Two hundred and nineteen over seventy-three

Finitely many is not few divided the number of crystallographic groups in each dimension by the number of arithmetic classes and found a ratio that climbs: 1.3 in the plane, about three in space, nearly seven in four dimensions and 339 in six. It then pointed out that a ratio is an average, and that two very different shapes produce the same average. The groups could sit in a few enormous classes with most classes carrying one or two, or the typical class itself could carry many. The essay could not tell which, because its space census covered six of the seventy-three classes, chosen for being enumerable rather than at random, and it said so.

The distribution is now computable for all seventy-three. Seventy-three Smith forms gave the cohomology group of every arithmetic class of space with an explicit representative for each of its elements. What turns cohomology classes into space groups is the third step of the classification: two assignments of translations are the same group when a change of basis that carries the point group to itself carries one onto the other. That is an orbit count. Run on every class, it gives 219 space-group types, and 230 when only changes of basis that keep the handedness are allowed. The Tables’ two famous numbers are reproduced from integer matrices and nothing else, class by class, and the distribution beneath them can be read off.

The last step, carried out #

Each arithmetic class arrives from the fourteen Bravais groups as a list of integer matrices on a primitive basis of its lattice. The changes of basis that could relabel its space groups are integer matrices of determinant $\pm1$ that carry the point group to itself: its normaliser in $GL(3, \mathbb{Z})$. The normaliser of an arithmetic class lies inside the normaliser of its Bravais group, because the Bravais group is determined by the class. The census uses every integer matrix with entries $-1$, $0$ or $1$ that normalises the Bravais group, keeping those that also normalise the class.

A normaliser element $N$ acts on a cohomology class by moving each translation with it: the new class attaches to $g$ the translation $N\,t(N^{-1} g N)$. The census applies every kept element to every class’s representative and joins classes into orbits by the class key the Smith census already uses. That key is unchanged by any change of origin, so it compares classes and not assignments. An orbit is a space-group type. With every kept element the orbits are the 219 affine types; with only those of determinant one, the orientation-preserving types, of which there are 230.

That the finite set of normaliser elements is enough is not proved here. The normaliser of a monoclinic or triclinic class is infinite, and a finite set could in principle miss an identification and report two types where there is one. It is tested instead, and the test is severe: a missed identification would make a total too large, an extra one would make it too small, and the totals are exactly 219 and 230. The classes checked against the Tables one at a time agree as well. Pmmm and P4/mmm carry sixteen groups each, Pmm2 ten, P4mm eight, and Amm2 four against Cmm2’s three. P422 carries six up to any change of basis and eight when the hands are kept apart, and Pm3̄m carries four, from Pm3̄m to Pn3̄m.

Twelve ones, thirty-five twos, and a short tail #

The picture at the head of this essay answers the question the earlier essay posed. In space the typical arithmetic class carries two groups. Twelve classes carry one, thirty-five carry two, five carry three and fifteen carry four. After that the tail is short: two classes with six, one with eight, one with ten and two with sixteen. The mean is three, the median is two, and no class carries more than sixteen.

So the plane’s shape has not simply been stretched. In the plane ten of the thirteen classes carry a single group, the symmorphic one, and being non-symmorphic is the exception, owned by three classes with a mirror. In space the same statement fails outright: sixty-one of the seventy-three classes carry at least one non-symmorphic group, and only twelve carry nothing but their symmorphic group. The tail has thickened, not lengthened. The largest class carries sixteen groups, only four times the median, where a picture of a few enormous classes would put the maximum at many times the median.

The sorted list shows where the groups live. The top of it is orthorhombic and tetragonal: Pmmm and P4/mmm with sixteen, Pmm2 with ten, P4mm with eight, Cmmm and P422 with six. Those are the classes with several independent mirrors or two-fold axes on a primitive lattice, each of which may or may not carry half a cell of glide or screw. The bottom of the list is the classes whose cohomology is trivial. Some have a point group with no fixed direction, such as P1̄. Some have a lattice whose centring already supplies every half-cell an operation could want, such as F222 and F23. The rest have a single operation with nothing of its own to slide along, such as P4̄ and R3̄. Between them sit the cubic classes. Their three-fold axes tie the three cell directions together, so most carry two and none more than four, although their point groups are the largest. The three cubic holohedries show it plainly. Pm3̄m carries four groups, Pm3̄m, Pn3̄n, Pm3̄n and Pn3̄m; Im3̄m carries two, Im3̄m and Ia3̄d; Fm3̄m carries four, Fm3̄m, Fm3̄c, Fd3̄m and Fd3̄c. The orthorhombic holohedry on a primitive lattice, with a point group a sixth the size, carries sixteen, because nothing there forces its three mirrors to make the same choice.

No handful of classes holds the list #

The concentration curve makes the same point quantitatively. The five classes carrying the most groups hold 26 per cent of them, the ten largest 36 per cent, and it takes twenty classes to reach half. If the picture of a few enormous classes were right, the curve would rise almost vertically and then flatten. It bends, but it stays far from vertical: the curve with every class carrying exactly three is the dashed diagonal, and the real curve never gets more than about twenty-five points above it. The last twelve classes, each carrying one group, hold under six per cent of the list.

For the space groups, then, the average describes the typical class, which is what a reader would hope an average does. Whether that stays true in higher dimensions is the question the earlier essay cared about, and one dimension of evidence cannot settle it. What space shows is the direction the plane pointed away from: from two dimensions to three, the median moved from one group to two, while the maximum moved from three to sixteen.

The third step is not small #

The earlier essay called the classification’s last step a subtraction that barely subtracts, and in the plane it is. Eighteen extension classes become seventeen groups, with one merge in the rectangular 2mm class where swapping the axes identifies two arrangements of glides. It guessed that one dimension up the merges would stay a detail, eleven more at most, the eleven enantiomorphic pairs of two hundred and thirty or two hundred and nineteen.

It is not a detail in space. The seventy-three classes carry 303 cohomology classes between them, and those become 219 groups, so the quotient removes eighty-four, more than a quarter of the list. Even keeping the hands apart it removes seventy-three. Almost all of that is the orthorhombic and tetragonal classes, where a permutation of the axes carries one arrangement of glides onto another. Pmmm alone has sixty-four cohomology classes and sixteen groups, and its orbits have sizes six, three, two and one, because the normaliser permutes the three axes and most choices of which mirror glides along which direction have five others that differ only by a relabelling. Pmm2 has sixteen cohomology classes and ten groups. The classes on the diagonal of the figure are the ones where no relabelling identifies anything. They are most of the list, and they are the classes with small cohomology. Of the 219 groups, 73 are the symmorphic ones, one to a class, and 146 are not. Two thirds of the space groups are non-symmorphic, against four of the seventeen plane groups, and the median class carries exactly one non-symmorphic group beside its symmorphic one.

That reverses one sentence of the earlier essay. There the second step, attaching translations, was the multiplication that makes the list long, and the third a correction that changes little. In space the second step multiplies the seventy-three classes into 303 extension classes, a factor of about four, and the third divides by about 1.4. The multiplication still dominates, but the division is no longer negligible, and the earlier essay’s guess that the four-dimensional count would come out “a little above” what the extension classes give is not safe on this evidence.

The sixteen of Pmmm, by their orbits #

The orbit sizes of the largest class are worth reading one at a time, because each says how much of a group’s symbol is a choice of axes. Pmmm’s cohomology is $(\mathbb{Z}_2)^6$: each of its three mirrors may glide by half a cell along either of the two directions lying in its plane, or along both, which is a diagonal glide. The normaliser that matters is the group of permutations of the three axes, of order six. So an arrangement of glides lies in an orbit of size six, three, two or one according to how much of that permutation group leaves it alone.

Two orbits have size one. One is Pmmm itself, with no glides at all. The other has every mirror gliding diagonally: Pnnn, whose three $n$-glides are indistinguishable under any relabelling. These are the two arrangements that treat the three axes identically.

Six orbits have size three. Each fixes one axis as special and is carried onto itself by exchanging the other two. Pban, with glides perpendicular to $c$ and $a$ that bring the two in-plane directions together, is one; Pnnm and Pccm, with two identical glides and one plain mirror, are others. Their standard symbols record which axis the Tables chose as the special one, and the other two choices are the same group with relabelled axes.

Seven orbits have size six. These arrangements single out all three axes. The symbol Pnma is one of them: an $n$-glide perpendicular to $a$, a mirror perpendicular to $b$ and an $a$-glide perpendicular to $c$. Any of the six relabellings of the axes gives a symbol such as Pbnm or Pmcn, and all six describe one group. That is why Pnma is quoted in six settings in the structural literature, and why a census that counted symbols rather than orbits would find six times as many.

One orbit has size two, and it is the most interesting. Its arrangement is cyclic: the mirror perpendicular to $a$ glides along $c$, the one perpendicular to $b$ glides along $a$, and the one perpendicular to $c$ glides along $b$. A cyclic permutation of the axes carries this arrangement to itself, and an exchange of two axes carries it to the same cycle run backwards. The two cyclic orders are the two settings, Pbca and Pcab, and no permutation merges them into one. They are nevertheless one space-group type, because the exchange of two axes is itself in the normaliser and identifies them.

So the sixteen are $1 + 1 + 6 + 7 + 1$ orbits carrying $1 + 1 + 18 + 42 + 2 = 64$ cohomology classes. The Tables’ list of the sixteen orthorhombic P holohedry groups is exactly this orbit decomposition, each group given in one standard setting chosen from its orbit.

Where 219 becomes 230 #

The census also locates the difference between the two counts. Restricting the normaliser to determinant one splits an orbit exactly when the only elements that identified two of its classes reverse orientation. That happens in eight classes, and every one of them has a screw available whose two hands are different: P3, P321, P312, P4, P422, P432, P6 and P622. The extra groups are one each in P3, P321, P312, P4 and P432, and two each in P422 and P6 and P622, which is eleven. They are the pairs 3₁ and 3₂, 4₁ and 4₃, 6₁ and 6₅, 6₂ and 6₄ with their decorations, and they are exactly the groups whose cohomology class has order three, four or six. A class of order two is its own negative and has no mirror partner to be split from. That is why no orthorhombic or cubic class other than P432 contributes, despite carrying most of the list.

What the census has to refuse #

The totals carry most of the weight, and they carry it in both directions. The census could fail by the normaliser set being too small, which would leave two relabellings of one group in separate orbits and push the total above 219. It could fail by the class key confusing two classes that differ, which would pull it below. It could fail by a Smith form reporting the wrong group, which would move a class’s contribution either way. That the plane gives seventeen, space gives 219 with every normaliser element and 230 with the proper ones, and the individual classes checked against the Tables agree, is the evidence that none of these happened. The origin-shift refusal protects the key directly, since a key that changed under a change of origin would count every group as many times as it has origins.

The convention under the whole census is the one that separates 219 from 230, and it should be named rather than left for the reader to infer. An affine space-group type allows any change of basis, including one that reverses orientation, and so identifies a crystal with its mirror image. The proper type does not. Neither is more correct. The first is the classification of groups, and the second the classification of groups together with a handedness that a chiral crystal actually has.

Still open: whether four dimensions follows #

The distribution in space answers the earlier essay’s question for one dimension. In four dimensions there are 710 arithmetic classes and 4,783 groups, a mean of about 6.7. Whether the median follows the mean upward, as it did from the plane to space, or stays small while a tail of large classes carries the list, is the next measurement. The method runs unchanged in any dimension: the cohomology is a Smith form of the Cayley graph’s conditions, and the types are orbits of normaliser elements on explicit representatives. What stops it here is the list of 710 classes, which is not derived here. Deriving it would need the four-dimensional Bravais groups, and the construction that produced the seventy-three from fourteen does not obviously scale to the four-dimensional lattice types.

There is also a question inside space. The normaliser elements with entries in $-1$, $0$ and $1$ were enough to reach both totals exactly, which is evidence rather than proof that they generate the whole action on the cohomology in every class. A proof would need either the full normaliser of each class, which is infinite for the monoclinic and triclinic ones, or an argument that its action on the cohomology factors through a finite group that these elements already generate. For the triclinic classes that argument is short, since P1 and P1̄ have trivial cohomology and nothing to act on. For the monoclinic classes it is the one place where the census is checked only by its totals.