The step a flat surface has no room for

There are seventeen ways to repeat a pattern across a plane and two hundred and thirty ways to repeat one through space. The second number is thirteen and a half times the first, and the obvious explanation — space has more directions, so there is more room for symmetry — accounts for almost none of it.

Count the obvious sources of growth first. The plane has five lattices; space has fourteen. The plane has ten point groups compatible with a lattice; space has thirty-two. Multiply the two and the answer is not 140, because most pairings are impossible — a four-fold rotation cannot act on a hexagonal lattice — and what survives is seventy-three compatible pairings, each of a point group with a lattice it can actually sit on. Those seventy-three are the arithmetic crystal classes, and they are the honest count of “a point group on a lattice” in three dimensions.

Seventy-three. Not two hundred and thirty. The remaining factor of a little over three is the whole subject of this field, and it comes from a single fact.

It is worth seeing where the seventy-three comes from, because the arithmetic is less tidy than the sentence above suggests. The thirty-two point groups distribute unevenly across the seven crystal systems — the triclinic system has two, the monoclinic three, the orthorhombic three, the tetragonal seven, the trigonal five, the hexagonal seven and the cubic five — and each system offers between one and four lattices to sit on. A point group does not get to choose freely among them: the cubic groups need a cubic lattice and there are three, the tetragonal groups need a tetragonal lattice and there are two, and the triclinic groups have exactly one available. Adding up point groups times available lattices, system by system, gives seventy-three pairings, and each pairing is a distinct way for a point group to act on a lattice by integer matrices.

That last phrase is doing the work. Two point groups can be abstractly the same and act differently — the group 2 acting on a monoclinic lattice with its axis along b is not the same arithmetic object as the group m acting with its mirror perpendicular to b, even though both have two elements. What is being counted is the action, not the abstract group, and that is why the count is seventy-three rather than thirty-two.

The operation the plane cannot hold #

In the plane there are four kinds of symmetry: translation, rotation, reflection and glide reflection. Three of them fix at least one point. The glide fixes nothing — it reflects across a line and slides along it, and no point stays put.

That is the plane’s one operation with a translation built into it, and it is the reason the plane has seventeen groups rather than thirteen. Four of the seventeen — pg, pmg, pgg and p4g — exist only because a glide is available.

Space has the glide too, in five varieties instead of one. It also has something the plane has no room for at all.

A screw axis turns and advances. Take a point, rotate it a quarter turn about a vertical axis, and lift it a quarter of the way up the cell. Do that four times and it has come back to its original orientation, one whole cell higher — which is a lattice translation, so the operation is a symmetry of something periodic. It has no fixed point. It is not a rotation, and it is not a rotation composed with anything a change of origin can remove.

The plane cannot do this because there is nowhere to advance to. A rotation in the plane turns about a point and the axis of that rotation sticks out of the page, into a dimension the plane does not have. Give it that dimension and the rotation can climb.

Why the count is not simply larger #

Here is the part that is easy to state and easy to get wrong. Adding screws and glides does not multiply the number of groups by the number of screws and glides available. What it does is ask a different question, and the different question has more answers.

Take a point group — say the group generated by a two-fold rotation, which has two elements. Put it on a primitive monoclinic lattice. Now ask: what translation goes with the rotation?

The answer is not “none”. It is whatever the geometry allows, subject to one condition: applying the operation twice must give a lattice translation, because the square of a two-fold rotation is the identity of the point group and so must be a pure translation of the crystal. That condition permits two answers. The rotation can come with no translation along its axis, in which case it is a rotation and the group is P2. Or it can come with half the cell, in which case it is a 2₁ screw and the group is P2₁.

Two groups from one point group and one lattice. Not because there is more space, but because there is more than one way to attach the translations.

That is the mechanism, and repeating it across all seventy-three arithmetic classes is what turns seventy-three into two hundred and thirty. Some classes admit only one attachment; the class with a single two-fold on a primitive lattice admits two; the class with two perpendicular mirrors and a polar axis admits ten.

Symmorphic, and the rest #

A group in which the translations can all be removed at once — where some choice of origin makes every operation a plain rotation, reflection or inversion — is called symmorphic. There is exactly one symmorphic group per arithmetic crystal class, because “no translations anywhere” is a single answer, so the symmorphic groups number seventy-three.

The other hundred and fifty-seven have at least one screw or glide that no origin removes.

The word symmorphic is worth a moment, because the distinction it draws is not cosmetic. In a symmorphic group there is a point of the crystal whose site symmetry is the whole point group — a place where every operation of the point group acts, all at once, about that one spot. In a non-symmorphic group there is no such place. The point group is still there, in the sense that dropping every translation leaves it behind, but it is a quotient rather than a subgroup: something the space group maps onto, not something sitting inside it.

That distinction has consequences a reader can check. A symmorphic group has a Wyckoff position whose multiplicity equals one; a non-symmorphic one need not. And non-symmorphic groups extinguish reflections, which is the only reason anyone can tell them apart from the outside.

Reading the symbol at the top of the page #

P2₁/c is four characters and it says everything above about one group. It is worth unpacking, because the two-dimensional notation already met on this site extends with one addition, and the addition is the interesting part.

The leading capital is the lattice: P for primitive, and the letters I, F, A, B, C and R for the centred cells. That character is the whole of the lattice’s contribution, which is a remarkable compression of fourteen possibilities into one letter — it works because the system is recoverable from what follows.

After it come the symmetry elements, one position per direction rather than one per element. A monoclinic group has one direction worth naming, the unique axis, so P2₁/c has one position and it reads: along b, there is a two-fold screw with an advance of one half, and perpendicular to b there is a c glide. The slash is what makes “perpendicular to” rather than “along”.

The subscript is the piece that has no two-dimensional counterpart. 2₁ is a two-fold axis whose advance is 1/2 of the repeat; 4₃ is a four-fold whose advance is 3/4. A subscript of zero is not written, so a bare 2 is a plain rotation. That single digit is the intrinsic translation, and this field spends most of its time on it.

There is one trap in the notation worth naming early, because it catches everybody. P2₁/c and P2₁/m differ in one character and are numbers 14 and 11 in the Tables — but P2₁/c, P2₁/n and P2₁/a are all the same group, written on differently chosen axes. The glide letter names a direction, and a monoclinic cell can be re-chosen by replacing c with a + c without disturbing the unique axis, which turns an n glide into a c glide. Three symbols, one group. The enumeration essay is where that stops being an annoyance and becomes the mechanism the count depends on.

Two counts, and only one of them is here #

This site does not enumerate the two hundred and thirty. It is worth being blunt about that, because the whole habit here is that a count is produced rather than quoted, and this is the largest number the field will use.

What the machinery does enumerate is the space groups in a stated arithmetic class: a finite brute-force search over the translations that can be attached, quotiented by the two things that do not make a new group. That search is exact, it is complete, and it is run on six classes across this field.

Getting from six enumerated classes to all seventy-three is not a matter of running the same loop more times. The obstruction is the equivalence: deciding when two space groups written on different axes are the same group is a question about the normaliser of the point group in the integer matrices, and doing it correctly for all seventy-three is the content of the classification rather than an application of it. So 230 appears in this field as a fact from the literature, and every essay that uses it says so.

The number this site does produce, and produces from scratch, is fourteen.

What the picture cannot show #

A space-group diagram is a projection, and the projection loses things.

Look again at the diagram at the top of this page. It is the cell seen from above, with heights printed as numbers beside the points. That is the Tables’ convention and it works, but it means the third coordinate is a label rather than a position — a reader cannot see that two points are at the same height unless they read two numbers and compare them.

The deeper limit is one this field will keep running into. Nobody can check a space-group diagram by looking at it. In the plane, a reader with patience can trace the operations of a wallpaper pattern and satisfy themselves that the caption is right. Here they cannot, and neither can the person who drew it.

That is not a reason to draw fewer diagrams. It is the reason every point set here goes through a round trip: the pattern is generated by applying a group’s operations, the group is then discarded, and every operation the lattice permits is tested against the bare point set. If what comes back is not exactly what went in, the figure does not appear at all.

Run on the group at the top of this page, that is a short computation with a long consequence. Four operations come out of the symbol; the orbit of three points in general position gives twelve points; the group is thrown away; and the detector tests each of the four matrices a monoclinic lattice permits against every translation that could carry one of the twelve onto another. It rediscovers four operations, and they are the four that went in. Nothing in that exchange is a matter of judgement, which is exactly the point — the diagram cannot be checked by looking and does not have to be.

Extending that machinery from two dimensions to three is the work this field opens with, and it is the next essay. Nothing in the decidability argument ever mentioned the number two — the operations are integer matrices because they map a lattice to itself, and that is as true of a 3×3 matrix as of a 2×2 one. The machinery here was two-dimensional for a long time, and that was a limit rather than a principle.

Three people, working separately, and one of them was wrong #

The classification was completed three times in four years by people who did not know what the others were doing, which is unusual enough to be worth the paragraph.

Evgraf Fedorov published in 1891, in Russian, in a mineralogical journal. Arthur Schoenflies published in 1891 in German, working from group theory rather than from crystals. William Barlow, an English businessman with no mathematical training who had made a fortune in property and taught himself crystallography, published in 1894 by an argument about how spheres pack.

All three arrived at essentially the same list, and none of them arrived at it correctly. Fedorov’s first list had 229, having merged a pair that are distinct; Schoenflies’ had errors of the opposite kind. The two corresponded, found each other’s mistakes, and both published corrections — after which the number was 230 and has stayed there. Barlow’s independent third pass, arriving from packing rather than from algebra, is the reason nobody has seriously doubted it since.

The detail worth carrying is which pair Fedorov merged. He had identified two groups as the same that are mirror images of one another and cannot be superposed by any rotation — one of the eleven enantiomorphic pairs. Whether they count as one group or two is a question about what “the same” means, and the answer the Tables settled on is the one that matters for a real crystal, which can be built one way round and not the other. Fedorov’s error was a defensible position stated as an arithmetic result, which is a specific and recognisable way to get a count wrong.

The classification was also, in a sense that took another eighty years to appreciate, only the beginning. The same question in four dimensions has 4,894 answers, computed by machine in 1978; in five, 222,018; in six, nobody has finished counting. The plane’s seventeen and space’s two hundred and thirty are small enough to be tabulated and large enough to be interesting, which is a narrow window and a piece of luck.

Where the ladder goes #

The two facts this essay leans on are both worth their own arguments. The screw axes are enumerated — there are exactly eleven, for a reason about divisibility — and so are the glide planes, where the fifth one exists only in centred lattices and the enumeration says why.

The fourteen lattices are counted here rather than displayed, and the counting turns on a single computable question asked of twenty-five candidate cells.

And the reason any of this is known about a real crystal is that screws and glides extinguish reflections — which is a claim about a sum over the group’s operations, computed independently of every diagram on this page, arriving at the same symmetry by a route that shares nothing with it but the atoms.

The factor of thirteen, split into its two causes #

The essay’s opening asks where the growth comes from and answers the screw. That answer can be made quantitative, and the arithmetic separates two effects that a single ratio runs together.

The plane: thirteen arithmetic classes, seventeen groups. Thirteen of the seventeen are symmorphic — one per class — so the extensions contribute four extra groups, a surplus of about thirty per cent.

Space: seventy-three arithmetic classes, two hundred and thirty groups. Seventy-three are symmorphic, so the extensions contribute a hundred and fifty-seven extra, a surplus of more than two hundred per cent.

So the factor of thirteen between the two totals is a product of two separate factors. Seventy-three against thirteen is a factor of about five and a half, and it is the ordinary consequence of having more lattices and more point groups — the “more directions” the essay’s dek sets aside.

And the surplus goes from thirty per cent to two hundred, which is a further factor of nearly two and a half. That second factor is the screw, and it is where the essay’s claim lives: the extensions are not merely more numerous because there are more classes, they are richer per class.

The reason is that space has more room to hide a translation. In the plane an operation’s intrinsic translation can only lie along a mirror line, so there is one kind of glide and no screws. In space it can lie along an axis or in a plane, giving eleven screws and five glides — and each of those is a way for a class to produce a group its symmorphic member does not cover.

So the honest reading is that both halves matter and they multiply. More classes because there are more lattices and more point groups; more groups per class because there are more places for a translation to survive an origin shift. Neither alone gives thirteen.