Twenty-five cells, and fourteen lattices

Seven crystal systems, four things that can be done to a cell — leave it alone, centre one face, centre the body, centre all the faces — and the answer is fourteen rather than twenty-eight.

Every textbook says so and almost none of them says why. The reason is that “why” needs a computation, and the computation is short.

It is worth being clear at the outset what kind of loss the missing fourteen represent, because there are two kinds and they get conflated. Some candidates produce a lattice that is not of the system it was built in — centring has broken the very symmetry the system is defined by, and what is left belongs somewhere else. Others produce a lattice that is perfectly good and has already been counted, under a different letter or in a badly chosen cell. Three of one, eight of the other, and the arguments for the two are completely different in character: the first is a computation of what survives, the second is an exhibited change of basis.

The question #

A lattice belongs to a crystal system when it has that system’s full point symmetry. Cubic means forty-eight symmetries; tetragonal, sixteen; orthorhombic, eight; monoclinic, four; triclinic, two; hexagonal, twenty-four; rhombohedral, twelve.

So the question to ask of a centred cell is: does the lattice it describes still have those symmetries?

That is decidable and it is the same computation the round trip uses. Write the primitive lattice’s metric — the matrix of dot products of a primitive basis — and search for every integer matrix that preserves it. The search is a bounded enumeration of lattice vectors with the right lengths, and it is complete.

The three that break #

Three candidates lose symmetry outright, and one of the three is worth the whole essay.

Cubic C. Centre one face of a cube. The threefold axes along the body diagonals are gone, because they permute the three axes and the centred face distinguishes one of them. The resulting lattice has sixteen automorphisms, not forty-eight — and sixteen is the tetragonal number, so the lattice is tetragonal, and tetragonal lattices are already on the list.

That is the whole of “there is no C-centred cubic lattice”, and it arrives as a number rather than as an assertion. Not nobody has found one; not a matter of convention; the lattice exists, it just is not cubic.

Hexagonal C and hexagonal I. Centre the ab face of a hexagonal cell and the threefold axis goes, for the same reason. Both come out at eight automorphisms, which is orthorhombic.

The eight that are already on the list #

The larger group of casualties is subtler: eight candidates keep the symmetry and describe a lattice that has already been counted, under another letter or in another cell.

A claim of that kind is only worth the transformation attached to it, so each of the eight comes with a change of cell that is written down and then checked. Three conditions have to hold: the new metric must have the shape the target system requires; every vector of the old lattice must be a vector of the new one; and the new centring must actually be reached, so that the new description is not a smaller lattice wearing the right letter.

The eight fall into four groups.

All three centrings of the triclinic cell. Triclinic imposes no shape condition at all — any three vectors form a triclinic cell — so the primitive cell of any centred triclinic lattice is itself a conventional triclinic cell, and the letters C, I-centring and F-centring are all describing the same lattice with a cell two or four times too big.

Two operations is the least any lattice can have, and it is worth being clear why the number is two rather than one. Every lattice contains −v whenever it contains v, because a lattice is closed under subtraction and −v = 0 − v. So the inversion is a symmetry of every lattice in every system, without exception and without any condition on the cell — which is the reason all thirty-two point groups sit inside eleven Laue classes, and the reason a triclinic bar has nowhere further to fall.

Monoclinic I-centring and F-centring. Both are monoclinic C in another cell. The transformation for the body centring is to replace c by a + c, which keeps b unique and moves the body centre onto a face; for F it is to halve the cell along a + c.

Tetragonal C and F. Both are the 45° turn: C becomes P, and F becomes I.

Hexagonal R. This one is different in kind and gets its own essay, because R-centring genuinely does lower the symmetry — from twenty-four to twelve — and the lattice it produces is the rhombohedral one, which is on the list under its own axes.

The count #

Twenty-five candidates. Three lose their system’s symmetry. Eight describe a lattice already listed. Fourteen remain.

The count is checked every time the enumeration is drawn, and it checks three numbers rather than one: the total kept, the number destroyed, and the number of duplicates. That matters because the three can compensate — a duplicate misclassified as destroyed would leave the total at fourteen and the reasoning wrong, and a check on the total alone would pass.

The difference between those two kinds of exclusion is not a technicality about how the table is kept. A short bar is a fact about the lattice and a full bar that is nonetheless struck out is a fact about the description. Cubic C is not a cubic lattice, and no cleverness with axes would make it one. Tetragonal C is a tetragonal lattice, in every sense, and the only thing wrong with it is that somebody has drawn a cell twice as large as the one that shows it. The first is a discovery about three-dimensional space; the second is a matter of bookkeeping — and the enumeration has to run both, because either alone gives the wrong number.

What centring actually is #

Underneath all of this is one idea worth stating plainly, because the word “centring” makes it sound like decoration.

A centred lattice is not a cell with extra points added. It is a lattice, and the conventional cell is a description of it that happens to contain more than one lattice point. The centring translations are the coset representatives of the conventional lattice inside the real one, and their number is the index — two for C and I, four for F, three for R.

That is the sublattice reading this site has already given in the plane, and every consequence of centring follows from the index rather than from where the extra points sit. The cell of a body-centred lattice has twice the volume of a primitive cell of the same lattice. Its diffraction pattern loses half its reflections. Its space groups have twice as many operations modulo the conventional cell as the primitive description would give.

The reason to prefer the larger cell, when the lattice permits a primitive one, is that the conventional cell shows the symmetry. A primitive cell of a face-centred cubic lattice is a rhombohedron with 60° angles, and nothing about looking at it suggests forty-eight symmetries; the cube does. The cell is a choice, and this is the choice being made for legibility at the cost of volume.

Why the shapes in the pictures are a lie #

Every drawing of the fourteen, here and everywhere else, is misleading in the same way, and it is worth being explicit.

The angles are the drawing’s, not the crystal’s. A monoclinic cell is drawn with a visible lean so that it does not look orthorhombic, and the amount of lean is chosen for legibility. Real monoclinic cells are often within a degree of ninety, which is a genuine difficulty in real crystallography and looks nothing like the picture.

The relative lengths are the drawing’s too. An orthorhombic cell is drawn with three obviously unequal edges, because the system is defined by not requiring them equal — but a particular orthorhombic crystal may have two edges within a percent of each other, and it is still orthorhombic, because the classification is about symmetry rather than about measurement.

The only real content is which points are there. The corner points and the centring points are the lattice; the box is a drawing aid. The whole classification is about the point set and none of it is about the shape of the box, which is why the automorphism counts underneath the pictures are the part worth reading.

That last point has a consequence in practice. A crystal whose metric is accidentally more symmetric than its structure — the monoclinic cell that measures β = 90.00° — presents a lattice with more symmetry than the crystal has, and the resulting ambiguity is a real and common problem in structure determination. This machinery has nothing to say about it, because deciding whether 90.00° means ninety is a question about measurement error, and this arithmetic has none.

Bravais, and the mistake that came first #

Auguste Bravais published the fourteen in 1850, and the interest is in what he was correcting.

Moritz Frankenheim had published a classification in 1842 with fifteen lattices. It was a serious piece of work by a serious person and the error was of exactly the kind this essay’s machinery is built to catch: he had counted two descriptions of the same lattice as two lattices. The pair he separated were two monoclinic cases — and the transformation identifying them is precisely one of the eight recorded here, the one that replaces c with a + c and turns a body-centred monoclinic cell into a face-centred one.

Bravais found it, and Frankenheim accepted the correction.

The moral is not that fifteen was careless. It is that the difference between fifteen and fourteen is a change of basis that nobody would find by looking at pictures of cells — the two lattices Frankenheim separated look completely different when drawn, and are the same set of points. A classification of this kind cannot be done by inspection, in 1842 or now, and the eight verified transformations in this essay are the modern form of the check Bravais ran by hand.

There is a small irony worth recording. The count is universally called the Bravais lattices, and Frankenheim’s name attaches to nothing — but the thing Bravais did was not the enumeration, which Frankenheim had substantially right, but the deduplication. The half of the work that gets remembered is the half that produced the list, and the half that made it correct was noticing that two entries were one.

The five, and the fourteen #

The plane has five lattices and the same argument produces them, which is worth checking because it is the case where a reader can verify the machinery against something they already believe.

Four systems in the plane — oblique, rectangular, square, hexagonal — and one thing that can be done to a cell, which is centre it. Centring an oblique cell gives an oblique lattice, so nothing new. Centring a square cell gives a square lattice turned 45°, so nothing new. Centring a hexagonal cell breaks the threefold, so nothing new and something lost. Centring a rectangular cell gives a lattice that is not rectangular-primitive and is not any of the others: the centred rectangular lattice, the fifth.

So four plus one is five, by exactly the argument that gives fourteen here — and the plane’s one survivor, the centred rectangular lattice, is the reason cm and cmm exist and the reason the plane has seventeen groups rather than thirteen.

The two counts have the same shape and the ratio between them is not obvious. Five to fourteen is not five times anything; it is the outcome of a longer version of the same enumeration, with more systems and more centrings and more coincidences to remove.

What the fourteen are for #

The lattices are not the classification and it is worth saying what they buy, because a list of fourteen boxes looks like an end in itself.

Each Bravais lattice is one factor in the pairing that produces the seventy-three arithmetic crystal classes: a point group acting on a lattice, where the pairing is legal only if the point group’s operations preserve the lattice. Fourteen lattices, thirty-two point groups, seventy-three legal pairings — and then the attachment of translations takes seventy-three to two hundred and thirty.

So the fourteen sit two steps below the answer and the whole structure rests on them. Getting the count wrong in either direction changes everything downstream: Frankenheim’s fifteen would have given a different number of arithmetic classes and a different number of space groups.

They also do work no classification needs. The lattice letter is the first character of every space-group symbol, which means the fourteen are compressed into seven letters — P, A, B, C, I, F, R — and the system is recovered from what follows. That compression works only because the enumeration came out at fourteen rather than at twenty-eight; a scheme with a letter per surviving centring per system would need many more characters and the symbols would be unreadable.

Where the argument stops #

Two limits worth stating.

The duplicate claims are positive and the distinctness claims are not. For each of the eight duplicates there is an exhibited, verified transformation. For the fourteen survivors there is no proof here that no transformation relates them — the argument is that their automorphism orders differ, or that they differ in centring index, and for the orthorhombic triple C, I-centred and F-centred those are genuinely different lattices for reasons this essay states rather than computes. An argument from the absence of a transformation would be an argument from a failed search, and the searches here are not run in that direction.

The seven systems are taken as given. This essay counts centrings within systems and does not derive the systems themselves. That derivation — why there are exactly seven possible holohedries in three dimensions, and not six or eight — is the same kind of argument as the crystallographic restriction and is a level below what is computed here. The seven orders this essay uses as targets (2, 4, 8, 12, 16, 24, 48) are stated rather than derived.

Fourteen is a count of lattice types, not of lattices. There are infinitely many monoclinic P lattices, one for each choice of three lengths and an angle, and they are all the same type. What is being counted is the ways the symmetry can be arranged, and a type is a class rather than an object. That is the same sense in which there are seventeen wallpaper groups and infinitely many wallpapers.

The two rungs above this one take the two most interesting rows of the table: why centring a cube destroys it, in detail, and the rhombohedral case, which is the only one where a centring produces a lattice of a different system that is nonetheless new.

Why the list of centrings has four entries #

The census takes “leave it alone, centre one face, centre the body, centre all the faces” as given, and the list deserves a derivation of its own, because it is not a list of things somebody thought of trying. It is a complete classification of a small finite group’s subgroups.

A centred lattice contains the conventional cell’s own lattice. Write $L_0$ for the lattice generated by the three cell vectors and $L$ for the actual lattice, which has $L_0$ inside it together with the centring points. The centring information is the finite quotient $L/L_0$ — the extra points, counted modulo the cell.

For the standard centrings that quotient consists of half-integer combinations, so it sits inside the group of the eight vectors $\tfrac12(\varepsilon_1\mathbf{a} + \varepsilon_2\mathbf{b} + \varepsilon_3\mathbf{c})$ with each $\varepsilon$ zero or one. That group is three binary digits — the same group of order eight the frieze classification runs inside — and the possible centrings are its subgroups.

Its seven non-zero elements split three ways. Three of them, $\tfrac12\mathbf{a}$ and its two companions, halve one axis and nothing else: the result is a lattice of the same kind with a shorter axis, so it is not a new lattice at all. Three more, $\tfrac12(\mathbf{a}+\mathbf{b})$ and its companions, centre one face — the A, B and C centrings, which are one centring under a relabelling of axes. The last, $\tfrac12(\mathbf{a}+\mathbf{b}+\mathbf{c})$, centres the body.

The subgroups of order four behave the same way. One of them contains all three face-centring vectors and no axis-halving vector: that is F, all faces centred. Every other order-four subgroup contains an axis-halving vector and therefore describes a rescaled cell rather than a new lattice. The whole group of order eight is every half-integer point, which is the primitive lattice at half the scale.

So four survive: nothing, one face, the body, all faces. Not because four things were tried, but because a group of order eight has exactly that many subgroups whose elements do not simply shorten an axis.

The centring that uses thirds #

The derivation above assumed half-integers, and one entry in the fourteen does not obey it — which is why R-centring keeps being described as different in kind rather than as a fifth column of the table.

R-centring adds points at $(\tfrac23, \tfrac13, \tfrac13)$ and $(\tfrac13, \tfrac23, \tfrac23)$. The quotient $L/L_0$ has order three, not two or four, so it is nowhere in the group of half-integers and could not have appeared in the census above however carefully it was run.

Order three is available for a specific reason. The centring points must be permuted among themselves by every operation the lattice keeps, and a threefold axis permutes a set of three. In a system whose highest rotation is twofold or fourfold there is nothing to permute three points, so an order-three quotient could not be symmetric; in the hexagonal system there is, and the two extra points sit on successive thirds of the body diagonal precisely so that the threefold rotation cycles them with the origin.

It is also the one centring that genuinely lowers the symmetry, from the hexagonal lattice’s twenty-four operations to twelve, which is what makes rhombohedral a lattice rather than a re-described hexagonal one. Every other entry in the table either keeps the system’s full symmetry or turns out to be another cell for a lattice already listed.

The general rule the two sections give together is that a centring is an index-$n$ supergroup of the conventional lattice which the system’s point group carries onto itself, and the arithmetic of which $n$ are available is decided by the rotations present. Two and four are available everywhere; three is available only where a threefold axis is.