Five hundred and thirty rows, four hundred and ninety-nine descriptions

Six ways to name one group and one matrix, four rules both end the same way. The 230 groups have 530 settings in the International Tables, they say, and deriving that list from the operations rather than reading it would be the strongest available test of what a setting is. It could not be done then, for want of every group, and two hundred and thirty or two hundred and nineteen is where the two ways of counting them are separated. Each of the 230 is now available with its operations and with a way to ask whether a change of basis carries it onto itself.

So the derivation has been run, and the outcome is not the one those essays expected. The operations do not give 530. They give 499, and the 31 rows beyond that are rows of the printed list that name descriptions already present. That is no error in the Tables. It reveals that their list is a list of something a little different from what it is usually said to be, a list of symbols with their cell choices rather than of different sets of operations. The difference is confined to six types, and the reason for it can be drawn.

There is a second result on the way. The Tables give 24 of the groups a second origin, and that is usually presented as a matter of convenience. A short rule about site symmetry reproduces 23 of the 24, with no false positives. The twenty-fourth is a tie that the rule cannot break and the Tables broke by judgement.

A setting is a coset #

Begin with the definition, since the answer depends on it. A setting here is a description of a group: its operations written on one choice of conventional axes, taken modulo an origin shift. Two choices of axes give one setting exactly when the change of basis between them, combined with some shift of origin, carries the group onto itself. That last condition is membership of the group’s affine normaliser.

So a count of settings is the size of a coset space. Take the finite group of axis choices that are available, ask how many of them the normaliser holds, and divide. In the orthorhombic system the available choices are the six orders of the three axes, as the earlier essay on naming explained, and the number held is a subgroup of the permutations of three things, so it has one, two, three or six elements. The settings are then six, three, two or one.

Each order of the axes is tested by trying it with every choice of signs and every origin shift on a grid of twenty-fourths of a cell, and requiring that the operations come back as the same set. Nothing in that test mentions a symbol. Of the 59 orthorhombic types, 11 have one setting, 1 has two, 26 have three and 21 have six. The single type with two is $Pbca$, the only one whose normaliser holds a three-cycle of the axes without holding a swap. Its three glides chase each other round the axes, $b$ then $c$ then $a$, so turning the axes cyclically gives back the same group, while swapping two of them reverses the sense of the chase and gives $Pcab$. That one bit, the direction of a cycle of glides, is a real distinction no origin can remove, and it is close kin to the sense of a screw. Nothing else on the list of 59 has that structure, which is why the column for two settings holds a single entry. $Ibca$ looks as if it should join it, and that is one of the six cases below.

The monoclinic system needs a second finite group. There are three choices of which axis is the unique one. Then, in the plane across that axis, the cell’s other two edges can be chosen in several ways, and the Tables take six of them: three “cell choices”, each with or without a swap. Those six form a group isomorphic to the permutations of three things, and it can be pictured as permuting the three non-zero half-vectors of the plane, $a/2$, $c/2$ and $(a + c)/2$. The normaliser only acts modulo two is the reason so small a group is enough. So a monoclinic type has three times six divided by how many of the six its normaliser holds, and that comes out as three or nine across all 13 types: three for the five with no glide and no centring, where every cell choice gives the same spelling, and nine for the eight others.

The remaining systems contribute nothing to think about. A tetragonal, trigonal, hexagonal or cubic group has its axes forced, so it has one setting per origin. The seven rhombohedral groups are written on both hexagonal and rhombohedral axes, by convention, and so they have two. An enantiomorphic pair counts as two types, as it does in the Tables.

The two origins, as a rule #

One more multiplier is needed. Some centrosymmetric groups are printed twice, once with the origin on an inversion centre and once at a point of higher site symmetry. Two origins for one group took one example and asked why. The usual answer is that a centre is convenient for computing structure factors, and that a point of high symmetry is convenient for describing the structure. That suggests a test that can be run on every group: a group should have two origins when some site is larger than every site holding an inversion.

The rule uses only the site orders, found from the operations by stabilisers on the same grid. It selects 23 types. Each is among the 24 that the Tables print twice, and there are no others. So what looked like a case-by-case editorial decision is, to within one group, a statement about point symmetry. The Tables give a second origin when choosing a centre would mean abandoning a point of strictly higher symmetry.

The twenty-fourth is $P4_2/ncm$. Its largest site with an inversion has order four, the site symmetry $..2/m$, and it also has sites of order four with no inversion, $\bar4..$ among them. That is a tie. So is $P4_2/m$, which has the same pair of orders and only one origin in the Tables. In all, 33 types have a tie of that kind: a site of the largest order without an inversion beside a site of that order with one. The Tables print two origins for exactly one of those 33. Whatever made them choose that, it was not the site order alone. The rule therefore has a real exception, one that should be reported rather than fixed by making the rule more elaborate. A rule that fits the 24th by adding a clause about which symmetry is more useful is no longer a rule, and it would be a description of a choice made by people.

The count #

Multiply them together for each type, and the result can be compared with the printed list.

The list used for the comparison is the one printed in volume B of the International Tables, with Hall’s symbols attached, which is also the table the spglib library uses, and that is where the row counts here were read from, type by type. It has 530 rows by system: 2 triclinic, 105 monoclinic, 241 orthorhombic, 81 tetragonal, 32 trigonal, 27 hexagonal and 42 cubic.

The count from the operations is $2 + 87 + 228 + 81 + 32 + 27 + 42 = 499$, with $P4_2/ncm$’s origin included by hand as the one exception. Five of the seven systems agree exactly. That is itself a substantial test: the tetragonal column depends on the origin rule 13 times, the cubic on it 6 times, and the trigonal on the two descriptions of the rhombohedral groups.

Two systems do not agree, and the excess is 31. It is natural to suspect an error in the count at this point, and most of the effort went there. The discrepancy turns out not to be spread about. It is in six types and nowhere else.

Each of the six has exactly twice as many rows as descriptions. So the printed list treats two descriptions as different where the operations say they are one, and the question is what those pairs look like.

Two rows, one set of operations #

Take the first pair. The printed list has the symbols $C1c1$ and $C1n1$ in two separate rows for space group number nine, under different cell choices. They look like different statements: the first has a glide of half of $c$, the second a glide of half of $a + c$.

The two are one group, for an elementary reason. The $C$-centring adds a translation $(\tfrac12, \tfrac12, 0)$. Compose that with a $c$ glide in the plane $y = 0$, and the result is a glide in the plane $y = \tfrac14$, with translation $(\tfrac12, 0, \tfrac12)$: an $n$ glide. So every crystal in $Cc$ already has both planes, alternating. Whether it is called $C1c1$ or $C1n1$ depends only on which of those planes the origin sits on. Move the origin by a quarter of $b$ and the name changes, and not a single operation does.

The computation found this by itself. Asked whether the change of cell between those two rows lies in the normaliser of this type, it returned a shift of six twenty-fourths along $b$, and 1,152 grid shifts in all, since the directions in the plane allow free movement in a polar group. Nothing told it about glides.

So a printed row is not simply a set of operations with a fixed origin convention. Each row carries a symbol, and each symbol is read at a stated origin choice, on the plane the first symbol position refers to. When a symbol can be read two ways at two origins, the list keeps both. That is a reasonable thing for a reference to do, since a user searching for the symbol $C1n1$ needs to find it. It means, though, that the number 530 counts names rather than descriptions. The difference matters only if the number is read as a measure of how many genuinely different ways a group can be written, which is how it is usually quoted.

The same mechanism accounts for the other five. In $C2/c$ the argument is identical. In $Cmme$ and $Ccce$, $C$-centring on a plane already carrying a glide produces a plane that carries two glides, which is the case the double-glide letter $e$ was introduced in 1992 to name, and the two letters it replaced are a pair of rows here. In $Imma$ and $Ibca$ the body centring does the same job as the $C$-centring did, relating a mirror or a glide on one plane to a different one on a parallel plane a quarter away. All six are centred types. No primitive type appears, as it could not: without a centring there is no second plane for a name to fall on.

Rows and descriptions, as a matter of what the question was #

The strongest thing to say is what this does and does not fault. The Tables are not wrong, and the derivation is not a correction of them. They list symbols, since a symbol is what a crystallographer has in hand, and any symbol a person could reasonably write should appear. The earlier essays said “the strongest available check on the symbol derivation” for the wrong quantity. The thing the operations can test directly is how many descriptions there are. The symbols are a further layer, and to reproduce all 530 one would have to add the rule that an origin may be stated in more than one way, which is exactly the 31.

So the two results together make a useful test of the whole structure. Five of seven systems match exactly with no adjustment. Two have an excess whose every row has been traced to a specific pair of symbols with the same operations. Nothing is left over in either direction. The one place where a choice had to be imported is $P4_2/ncm$, and that is stated as an exception.

The orthorhombic counts in particular give an internal check on the normaliser calculations of one cell reaches every Wyckoff set, because the same list of shifts is used there for an entirely different purpose. If the shifts were wrong, Wyckoff sets and the settings would be wrong together, and these two comparisons with printed tables are independent.

There is also a historical point, of modest interest. Hall’s 1981 system of symbols, which this list was organised around, gave each row a distinct Hall symbol built from the generators, and on that level of description several of these pairs have different symbols because the generators are written at different origins. A group with an origin shift has a different Hall symbol, by design. So Hall’s list is exactly what would be expected if the unit of counting is a description with a stated origin, and the 530 is a count of those. The count here takes the shift out. Both are right, and they answer different questions.

What the count does not settle #

Only the number of settings has been reproduced, not the symbols. This derivation counts cosets and does not spell the names of the rows. Six ways to name one group did derive symbols for a sample. Producing every one of the 530 strings from the operations would require the rule for which position of a symbol reports which plane, on every setting, which is a large and separate job.

The comparison rests on an external list read rather than derived, and on the row counts as printed. It is one library’s reading of the printed list, compared here type by type only where the types could be identified. The count of 499 does not depend on that list, since it comes from operations, but the 31 does.

The origin rule uses site order alone. A site of order four with an inversion and a site of order four without are treated as a tie, though a person might value one kind of site over the other, and that is presumably why the $P4_2/ncm$ choice went as it did. No attempt is made to reconstruct their reasoning.

Finally, a shift on a grid of twenty-fourths cannot express every possible origin. For the shifts that join descriptions of space groups that grid is known to be fine enough, since every normaliser’s translations in the Tables have denominators that divide it. It is not derived here as a theorem.

What the count had to refuse #

The refusals exist to keep the two numbers apart. The printed row counts are held as data and never used to produce the count, so a comparison between them cannot be circular. The computed total is not allowed to pass as 530. A swap of axes on an $Amm2$ type must fail, because a test that allowed everything would give every type one setting, and the census would then read 230.

Still open: the normaliser in the other direction #

Every setting here starts from one chosen description and asks what the normaliser holds. The converse question has not been asked. Given two of the 499 descriptions, each a set of operations, what is the smallest integer change of basis and shift that carries one to the other? In the orthorhombic and monoclinic cases those are the entries of a transformation table, and the Tables print them as the transformation from the standard setting to each listed one. Whether the printed matrix is the shortest available, in a sense such as the largest entry in its integer change of basis, or whether a smaller one exists that nobody needed to look for, is a clean and finite question, and one matrix, four rules gives everything needed to apply the answer. A short transformation matters in practice, since it is what lets a coordinate list be moved between settings with the fewest fractions. It has not been computed here.