The plan contains the group

Every pattern figure on this site makes a round trip. A group generates an orbit, the group is then forgotten, and a detector rediscovers it from the bare point set; the two must agree exactly or the figure is not drawn. In three dimensions the same trip runs on a set of points in a cell.

The diagrams have never been part of that. A plan is produced from the operations — each element located, each mark placed — and then it is drawn, and nothing has ever asked whether the drawing is a complete description of what produced it.

So: is the drawing enough? Take the marks, throw the operations away, and try to get the group back.

What counts as the drawing #

The whole difficulty is being strict about what a reader can see, because it is easy to smuggle the answer in.

From each mark, four things are kept:

• what kind of thing it is, and of what order — a four-fold axis, a mirror, an inversion centre;
• which way it points — the axis direction, or the plane’s normal;
• where it sits — the located position, exactly as drawn;
• how far it slides — the screw’s advance along its axis, or the glide’s vector in its plane, which is what the tails and dashes on the symbols encode.

The matrix is thrown away. A reader does not see a matrix, and a reconstruction that kept one would be reading the answer rather than the drawing. What replaces it is the lattice: given a direction and an order, the lattice’s own holohedry supplies the candidate matrices, and there is generally exactly one. That is precisely what a reader brings to a diagram — the symbol at the top of the page says which lattice, and the marks are read against it.

Forty-five out of forty-five #

The centred groups are the result worth pausing on. A centring translation is not drawn on any diagram: it is a pure translation, it has no location and no direction, and there is nothing on the page to mark. The lattice letter appears in the symbol above the plan, not in the plan.

And the marks turn out to imply it anyway. Feeding the closure nothing but the drawn elements of Fddd or of Fm3̅m returns the full group, centring included — because the marks sit at positions that force it. Two glide planes a quarter of a cell apart compose to the centring translation, and the diagram cannot show the pair without showing what they generate.

That is a stronger statement than the one this essay set out to check, and it says something about the Tables’ design: the diagram is not an abbreviation of the operation list, it is an alternative encoding of it.

The misreadings, which are the refusals #

A reconstruction that returned the right group whatever it was fed would prove nothing at all. So the same diagram is read five wrong ways, each corresponding to a mistake a reader could actually make.

Rub the tails off the screw axes. Every screw becomes a plain rotation, and the group that comes back is a different one — for all thirty groups here whose diagrams have a screw. The tails are the difference between P2₁/c and P2/m, and the reconstruction knows it.

Read every glide as a mirror. The dashes are what distinguish them, and ignoring them gives the wrong group in all twenty-one groups with a glide. This is the difference between pm and pg restated in three dimensions: a slide of half a cell is not a decoration.

Keep only some of the marks. Pnma with only its axes drawn rebuilds a group of four rather than eight — which is P2₁2₁2₁, sitting inside it exactly as a subgroup that keeps every translation should. P2₁/c with only its inversion centres rebuilds P1̅, of order two. Both are proper subgroups, and both are what the retained marks genuinely support.

Move one mark. Displacing a single element by a third of a cell — an amount no lattice vector and no choice of origin can absorb — produces operations the group does not contain. The position of a mark is as load-bearing as its kind.

Two defects the round trip found #

The reconstruction failed on eight of the forty-five when it was first run, and both reasons turned out to be in the element locator rather than in the reconstruction. Both are the same shape of defect as the one the double-plane essay reports: something the diagrams had been drawing wrongly, invisibly, for a long time.

The screw index was computed from a reduced vector. A screw’s index — the 1 in 2₁, the 3 in 4₃ — is how far one turn advances along the axis, as a fraction of the repeat. The locator computed it by projecting the intrinsic translation onto the axis direction, having first reduced that translation into the unit interval component by component.

For an axis along a cell edge, reducing changes nothing. For an axis along a diagonal, it destroys a sign: the 2₁ screw of P4/mmm along [1, −1, 0] has an intrinsic translation of (½, −½, 0), the reduction turns it into (½, ½, 0), and the projection onto the axis then comes to zero. A genuine 2₁ was reported as a screw of index nought.

The quantity being got wrong is the one the tails on a mark encode. There are eleven screw axes in all, and each is named by an order and an index — 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁ through 6₅ — where the index is how far one turn advances along the axis as a fraction of the repeat. Along a cell edge the advance is a single positive component and reducing it into the unit interval leaves it alone. Along a diagonal it is not, and the reduction is the whole of the damage.

Every diagonal screw in a tetragonal, hexagonal or cubic group carried the wrong tail, and no gate could see it. The mark was drawn, in the right place, of the right kind; only its index was wrong. Every check this fleet has asks whether what is drawn fits, contrasts and sits inside the canvas.

The second defect is smaller and the same in kind. A rotoinversion’s axis is the direction it reverses, whatever its order — the rotation leaves the axis alone and the inversion negates it. The locator asked for the direction the operation fixes, which is right for odd orders and empty for even ones, so 4̅ and 6̅ came back with no direction at all. Nothing drawn was wrong, because a plan marks a rotoinversion by its point; what could not be done was read the mark back, which is how it was found.

Both are fixed, and the reconstruction that found them is now the check that would find them again.

How much of a plan is redundant #

The first version of the refusals asked a different question — does removing a mark break the rebuild? — and the answer is no, almost always.

A plan is enormously over-determined. P2₁/c carries twenty-seven marks for four operations; Fm3̅m carries over a thousand for a hundred and ninety-two. That is not waste — the marks a reader needs are the ones near where they are looking, and a diagram that showed a minimal generating set would be unusable — but it means that a missing mark is a poor test of whether a drawing is complete.

A misread mark is a good test, and that is why the refusals above are misreadings. It is the same distinction the double-plane essay ran into from the other side: the failure that no check could see was an element that was absent, and absence is exactly what redundancy hides.

The diagram says more than the symbol does #

There is a third description in play and it is the one a reader meets first: the Hermann–Mauguin symbol. Setting the three side by side is worth doing, because they carry different amounts.

The symbol names the group and not its placement. P2₁/c is a group; it does not say where the origin is, and the same group has two inequivalent origins in a good many cases — one on an inversion centre, one on the axis — with the Tables printing both settings on the same page. It does not say which axes were chosen either, which is why one group answers to three symbols and, with axis permutations, to as many as six.

The diagram names the placement too. Every mark carries a position, so a plan drawn on the second origin is a different drawing from one drawn on the first, and the reconstruction returns the operations with their translations as drawn rather than up to a shift. That is why moving one mark by a third of a cell breaks it: the reconstruction is not free to slide the answer.

And that is exactly what the normaliser measures. The operations that move a description without moving the pattern are the ones that carry one legitimate diagram to another; two plans related by a normaliser element rebuild to conjugate groups, which are the same group described twice. So the ordering is: the symbol determines the group up to conjugacy, the diagram determines it up to nothing, and the operation list is what the diagram is a picture of.

A reader who has met a group only as a symbol has met less of it than a reader who has met the plan — which is the argument the Tables have been making by printing the diagram first.

The other round trip, for comparison #

It is worth putting the two trips side by side, because they check different things and the difference is the point of having both.

Pnma is the case where that blindness is easiest to feel. Its elements alone — screw axes along all three directions, glide planes perpendicular to them, inversion centres between — come to sixty-three marks in one cell, for a group of eight operations, four of which suffice to generate it. Every one of those sixty-three is drawn from an operation, and until the marks could be read back, not one of them was checked by anything. The orbit does not depend on them. A reader does.

The point-set trip checks the pattern. It asks whether the drawn points really have the claimed symmetry and no more, and it catches the failure this site was built around: a motif placed too symmetrically, so that the picture illustrates a different group from the one in the caption.

The diagram trip checks the description. It asks whether the marks say enough, and it catches a different failure: a symbol drawn with the wrong tail, a plane drawn without its dashes, an element in the wrong place. Neither trip can catch the other’s failure, and for most of this collection’s life only one of them was being run.

The pairing also explains why the two defects above survived so long. A wrong screw index changes no point in any orbit — the operations are generated from the group’s own list, not from the diagram — so the point-set trip is completely blind to it. The mark is decoration as far as the pattern is concerned, and a decoration that nothing reads is a decoration that nothing checks.

Where the exactness stops #

The reconstruction is exact and integral. Positions and slides are exact rationals; matrices are found by search over the holohedry; the comparison of the closure with the group is a set comparison on canonical keys, both directions checked.

The sense convention is quoted, not derived. An axis carries an operation and its inverse, drawn as one mark, and reading the mark requires knowing that the screw index counts a right-handed advance. That is the International Tables’ convention, it is what a reader has to bring to the page, and it is the one input here that is not computed. It is also what distinguishes P4₁ from P4₃, so it is not a technicality.

The advance is carried as a vector, not as an index, and the reason is a genuine subtlety about centred cells. The index counts fractions of the lattice repeat along the axis, and in a centred lattice that repeat is shorter than the conventional direction vector — an F-centred cubic group’s diagonal axes advance by a quarter of the conventional vector, which is half of the real repeat. Carrying the vector needs no convention and no case analysis.

Forty-five is not two hundred and thirty. This site defines forty-five space groups, chosen to cover the phenomena; the census is over those. Whether every one of the two hundred and thirty rebuilds from its plan is not established here, and the honest statement is that no group among the forty-five fails and that they include every lattice type, every kind of glide and screw, and the two cubic holohedries.

The diagram is one cell. The Tables draw one cell and this reads one cell. A rebuild that needed two would be a different claim about diagrams, and would be worth saying so.

The heights, which the projection cannot draw #

A plan is a projection down one axis, so two elements at different heights land on the same place on the page. The Tables’ answer is a fraction printed beside the mark, and those fractions are not decoration: they carry a coordinate the drawing has thrown away.

That means the reconstruction is not reading a picture in the ordinary sense. It is reading a picture plus a list of numbers, and the numbers are exactly the information the projection destroyed. A plan with its heights rubbed out would be a genuinely weaker input, and it would fail: an inversion centre at height zero and one at a quarter are different elements, and a group with the first is not the group with the second.

So the essay’s claim has a qualification worth stating precisely. What contains the group is the plan as the Tables print it — marks, positions, kinds and heights — and not the black-and-white pattern of symbols on the page.

That is also why the axonometric diagram exists alongside the plan. It shows the third dimension directly, at the cost of being harder to measure positions on, and the pair is the Tables’ resolution of a trade every projection makes. The plan is exact and needs annotation; the axonometric is legible and needs care.

How few marks would do #

The redundancy measurement asks what happens when a mark is removed and finds that almost nothing does. The complementary question — how few marks suffice — has a short answer, and it says what the extra marks are for.

The reconstruction closes whatever it is given, so it needs exactly enough marks to carry a generating set. For most groups that is two or three operations, so two or three marks are enough, and a plan carrying twenty-seven is carrying eight times what the reconstruction requires.

The surplus is for the reader and not for the algorithm. A diagram showing only the generators would be correct, unreadable, and useless for the job a plan is actually used for — which is to answer, at a glance, what sits at this position. Every mark is there to be looked up rather than to be closed over.

And the surplus is what makes the misreadings detectable. A drawing with three marks and a wrong one produces a wrong group with nothing to contradict it. A drawing with twenty-seven produces a set of operations that do not close, or that close into something whose own diagram would carry different marks — so an error is visible as an inconsistency rather than only as a wrong answer.

That is the general argument for redundancy in a description, and it is the same one behind measuring every reflection several times: the extra copies buy nothing when everything is right, and they are the only thing that says so.

Who found it, and when #

The plan diagrams are older than the space groups they describe are old in print. The graphical symbols — the lens for a two-fold, the tailed shapes for screws, solid and dashed lines for planes — settle into their modern form in the 1935 first edition of the Internationale Tabellen, drawing on Schoenflies’s and Fedorov’s own figures from the 1890s.

They were designed to be read, not merely looked at. A crystallographer working from the Tables reads a plan to find out what operations sit where — which coordinates are equivalent, which positions are special, where an atom may go. That the marks determine the group is therefore something the design has always assumed, and this essay is the first place on this site where it is checked.

The convention that made the check possible is the one that nearly broke it. Screw tails are drawn for a right-handed advance, and enantiomorphic pairs like P4₁ and P4₃ differ in their diagrams by exactly those tails. A diagram drawn without them would leave eleven pairs of groups indistinguishable, which is the fact the enantiomorphic pairs essay is about.

Where this leaves the drawings #

This site’s rule has been that a figure is decoration unless its point set came through the machinery and back. That rule has now been applied to the machinery’s own output in the other direction: the drawing is not a lossy rendering of the group, it is the group in another notation.

Which raises the standard the diagrams are now held to. A plan that omits a mark is redundant enough not to notice; a plan that draws a mark wrongly — one line where there are two, a tail with the wrong index, an axis in the wrong place — is a different group, and there is now a check that says so.