The holohedry is the ceiling

A crystal is a motif repeated on a lattice, and it has two symmetry groups that are easy to confuse. The lattice has one — the set of rotations and reflections carrying the array of points onto itself, which is its holohedry. The crystal has another, which is the set of operations carrying the whole decorated structure onto itself.

The second is a subgroup of the first, always, and the reason takes one line: an operation that carries the crystal onto itself carries its set of lattice translations onto itself, so it is a symmetry of the lattice.

A crystal never has more point symmetry than the lattice it sits on. That is the containment this essay is about, and almost everything in the classification is downstream of it.

Seven ceilings #

There are seven holohedries because there are seven distinct lattice metrics, and this site derives them rather than listing them: for each system’s metric, search for every integer matrix preserving it. Triclinic gives 2 operations, monoclinic 4, orthorhombic 8, rhombohedral 12, tetragonal 16, hexagonal 24, cubic 48.

Those orders are asserted every time a figure on this page is drawn, and the assertion is not decoration. automorphisms is a bounded search — it enumerates integer vectors of the right squared length, using a bound from Cauchy–Schwarz applied to the metric — and the failure mode of a bounded search is quiet. It returns a group that is correct as far as it went. Requiring the cubic answer to be 48 is the statement that the bound was right.

In point-group symbols the seven ceilings are 1̅, 2/m, mmm, 3̅m, 4/mmm, 6/mmm and m3̅m. The enumeration checks that all seven appear among the thirty-two, which sounds like a formality and is not: it is the only test of the containment the whole method assumes, since the search only ever looks inside two of the seven.

Which system a class belongs to #

Every class sits inside several holohedries. Class 2 is a subgroup of 2/m, of mmm, of 4/mmm, of 6/mmm and of m3̅m. Saying which system it belongs to therefore needs a rule, and the rule is: the smallest holohedry containing it.

“Smallest” is doing real work. Reading the system off the holohedry a class happened to be found in would put half the classes in the cubic system, since the enumeration searches m3̅m first. Reading it off the containment relation without the minimum would put every class in every system it fits into.

With the minimum, the assignment is forced and the seven columns come out with two, three, three, five, seven, seven and five classes in them.

The cubic system is identified by a count #

Six of the seven systems are recognised by their highest-order axis. A six-fold means hexagonal; a four-fold with nothing above it means tetragonal; a three-fold means trigonal; three perpendicular two-folds mean orthorhombic; one two-fold means monoclinic; none means triclinic.

Cubic breaks the pattern, and the way it breaks it is the second place in this field where the obvious test is the wrong one.

A cubic group has no axis higher than four — m3̅m has three four-folds and nothing above them, and 23 has no four-fold at all. What marks the system is having four three-fold axes rather than one, running out along the body diagonals of the cube. Four axes, each carrying two rotations, is eight operations of type 3.

A classifier that asked only for the highest order would file 23 and m3̅ as trigonal, and both are cubic. So the test is a count — eight or more three-folds — and it is checked before the axis-order tests rather than after.

The underlying fact is worth the sentence: cubic symmetry is about the diagonals, not the edges. A cube’s four-folds are the familiar part and they are not what distinguishes it from a square prism. What does is that a cube can be turned about a body diagonal and land on itself, and no prism can.

Merohedry is the ordinary case #

Of the thirty-two classes, seven are holohedries and twenty-five are not. A crystal picked at random has strictly less symmetry than its own lattice, and by a wide margin: the cubic system has five classes and only one of them is m3̅m, so four cubic crystals in five have a lattice with more symmetry than the structure on it.

This is the point-group version of a fact the site has met twice already. The cell is a choice and the lattice is not makes it about descriptions; why the bigger cell wins makes it about conventional settings. Here it is about the structure itself: the scaffold is symmetric and the thing built on it need not be.

And it has an experimental consequence that is the reason the word merohedry exists at all. A crystal whose class is smaller than its holohedry can twin — grow as two orientations related by an operation the lattice has and the crystal does not — and the two orientations share a lattice exactly, so the diffraction patterns superimpose with no splitting. Merohedral twinning is undetectable by looking at the spot positions, and it is one of the standard ways a structure determination goes wrong.

Seven ceilings, fourteen lattices, thirty-two classes #

Three counts, and readers meet them in three different places and are rarely told how they fit. They fit like this.

Seven is the number of distinct lattice metrics — shapes of cell, up to the relations among lengths and angles that a symmetry forces. It is the number of holohedries because the metric determines the holohedry.

Fourteen is the number of distinct lattice types, which is larger because a metric can be centred. A cubic metric supports a primitive lattice, a body-centred one and a face-centred one, and those are three lattices with one holohedry between them: m3̅m is the point symmetry of all three. So the fourteen fall into the seven with two, two, four, one, two, one and three in each.

Thirty-two is the number of point groups, and it is bigger than seven for the entirely different reason that a crystal may use less of its lattice’s symmetry than is on offer.

So the seven is a fact about shapes, the fourteen a fact about centrings, and the thirty-two a fact about motifs — and none of the three is derivable from either of the others.

The cubic system is the clearest place to see that the two effects are independent. There are three cubic Bravais lattices — primitive, body-centred and face-centred — and all three have the same holohedry, m3̅m with its forty-eight operations, because adding lattice points at the cell centre or at the face centres does not change which rotations carry the array onto itself. So centring multiplies the lattices without touching the ceiling. Merohedry does the opposite: it multiplies the classes under a ceiling without touching the lattice. The fourteen exceed the seven by the first, the thirty-two exceed the seven by the second, and the two multiply into the two hundred and thirty.

What the containment does not say #

It does not say the lattice is more symmetric than the crystal. It says it is at least as symmetric. Seven classes are holohedries, and for those the two symmetries coincide.

It does not force the lattice to be the least symmetric one that works. A monoclinic crystal may sit on a lattice whose metric is accidentally orthorhombic, and then the lattice has eight operations and the crystal four. Accidental symmetry is exactly this: a metric relation that holds numerically without being required, and it is the reason the site’s own machinery decides symmetry in fractional coordinates against a stated system rather than by measuring a cell and inferring one.

And it says nothing about the space group. A crystal’s point group is not a subgroup of its space group in general — that is what symmorphic means, and the non-symmorphic groups are exactly the ones where the point group appears only as a quotient. The containment here is between the point group and the lattice’s point group, and both are groups of linear parts with the translations discarded.

The ceiling is what makes the enumeration finite #

There is a practical payoff to the containment that is easy to miss because it is doing its work before the classification starts.

Without it, “enumerate the crystallographic point groups” is a search over the finite subgroups of the orthogonal group — an infinite family, since a rotation through any angle generates one. The restriction cuts that down but does not by itself bound the search: knowing that every operation has order 1, 2, 3, 4 or 6 does not say how many of them can be combined, and the free product of two order-two rotations is infinite unless the angle between their axes is right.

The containment closes it. Every candidate is a subgroup of a group of order 48 or of order 24, and a subgroup of a finite group is found by walking chains. That is the difference between a classification and an argument that one exists, and it is why the whole enumeration fits in a page.

That last observation is worth carrying into the next essay. Every lattice is centrosymmetric, whatever its system — inversion through a lattice point maps the array onto itself for the trivial reason that lattices are closed under negation. So the inversion is in all seven ceilings, and whether a crystal has it is entirely a fact about the motif.

Where this comes from #

The containment is why crystallography settled on seven systems rather than thirty-two, and the history shows the order the ideas arrived in. Bravais had the fourteen lattices in 1850; Hessel and Gadolin had the thirty-two classes independently in 1830 and 1867. The synthesis — that each class sits under exactly one lattice type, and that the two classifications are one classification seen from two sides — is what made the step to the 230 space groups possible for Fedorov, Schoenflies and Barlow in the 1890s.

The modern statement is smaller than the historical route to it. Take the seven metrics; find the integer matrices preserving each; take the subgroups; that is the whole of it, and it fits in a page of code because the containment does the organising.

The word the subject uses, and what it hides #

Holohedry and merohedry are nineteenth-century words — whole-facedness and part-facedness — and they come from the era when a crystal’s symmetry was inferred from the faces it grew rather than from anything inside it.

The etymology records a real observation. A crystal in a merohedral class grows fewer distinct kinds of face than its lattice would permit: a quartz crystal shows the trigonal 32 rather than the hexagonal 6/mmm its lattice metric might suggest, and it was the absent faces that gave the class away long before X-rays. Haüy’s contemporaries were reading point groups off morphology, which works and is treacherous, because growth conditions decide which permitted faces actually appear.

The treachery is worth being explicit about, since it is the same distinction this whole field turns on. The class permits a set of face forms; it does not produce them. A crystal can be grown so that a permitted face never develops, and then the morphology reports a smaller class than the structure has. Symmetry gives an upper bound on what can appear and says nothing about what will.

That asymmetry — permitted is not present — comes back as the central caution of what a property count does not say, and it is the same sentence about a different observable.

The containment in the other direction #

Everything above reads the containment downward: a crystal has at most its lattice’s symmetry. Read upward it says something a structure determination uses constantly.

Knowing the crystal’s point group bounds the lattice from below. A crystal in class 4mm must sit on a lattice whose holohedry contains 4mm, and the smallest such holohedry is 4/mmm, so the lattice is at least tetragonal. Its metric therefore satisfies a = b and all angles right, and those are constraints on the cell parameters that hold exactly rather than approximately.

That is why a structure refinement does not fit six independent lattice parameters when the symmetry is known. It fits one for a cubic crystal, two for a tetragonal or hexagonal one, three for an orthorhombic one, and the reduction is not an approximation or a convenience — a tetragonal crystal whose refined a and b differ is a crystal that has been assigned the wrong class, or one that is twinned, or one whose data are worse than the difference.

The two readings together are what makes the seven systems useful rather than decorative. Downward, they say a crystal cannot exceed its scaffold. Upward, they say a crystal’s own symmetry forces its scaffold’s shape. The first is what stops a classification of point groups being infinite; the second is what turns a symmetry assignment into a set of exact numerical constraints on a measurement.

A closing note on the word system, since it carries more weight than it looks.

A crystal system is not a shape of cell and not a set of lattice parameters. It is a holohedry — a group — and the cell shape follows from it rather than defining it. That order matters because a cell can accidentally have a shape its symmetry does not require: a monoclinic crystal whose β happens to measure ninety degrees has an orthorhombic-looking cell and monoclinic symmetry, and the symmetry is what it is.

Reading the system off the numbers rather than off the group is the standard way that goes wrong, and it is why this site decides symmetry in fractional coordinates against a stated metric rather than by measuring a cell and inferring one.

How many numbers a system leaves free #

The upward reading — that the group bounds the cell — has an exact form worth writing down, because it is the number a refinement actually fits and it is fixed entirely by the system.

A general cell is six numbers: three lengths and three angles. Each system imposes equalities among them, and what is left is:

Those are the parameters a refinement varies, and the constrained ones are not fitted and not reported with uncertainties — they are equal by symmetry, exactly, and a refinement that varied them independently would be fitting a model its own space group forbids.

The table also says how much a symmetry assignment is worth as evidence. Going from triclinic to cubic removes five numbers from the model, and each of them was a number the data had to determine — which is one more reason the observations-to-parameters ratio improves with symmetry from both ends at once.

The lattice that is more symmetric than it has to be #

The containment permits a crystal’s lattice to be more symmetric than its system requires, and that case is worth its own paragraph because it is where a determination most often goes wrong.

A monoclinic crystal has a β angle that is not constrained. Nothing prevents it from being 90.00° to within any measurement, and then the metric is orthorhombic while the structure is monoclinic: the lattice has eight operations, the crystal has four, and the four extra are symmetries of the points and not of what sits on them.

That is not a rare accident. It happens whenever a structure is a slightly distorted version of a higher-symmetry parent, which is most structures with a phase transition above them — and the closer the material is to its transition, the closer the metric is to the higher system.

The consequences are the ones this collection keeps meeting. The crystal can twin on the operations the lattice has and the structure has not, with an obliquity that shrinks as the metric approaches the higher symmetry. Automatic indexing will propose the higher lattice system, because indexing sees the metric. And a determination that accepts the proposal refines in a group with too much symmetry, averaging the structure over an operation it does not have.

The metric is a measurement and the group is not, which is the distinction the whole of this essay’s second reading rests on: the lattice bounds the group from below only as far as the metric is trusted, and how far a metric is from a higher one is a number rather than a verdict.

Where this goes #

The ceiling decides what a crystal can have. The next question is what an experiment can see, and the answer is smaller still: diffraction adds an inversion centre it cannot remove, so the thirty-two classes collapse to eleven Laue classes before a single reflection is indexed. A structure determination therefore starts by ruling out twenty-one of the thirty-two on grounds that have nothing to do with the crystal.