Crystallography & Symmetry
patterns derived from their groups
The classification

Reading Hermann–Mauguin

p4g looks like a licence plate and is in fact a set of instructions. Half an hour with the rules turns the seventeen from a list to be memorised into a notation that can be read.
14 min read 7 figures Exactly this manySymmetry is decidable

The symbol p4g contains four pieces of information and looks like it contains none. It is not a name; it is a set of instructions for building the group, and the instructions can be read off left to right.

The wallpaper group p4mA pattern with the symmetry of p4m, generated by applying the group's 8 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.p4msquare lattice · 8 operations per cellelements marked
Fig. 1 The group p4m, with its elements marked. Reading its symbol gives, in order — primitive cell, fourfold rotation, mirror along the first axis, mirror along the diagonal. Every one of those is visible in the drawing, once a reader knows to look for it.

Half an hour spent on the rules converts the seventeen from a list to be memorised into something that can be reconstructed, and the same rules scale unchanged to the two hundred and thirty space groups.

The first character: the cell

The symbol opens with a lower-case letter naming the lattice centring.

p means a primitive cell — one lattice point per cell, counting corners as quarters. Fifteen of the seventeen begin with p.

c means a centred cell — a rectangular cell with an extra lattice point at its centre, so two lattice points per cell. Two groups begin with c: cm and cmm.

That is the whole of the first character, and it is a statement about the description rather than about the pattern. Both cm and cmm could be described with a primitive rhombic cell instead, at the cost of axes that no longer lie along the mirror directions — the trade discussed in the essay on unit cells, which crystallography resolves in favour of clarity every time.

The first number: the rotation

Next comes a digit giving the highest order of rotation present: 1, 2, 3, 4 or 6, and nothing else, by the crystallographic restriction.

A digit of 1 is usually omitted where it would be uninformative — the group written pm is strictly p1m1 — and this abbreviation is the single largest source of confusion in reading the symbols. The full symbol always has the same number of positions; the short symbol drops the ones that say nothing.

So p4 is a primitive cell with fourfold rotation and no reflection at all. p6 is a primitive cell with sixfold rotation and no reflection. p2 is a primitive cell with half turns and no reflection. Three groups, fully described by two characters each.

The wallpaper group p4A pattern with the symmetry of p4, generated by applying the group's 4 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.p4square lattice · 4 operations per cellelements marked
Fig. 2 The group p4, whose symbol is complete after two characters. Fourfold and twofold centres, no mirror anywhere, and no glide either — everything the symbol does not mention is absent.

The remaining characters: reflections, positionally

Here is where the notation earns its keep and where readers most often give up.

After the rotation come one or two characters describing reflections, and their position in the symbol says which direction of the lattice they refer to. An m means a mirror perpendicular to that direction; a g means a glide perpendicular to it; a 1 means neither.

The directions are fixed by convention for each lattice type. For a rectangular lattice the first position refers to the xx axis and the second to the yy axis. For a square lattice the first refers to the axes and the second to the diagonals. For a hexagonal lattice the first refers to the axes and the second to the directions bisecting them.

So pmm has mirrors perpendicular to both axes. pmg has a mirror perpendicular to one axis and a glide perpendicular to the other. pgg has glides perpendicular to both, and no mirror. p4m has mirrors along the axes and along the diagonals. p4g has a mirror in one family and a glide in the other.

The wallpaper group pmgA pattern with the symmetry of pmg, generated by applying the group's 4 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.pmgrectangular lattice · 4 operations per cellelements marked
Fig. 3 The group pmg, read from its symbol — primitive cell, half turns, a mirror perpendicular to the first axis and a glide perpendicular to the second. The solid lines and the dashed ones in the drawing correspond exactly to the m and the g.

The two pairs, and why the symbols differ

The positional convention is what makes the classification’s two awkward pairs describable at all.

p3m1 and p31m have the same lattice, the same point group and the same number of operations. The symbols place the m in different positions: p3m1 has mirrors perpendicular to the axes and nothing in the second family, and p31m has nothing perpendicular to the axes and mirrors in the bisecting family. The digits are not decoration — they are the notation stating which of two genuinely different arrangements is meant.

The wallpaper group p3m1A pattern with the symmetry of p3m1, generated by applying the group's 6 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.p3m1hexagonal lattice · 6 operations per cellelements marked
Fig. 4 The group p3m1, whose mirrors pass through the threefold centres. Its partner p31m has mirrors in the other family of directions, and no relabelling of axes converts one into the other.

p4m and p4g are the same story on a square lattice. Both have mirrors and glides available; the difference is which family carries the mirror. Their own essay works through the threefold pair in detail, and the fourfold pair follows the identical logic.

A reader who treats the symbols as arbitrary names finds these pairs inexplicable. A reader who knows the symbols are positional finds them obvious, which is the argument for learning the rules rather than the list.

The seventeen, decoded

With the rules in hand, the whole list can be read rather than recalled. Taking them in order of rotation:

p1 — primitive cell, no rotation, no reflection. Translations only. p2 — primitive cell, half turns, no reflection. pm — mirror perpendicular to the first axis; strictly p1m1. pg — glide perpendicular to the first axis; strictly p1g1. cm — the same as pm on a centred cell, which forces glides between the mirrors. pmm — mirrors perpendicular to both axes. pmg — a mirror one way, a glide the other. pgg — glides both ways, no mirror. cmm — the pmm arrangement on a centred cell. p3 — threefold rotation, no reflection. p3m1 — mirrors perpendicular to the axes. p31m — mirrors in the bisecting directions instead. p4 — fourfold rotation, no reflection. p4m — mirrors along both the axes and the diagonals. p4g — a mirror in one family, a glide in the other. p6 — sixfold rotation, no reflection. p6m — sixfold rotation with mirrors in both families.

The seventeen wallpaper groupsEvery way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.p1p2pmpgcmpmmpmgpggcmmp4p4mp4gp3p3m1p31mp6p6m17 groups, each generated and verified
Fig. 5 All seventeen with their symbols. Read as instructions rather than as names, the list has a visible structure — the groups with no letters after the digit have no reflections at all, the ones with a g are the four that are non-symmorphic, and the two with a leading c are the two on a centred cell.

Reading the list this way also makes the gaps informative. There is no p6g, because a glide compatible with sixfold rotation composes with it to force a mirror, so the arrangement collapses into p6m. There is no p3g1 for the same reason. Every absence in the list is an absence the composition rules produce, not a case somebody forgot.

Reading the four glide groups

The four non-symmorphic groups — pg, pgg, pmg, p4g — are exactly the ones whose symbols contain a g, and the correspondence is not a coincidence.

A g in the symbol means the group contains a glide whose axis is not also a mirror. That is precisely the condition for non-symmorphism: no choice of origin removes the translation part from every operation at once. So the symbol announces, in one character, the structural property that makes a group awkward.

This is worth knowing because those four are the ones most often mislabelled by hand. A pattern in pg presents reversed motifs and the eye reads reversal as evidence of a mirror; the symbol pm is written; and the resulting figure is quietly of the wrong group. The g is the notation warning that the reflection has a slide attached and that checking is required.

Why positional and not descriptive

It is worth asking why crystallography chose a notation this terse, since the alternative — spelling out the operations — would be easier to read cold.

The answer is that the symbol has to be short enough to appear in a table header, on a diagram, and in the running text of a structure paper, thousands of times. Two hundred and thirty space groups, each appearing constantly, in a literature where every paper reports one: economy is not a stylistic preference but a practical requirement.

The second answer is that the positional convention encodes precisely the information a diffraction experiment can measure. The directions in a crystal are what an experiment sees — a reflection is indexed by three integers naming a direction in reciprocal space — so a notation organised by direction maps directly onto the observable. A notation organised by abstract group type, as Schoenflies is, does not.

That is why the two notations survive side by side rather than one displacing the other. Chemists asking what a molecule’s symmetry is want the abstract type. Crystallographers asking which reflections will be absent want the directions. Both questions are legitimate and they want different symbols.

What the symbol does not tell

Two things a group symbol is silent about, and both cause trouble.

It does not describe the pattern. Infinitely many patterns share the group p4m, and knowing the symbol tells a reader what symmetries the pattern has and nothing about what it looks like. The group determines the symmetry; the motif determines the appearance.

It does not fix the origin. The symbol says which operations exist and in which directions, not where the origin sits. Two descriptions of the same structure under different origin conventions differ by a fixed shift in every coordinate, and the International Tables list two origin choices for many space groups precisely because the symbol cannot settle it.

There is also a subtlety about settings. Because the symbol is positional, exchanging which axis is called first can change the symbol without changing the group. In the plane this is mild — pmg and pgm are the same group with the axes swapped, and only one is standard. In three dimensions it is not mild at all, and the same space group appears in the literature under more than one symbol.

Reading a symbol backwards

The notation is most useful when run in reverse: given a pattern, produce the symbol. The procedure is short and it is exactly the decision tree the classification supplies.

Find the highest rotation order. One, two, three, four or six — nothing else is available. This fixes the digit and narrows the candidates to at most five.

Decide whether any reflection is present. Not a reversal of handedness, which glides also produce, but a pure reflection with no slide attached. If there is none, the symbol ends at the digit unless a glide is present.

For each family of directions, decide mirror, glide or neither. The families are fixed by the lattice type. This fills the remaining positions.

Decide whether the cell is centred. A centred cell shows itself as mirrors alternating with glides at half-cell spacing, and it is the last thing to check because it is the easiest to mistake for something else.

Why p4 cannot be drawn with dotsThe same group applied to a single dot and to a motif with no symmetry of its own. The dot's orbit turns out to have more symmetries than the group it was made with, so a figure drawn that way illustrates a different group from the one in its caption.a dot at (1/12, 1/12)8 symmetries per cella motif with no symmetry4 symmetries per cell57 of 121 dot positions give more symmetry than p4both generated with the 4 operations of p4the dot gains 4
Fig. 6 Why the second step is the one that goes wrong. The same group applied to a dot and to an asymmetric motif — and the dot’s pattern has twice as many symmetries as the group used to make it, so a reader identifying its rotation order by eye would take the wrong branch immediately.

Each step is a question about the pattern rather than about the picture, and each can be answered by applying an operation and comparing rather than by judging. That distinction is the difference between a classification a reader can trust and one they merely believe, and it is why every figure here is generated and checked rather than drawn and labelled.

The same rules in three dimensions

The notation scales without modification, which is its real achievement.

A space-group symbol opens with a centring letter — P, C, I, F, R for primitive, base-centred, body-centred, face-centred and rhombohedral — and continues with up to three positions describing symmetry along the three principal directions of the crystal system. The characters are richer: as well as m and g there are 212_1, 313_1, 414_1 and their relatives, denoting screw axes, which are rotations combined with a slide along the axis.

So P2₁/c, one of the commonest space groups in organic crystallography, reads as: primitive cell, a twofold screw axis along bb, and a glide plane perpendicular to bb with its slide along cc. Four pieces of information in six characters, and every one of them checkable against a diffraction pattern.

What pg scattersThe diffraction pattern computed from the atom positions alone. Where a glide plane is present, alternate reflections along a row cancel exactly — and those missing spots are how the glide is identified in an experiment, since the glide itself is never seen.115 present6 absentthe glide's signaturecomputed from the atom positions, not from the grouppg
Fig. 7 Why the notation is checkable. A glide removes alternate reflections along a row of the diffraction pattern, exactly and completely, so the g in a symbol makes a prediction about which spots will be missing — and the missing spots are how the symbol is confirmed in an experiment.

Where the symbols come from

The notation is named for Carl Hermann and Charles Mauguin, who introduced it independently in the late 1920s — Hermann in 1928, Mauguin in 1931 — and it was adopted as standard by the International Tables for X-ray Crystallography in 1935.

What it replaced was Schoenflies notation, which describes abstract group types rather than generator positions. Schoenflies symbols are more compact and remain standard in chemistry and spectroscopy, and they have a specific weakness for this subject: because they name the abstract group, they cannot distinguish the pairs where position is the entire difference. p4m and p4g have the same Schoenflies designation in the point-group sense, and the crystallographic distinction is invisible in it.

That is the reason crystallography went its own way. A notation for crystals has to record where the elements sit, because where they sit is what a diffraction experiment measures.

What this site does with it

Every pattern figure here carries its Hermann–Mauguin symbol, and every symmetry element is drawn in the International Tables marks — lens for twofold, triangle for threefold, square for fourfold, hexagon for sixfold, solid line for a mirror and dashed for a glide.

That is a deliberate choice, and the alternative would have been worse. Inventing a friendlier notation would make this site’s figures unreadable everywhere else, and a reader who learns the marks here meets the same marks in every other source, from a textbook to a structure paper to the Tables themselves.

The marks are placed by computation rather than by hand: the detector finds an operation, reads its determinant to decide handedness, reads its trace to get the rotation order, and locates the element from the operation’s own matrix and translation. So the mark on the picture and the character in the symbol are two readings of the same object, and they cannot disagree.

Where the ladder goes next

The obvious next step is the pair the notation exists to distinguish: p3m1 and p31m, where the position of a mirror is the whole difference between two groups.

The obvious hazard is that a figure can display the wrong symbol without looking wrong, which is the comma problem.

And the confirmation route is diffraction, where a g in a symbol becomes a testable prediction about which reflections vanish.

What the pictures here cannot show. A symbol is a claim about which operations exist and where; a figure displays a pattern. The correspondence between them is checked by comparing the detected operations with the symbol’s, which is arithmetic rather than looking, and no drawing on this page establishes it.

Where this goes next

p1p2pmpgcmpmmpmgpggcmmp4p4mp4gp3p3m1p31mp6p6m17 groups, each generated and verified The seventeen Every way of repeating a pattern across a flat surface, exhibited rather than tabulated. The number is a theorem with a finite proof, and the proof is walkable in an afternoon. p3m1hexagonal lattice · 6 operations per cellelements marked p3m1 and p31m Two groups with the same lattice, the same point group and the same number of operations, differing only in where the mirrors sit. The pair is the clearest evidence that position is as much a part of a symmetry as presence. centred rectangularequal lengths, angle freethe arrows are the basis; the shaded region is one unit cell The cell is a choice, the lattice is not Every lattice has infinitely many unit cells and infinitely many bases, and crystallography picks one by convention. Knowing which convention is in force is the difference between a symbol that means something and a symbol that means nothing.