exactly an element of the normaliser that is not in the group. The distinction between conjugate-in-the-group and conjugate-in-the-normaliser is what tells a crystallographer whether an observed relationship is a symmetry or a growth accident.

The surprising part #

Conjugacy classes are the reason a pattern can have fewer symmetries than it looks like it has, and also the reason it can have more.

Here is the connection worth carrying away. In the group p1 — no rotations, no mirrors, nothing but translations — every element is in a class by itself, because the group is abelian and $g h g^{-1} = h$ whenever $g$ and $h$ commute. So p1 has as many classes as elements, and a pattern with no symmetry beyond repetition has the most “different symmetries” of any group in the list, by this count.

That sounds like a paradox and it is a definition working correctly. Conjugacy measures how much the group can move its own operations around, and a group with nothing to move things with cannot relate any two operations. High symmetry means few classes and many elements per class. The number of classes is a measure of rigidity, not of richness, which is precisely opposite to the intuition the word “symmetry” produces.

Who worked it out #

Conjugation as an idea belongs to Galois, who used it to distinguish the subgroups that behave well under quotients from the ones that do not — the normal subgroups, defined precisely by being closed under conjugation by everything.

Its arrival in crystallography came through the classification. Fedorov, Schoenflies and Barlow independently derived the two hundred and thirty space groups in the early 1890s, and the enumeration cannot be organised without it: the question “have these two arrangements already been counted” is the question of whether one is conjugate to the other — the same question that keeps the seventeen from being eighteen, and the two derivations that disagreed at first disagreed about exactly that. Fedorov and Schoenflies corresponded, compared lists, and reconciled them, which is one of the more amiable episodes in the history of a contested result.

The modern formulation, in which a space group’s classification is a statement about conjugacy classes of subgroups of the affine group, was settled by Bieberbach in 1911 in the course of proving that the number of groups in each dimension is finite — a result Hilbert had asked for as the eighteenth of his problems.

Where the ladder goes next #

The immediate companion is the fundamental domain, which is what the orbit of a point looks like when the group’s action is drawn rather than composed.

The immediate application is the pair p3m1 and p31m, where the difference between two groups is a difference in what the conjugacy classes of the mirrors look like, and the notation that records it position by position.

The structural sequel is the classification proof, which is an exhaustive search over arrangements that would not terminate without a way to recognise two arrangements as the same.

What the pictures here cannot show. A conjugacy class is a set of operations, and an operation is not a visible feature of a pattern. Every mark on these figures is placed by the computation, and a reader comparing two classes is comparing two annotations. That the mirrors in one class are unreachable from the mirrors in the other is a statement about a completed search through the group, and no drawing can exhibit the absence of a path.