The motif must be a comma

Take the group p3. Apply its three operations to a single dot. Tile the result across the plane. The pattern that comes out has six symmetries, not three, and the three extra ones are mirrors and glides that were never applied to anything.

The generation was correct. The picture is of p31m, and the caption says p3.

This essay is about how often that happens, why it cannot be seen, and what nineteenth-century ornamentalists did about it without being able to say why.

The mechanism #

A group applied to a motif produces the motif’s orbit: every image of every point under every operation. If the motif has symmetry of its own, or sits at a position that some operation leaves alone, the orbit is smaller than the group and the resulting point set can have coincidences that the group does not.

The clean case is a lone dot. Under p1 — the group of translations alone — its orbit is a copy of the lattice. And a lattice always has an inversion centre: the midpoint between any point and its own translate is one, because inversion through that midpoint swaps the two and does the same for every other pair. So the orbit of a single dot under p1 is always a p2 pattern.

That is not a near miss or a matter of degree. It is exact, it holds for every dot position and every lattice, and it means a lone dot can never illustrate the group with no symmetry at all.

How often, measured #

The obvious next question is whether p1 is a special case. It is not, and the numbers are worth having in full.

For each of the seventeen groups, take a grid of 121 candidate dot positions across the unit cell — every point $(i/12, j/12)$ with $i$ and $j$ from one to eleven — generate the orbit, and ask how many of the resulting patterns have more symmetry than the group used to make them.

p1, pm, pg and cm: all 121. Every dot position fails. These four groups can never be illustrated with a dot at all.

p3: 85 of 121. Seven times in ten.

p3m1: 67. p4: 57. pgg: 57. pmg: 41. p4g: 41. pmm: 40. p6: 37. p31m: 29. cmm: 20. p4m: 20. p2: 8. p6m: 2.

Every group fails somewhere. Thirteen of the seventeen can be illustrated with some dot position, and the four that cannot are among the simplest in the list.

Two different counts are in play here and they are worth keeping apart. The numbers above ask, for a fixed group, how many dot positions spoil it. The other question — how many groups a particular dot manages to illustrate — has a different answer: a dot placed at $(1/3, 1/5)$, a position chosen without any care at all, verifies eleven of the seventeen and fails the other six. Neither number is the more important one. The first says the hazard is widespread; the second says a careless choice runs into it immediately.

The counterintuitive entry #

The last number in that list deserves attention, because it runs opposite to intuition.

p6m, the most symmetric of the seventeen, is the hardest to spoil: only two of 121 dot positions produce anything more symmetric. The naive expectation is the reverse — that a group with twelve operations offers the most opportunities for an accident.

The reason is that p6m already has everything the hexagonal lattice permits. Its point group is the lattice’s full holohedry, so there is no further rotation or reflection available to acquire. The only way to gain symmetry is to gain translations, which happens when the dot lands exactly on a rotation centre and its orbit collapses onto a finer lattice. That requires a special position, and only two of the 121 grid points are one.

So the hazard is not proportional to a group’s size. It is proportional to how much room the group leaves between itself and the maximum its lattice allows, which is a much better way to think about it.

Stabilisers, which is what is really going on #

The hazard has an exact formulation, and having it removes the impression that this is a quirk of drawing.

The stabiliser of a point is the set of operations that leave that point exactly where it is. For a point in a general position the stabiliser is trivial, and the orbit then has as many members as the group has operations. For a point on a rotation centre or a mirror line the stabiliser is larger, and the orbit is correspondingly shorter — the relation is exact:

$$|\text{orbit}| \times |\text{stabiliser}| = |\text{group}|.$$

A short orbit is where the trouble starts. Fewer points means more chance that some operation outside the group happens to permute them, and the extreme case is an orbit of one, which is invariant under everything the lattice permits.

That is why the numbers in the previous sections vary so much between groups. A group with many operations and a lattice offering nothing further — p6m — has few ways to gain. A group with few operations on a lattice offering a great deal more — p3 on a hexagonal lattice whose holohedry has order twelve — has many.

Why looking does not work #

The failure has a specific shape, and it is the shape that makes it invisible.

Both pictures are patterns. Both repeat correctly. Both have the motif at the positions the group sends it to. The wrong one has more structure than intended, and more structure reads to the eye as a better drawing rather than as an error. A reader inspecting the p3-with-dots figure sees a pleasing hexagonal arrangement and has no reason to suspect anything.

Worse, the extra symmetry is usually the pleasant kind. Mirrors make a pattern look more regular; the accidental version is frequently prettier than the correct one. Anybody drawing by eye and choosing the more satisfying result is systematically selecting for the error.

And there is no visual cue to check against. The claim “this pattern has no mirrors” is a claim about a completed search, of the same kind as the claim that only five rotation orders exist. A drawing can display a mirror; it cannot display the absence of every possible mirror, and the eye has nothing to look at.

What fixes it #

Adding points to the motif removes the coincidences, and the arithmetic of how many are needed is tidy.

One point, at a position chosen without care, verifies eleven of the seventeen.

Two points verify sixteen, whatever positions are chosen. The one that still fails is p1, and it fails for the midpoint reason: inversion through the midpoint of the two points, composed with the translations, maps the pair onto itself for any two points whatever. That is a fact about pairs, not about drawing.

Three points in no particular arrangement verify all seventeen. Three points have no forced coincidence: a scalene triangle has no mirror, no rotation and no inversion of its own, and a general position gives it no accidental relationship with the lattice.

So the motif used throughout this site is three points forming a small scalene, chiral cluster, drawn as a short flag with a head so its orientation and handedness are both visible.

Handedness, and why the motif is drawn in two colours #

There is a second reason the motif has to be asymmetric, and it is about communication rather than correctness.

Of the four motions, two preserve handedness and two reverse it. That division is the most informative thing a pattern figure can convey — it is what distinguishes a rotation from a reflection at a glance — and it is entirely invisible if the motif is its own mirror image.

So every figure here draws the motif as a small flag with a head, and draws reflected copies in a second colour. The colour is not a colouring in the technical sense; it records the sign of the determinant of the operation that produced the copy, which is a property of the operation rather than a property assigned to the mark.

pgg is the case that most rewards the convention. Without handedness shown, its pattern is a pleasant arrangement of marks. With handedness shown, the alternating reversals are obvious, and the question “where is the mirror” becomes askable — at which point the answer, that there is none, becomes interesting rather than invisible.

The comma, which was right all along #

Look at any nineteenth-century ornament plate — Owen Jones’s Grammar of Ornament of 1856, or the crystallographic plates that followed Fedorov’s enumeration — and the motif is not a dot. It is a comma: a small asymmetric mark with a head and a tail.

The convention long predates the algebra that explains it. Draughtsmen used the comma because it shows what happened: a comma turned is visibly turned, a comma reflected is visibly reversed, and a reader can follow the operations across the plate. A dot shows nothing, because a dot looks the same after every motion.

What the numbers above add is that the convention is not merely helpful for exposition. It is necessary for correctness. A plate drawn with dots is not a less legible illustration of the seventeen groups; for four of them it is an illustration of different groups entirely.

That is a satisfying kind of result. A practice adopted for one reason turns out to have been required for another, and the second reason was not available to the people who adopted it.

The same problem, with real atoms #

Crystallography meets this constantly, and the vocabulary for it is a century old.

A special position is a point whose stabiliser is non-trivial — a point lying on a rotation axis, a mirror plane, or an inversion centre. An atom placed at a special position has a shorter orbit than the general one, and the International Tables list every special position of every space group explicitly, with its site symmetry and its multiplicity.

The listing exists because the consequences are practical. An atom at a special position contributes differently to the diffraction pattern, has fewer free coordinates to refine, and must have a local environment compatible with its site symmetry. Placing an atom at a special position it does not actually occupy is a standard way to refine a structure into a symmetry it does not have — which refines perfectly well and gives slightly wrong answers everywhere.

So the dot problem is not an artefact of drawing pictures, and it is not confined to the plane — the same bookkeeping about site symmetry governs which coordinates a structure refinement is allowed to vary.

The dot problem It is the same phenomenon that makes structure determination require care, and the same remedy applies: check the symmetry the arrangement actually has, rather than the symmetry it was built with.

How the check works here #

Every pattern figure on this site makes a round trip, and this hazard is what the round trip is for.

The pattern is generated by applying the group to the motif. The group is then discarded, and the bare point set is handed to a detector that enumerates every operation the lattice permits — at most twelve integer matrices, each with its translation determined exactly by a pair of pattern points — and keeps the ones that map the set onto itself. The detected set must equal the generating set exactly.

Both directions matter and the second is the one at issue here. Too few detected symmetries means the motif was not invariant. Too many means this essay’s problem, and the figure throws.

The detector was not written for this essay. This essay exists because the detector found the problem in the site’s own default motif — the intention had been to draw dots, and the round trip refused eleven of the seventeen groups outright.

What the rule costs #

Insisting on an asymmetric motif is not free, and the costs are worth naming rather than leaving as an unexplained preference for cluttered figures.

It costs legibility. A dot is the clearest mark available. A three-point comma has an orientation, a handedness and a size, and every one of those is a visual channel spent on something other than where the copies are. A reader who only wants to see the lattice would be better served by the dot, and the figures on this site are harder to read than the nineteenth-century plates partly for this reason.

It costs page weight. Each copy of the motif is a polyline of three points and a filled circle rather than a single circle, and the orbit contains one copy per operation per cell. For p6m over four cells that is twelve operations times four cells times two elements — a few hundred drawn objects where a dot pattern would use a few dozen. The plate of all seventeen is the worst case, and its size is dominated by this decision.

It costs arithmetic. The check is quadratic in the number of points: for each of the lattice’s linear parts, every pair of points supplies a candidate translation, and each candidate is tested against the whole set. Tripling the motif multiplies the point count by three and the work by nine. That is the single largest cost of making these figures, and it is paid on every one of them, every time.

None of the three is a reason to stop. They are the price of the one thing the convention buys — that a caption naming a group is a statement the machinery has tested rather than a statement the author believes — and a picture that is slightly harder to read and demonstrably of the right group is a better picture than the reverse.

What the check cannot do #

Three limits, stated plainly because the method is easy to over-trust.

It does not check the drawing. The round trip proves that the point set has the claimed symmetry. It says nothing about whether a label is in the right place, whether a mark is legible, or whether the caption describes what a reader will see. Those failures are caught by looking, and looking is still part of the process.

It does not apply to aperiodic patterns. A Penrose tiling has no lattice, so there is no finite holohedry to enumerate and no exact coordinates to compare. Its properties are measured rather than decided.

It does not choose the motif. Nothing in the machinery suggests three points; the machinery only refuses fewer. The choice came from running the check against one point, then two, and reading the failures.

Where the ladder goes next #

The pair most vulnerable to this hazard is p3m1 and p31m, where a careless motif collapses either group into p6m and the two become indistinguishable.

The framework it sits inside is the orbit, which is where stabilisers and special positions are set up properly.

And the independent confirmation route is diffraction: a pattern with accidental extra symmetry scatters differently, so the error shows up in a calculation that never looks at the point set directly.

What the pictures here cannot show. Two of the figures above do display the frequency — the table of which groups a dot can illustrate anywhere, and the map of where an orbit collapses — so the counts are no longer only in the prose. What no drawing here shows is the reason a particular dot fails, since that is a statement about an operation the pattern has and the group does not, and a picture can display an operation only by displaying the arrangement it maps onto itself. The pair figures exhibit one instance each; the tables exhibit how often; neither exhibits why, and the why is the stabiliser arithmetic in the text.

An atom is a dot #

The hazard on this page looks like a matter of illustration, and in three dimensions it stops being one, because nature supplies the offending motif.

An atom, to a first approximation, is a sphere. Its scattering density is spherically symmetric, so it has the symmetry of the dot and every accidental operation the dot invites. A structure built from one kind of atom on one orbit is therefore not merely drawn in too high a group; it genuinely has the higher symmetry, and no amount of care in the description will lower it.

What lowers it is chemistry, not drawing. Two different elements on what would otherwise be one orbit break the coincidence in exactly the way a second point in the motif does here — which is what an ordering transition arranges, and why the resulting structure has a lower group than the disordered one it came from. A molecular unit does the same job more emphatically, since a molecule is an asymmetric mark by construction.

And the anisotropy of thermal motion is the residual case. An atom vibrating in a site of low symmetry is described by an ellipsoid rather than a sphere, and the ellipsoid’s axes must be compatible with the site symmetry — so a refinement that reports an ellipsoid more symmetric than its site is reporting the same failure this essay is about, arriving from the data rather than from the pen.

The rule that comes out of all three is the one the plane case teaches. Symmetry belongs to the orbit, and the orbit is decided by what was repeated.