Crystallography & Symmetry
patterns derived from their groups
The classification

Seven friezes

The same classification argument on a strip instead of a plane, where it is short enough to check by hand. Seven ways to repeat a motif along a line, with names like hop, step and sidle.
13 min read 7 figures Exactly this manyGenerated, not drawn

A pattern that repeats along a line and not across it is a frieze — a border, a moulding, a row of footprints, the decorative band round a Greek vase. There are exactly seven kinds, and the argument that there is no eighth is short enough to follow in one sitting.

The seven frieze groupsEvery way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.p1hopp11gstepp1m1sidlep2spinning hopp2mgspinning sidlep11mjumpp2mmspinning jumpsolid line: a mirror · dashed: a glide · lens: a half-turn centreseven, and no eighth
Fig. 1 All seven, each generated from its own operations. Reading down, the extras accumulate — nothing, a glide, a vertical mirror, a half turn, a half turn with a vertical mirror and a glide, a horizontal mirror, and finally everything at once.

This is the wallpaper classification in miniature, with one translation direction instead of two and one lattice instead of five, and it is the right place to watch the argument work.

What a frieze is

Formally: a pattern whose group of translations is generated by a single vector. Slide by that vector or any whole-number multiple of it and the pattern is unchanged; slide by anything else and it is not.

That single direction is a severe constraint, and it does most of the classifying. Any other symmetry must map the translation direction onto itself, because it maps translations to translations and there is only one direction of translation available.

So a rotation must send the direction to itself or to its reverse. A rotation by anything other than a half turn would tilt it into a direction the pattern does not have, so the only rotation available is a half turn. That single observation eliminates most of the possibilities before the enumeration begins.

The four extras

With the translation fixed, four further operations are candidates.

A half turn, about a point on the strip’s centre line.

A vertical mirror, running across the strip.

A horizontal mirror, running along it.

A glide, reflecting across the centre line and sliding along it.

Four candidates, so sixteen subsets — and seven survive. The collapse from sixteen to seven is the whole content of the classification, and it happens by forcing.

How the forcing works

Symmetries compose, and the compositions are not optional. Three rules do the work.

A horizontal mirror and a vertical mirror force a half turn. Reflecting across two perpendicular lines is a half turn about their crossing point. So any candidate containing both mirrors and no half turn is inconsistent, and collapses into the one that has all three.

A horizontal mirror forces a glide. Composing the horizontal mirror with the translation along the strip gives a reflection-plus-slide, which is a glide. So the horizontal mirror never appears alone: it always brings a glide with it, and the candidate that has a horizontal mirror and no glide does not exist.

A half turn and a vertical mirror force either a horizontal mirror or a glide, depending on whether the twofold centre sits on the mirror line or halfway between two of them. Two positions, two outcomes, and this is the one place in the frieze classification where placement rather than presence decides the answer.

Why sevenThe ingredients available on a strip are a translation, two kinds of mirror, a half-turn and a glide. Combining them produces seven distinct groups, because several combinations generate operations they were not given and collapse onto each other.what is addedgivesnothing but translationp1a glidep11ga vertical mirrorp1m1a half-turnp2a horizontal mirrorp11mbrings a glide with itvertical mirror + half-turnp2mgbrings a glide with itboth mirrorsp2mmbrings a half-turn with itthe collapses are why the count is seven rather than larger7
Fig. 2 The enumeration, run rather than recalled. Each row is a combination of extras, and each is either consistent — giving one of the seven — or forced into a combination already listed.

Apply those three rules to the sixteen subsets and nine of them collapse. Seven remain.

The sixteen candidates, one by one

Since the argument is small enough to run in full, it is worth running.

Label the four extras H (horizontal mirror), V (vertical mirror), R (half turn) and G (glide). Sixteen subsets exist; here is what happens to each.

The empty set gives p1. G alone gives p11g. V alone gives p1m1. R alone gives p2.

H alone is impossible: the horizontal mirror composed with the translation gives a glide, so H forces G, and the set collapses to HG — which is p11m.

HG is p11m. HV forces R by composition, and R with H forces G, so HV collapses all the way to HVRG — p2mm. HR forces V for the same reason, so it also collapses to p2mm. VG gives either p2mg or p2mm depending on placement, and the case where it gives p2mg is the one where the glide axis and the mirror do not intersect at a twofold centre.

VR gives p2mg or p2mm, again by placement. RG forces V, giving p2mg. And every subset of size three or four either already contains a forced consequence or collapses into p2mm.

The frieze group p2mmEvery way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.p2mmspinning jumpsolid line: a mirror · dashed: a glide · lens: a half-turn centre
Fig. 3 The spinning jump, which is where nine of the sixteen candidates end up. Any combination that forces both mirrors and the half turn collapses here, which is why the count falls so far below sixteen.

Seven distinct outcomes: p1, p11g, p1m1, p2, p2mg, p11m, p2mm. Nine of the sixteen collapse, and the collapses are all instances of the three composition rules above.

The seven, with their nicknames

Crystallographic notation for friezes follows the same positional logic as the wallpaper symbols, and it is opaque enough that the informal names are genuinely useful. John Conway supplied a set based on how a person would walk the pattern out.

p1 — hop. Translation only. Step forward, step forward, step forward.

p11g — step. A glide. Left foot, right foot: a trail of footprints, whose only symmetry is reflect-and-slide.

p1m1 — sidle. A vertical mirror. Facing a wall and shuffling sideways, so each step mirrors the last.

p2 — spinning hop. A half turn, and nothing else.

p2mg — spinning sidle. A half turn, a vertical mirror, and the glide the two of them force.

p11m — jump. A horizontal mirror, and the glide it brings with it. The pattern and its reflection in the centre line coincide.

p2mm — spinning jump. Everything at once: both mirrors, the half turn, the glide.

The frieze group p11gEvery way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.p11gstepsolid line: a mirror · dashed: a glide · lens: a half-turn centre
Fig. 4 The step frieze, whose only symmetry beyond translation is a glide. There is no mirror here, though the alternating reversal makes it look as if there ought to be — which is the same trap that makes pg and pgg the most-mislabelled wallpaper groups.

Why the count is not sixteen and not four

Two miscounts are common, and each is instructive about a different part of the argument.

Sixteen comes from treating the four extras as independent switches. They are not, because symmetries compose; the point of calling the collection a group rather than a list is exactly that the switches constrain one another.

Four comes from classifying by point group alone — by which rotations and reflections appear, ignoring where they sit. That merges p2mg with p2mm and p11m with p11g, and it misses the one case where placement decides the answer. It is the frieze version of the mistake that would merge p3m1 with p31m.

Seven is what comes out when composition is respected and placement is counted, and both are needed.

The frieze as a rehearsal

Everything structural about the wallpaper classification appears here in miniature, which makes the frieze case worth doing properly rather than skipping.

The rotation restriction appears, in its simplest form: only a half turn is compatible with a single translation direction. In the plane the same reasoning, applied to two directions, gives the five permitted orders.

Forcing appears, in exactly the form it takes later: a proposed combination generates consequences, and the consequences either fit or collapse the case into another.

Essential glides appear. p11g has a glide and no mirror, and no choice of origin removes the slide. It is the frieze analogue of the four non-symmorphic wallpaper groups, and it is the one most often misidentified.

Placement appears, once, in the p2mg-versus-p2mm distinction. In the plane it appears twice, and among the space groups it does most of the work.

What does not appear is lattice variety. A frieze has one lattice — the integers along a line — and there is nothing to classify. That is why seven is so much smaller than seventeen: five lattice types are missing from the count.

Recognising one in the wild

The classification is most useful as a decision procedure, and for friezes the procedure is short enough to carry in the head.

Is there a half turn? Look for a point about which the whole strip rotates onto itself. If yes, the group is one of p2, p2mg, p2mm.

Is there a vertical mirror? A line across the strip, reflecting left into right. If yes and there is a half turn, the group is p2mg or p2mm; if yes and there is no half turn, it is p1m1.

Is there a horizontal mirror? A line along the strip, reflecting top into bottom. If yes with nothing else, p11m; if yes with everything else, p2mm.

Is there a glide but no horizontal mirror? Then p11g, the trail of footprints.

If none of the above, p1.

The frieze group p1m1Every way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.p1m1sidlesolid line: a mirror · dashed: a glide · lens: a half-turn centre
Fig. 5 The sidle, whose vertical mirrors run across the strip. Distinguishing this from the step — where the reversal comes from a glide rather than a mirror — is the one judgement in the procedure that regularly goes wrong.

The step that fails is the third and fourth together, and the reason is the one that recurs throughout this subject: a glide reverses handedness and so does a mirror, and the eye reads reversal as evidence of a mirror. The test that settles it is whether the pure reflection, with no slide attached, maps the strip onto itself — which is a question to be answered by applying the operation rather than by looking at the picture.

What a frieze does not have

The comparison with wallpaper is sharpest in what is missing, and two absences are worth stating.

There is no lattice classification. A frieze’s translations are generated by one vector, and one vector is one vector; there is no analogue of the choice between oblique, rectangular, square and the rest. Everything the five plane lattices contribute to the wallpaper count is simply absent here.

There is no rotation beyond a half turn. The strip has a distinguished direction, and any rotation must preserve it. In the plane the two independent translation directions permit rotations of order three, four and six as well, and those three extra orders account for eight of the seventeen groups.

Between them, those two absences explain the whole gap from seven to seventeen. Add lattice variety and higher rotations to the frieze argument and the wallpaper argument is what results — the mechanism does not change, only the number of branches.

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.p1pgpmp2pmgpmmcmm7 groups, each generated and verified
Fig. 6 The wallpaper groups whose highest rotation is a half turn — the closest planar relatives of the seven friezes. Everything beyond this row of the classification exists because the plane, unlike a strip, permits threefold, fourfold and sixfold rotation.

Friezes in three dimensions, and the rod groups

The natural generalisation of a frieze to space is a pattern periodic in one direction and bounded in the other two — a helix, a polymer chain, a column.

These are the rod groups, and there are seventy-five of them. The step up from seven is larger than it looks, because space allows screw axes and rotations of every permitted order about the periodic direction, where the plane allowed only a half turn.

Rod groups are the natural language for helical biological structures. A DNA double helix, a microtubule, a bacterial flagellum and an amyloid fibril are all periodic in one direction and finite across, and fibre diffraction — the technique that produced Rosalind Franklin’s Photograph 51 — is the experimental method that reads them. The cross pattern in that image is the diffraction signature of a helix, and reading it required exactly this classification.

How the seven are checked here

The frieze figures on this site are generated from their operations rather than drawn, and the count is asserted rather than stated.

Each of the seven is specified by which extras it contains. The generator applies them to an asymmetric motif — a comma, never a dot, for the same reason as everywhere else — repeats along the strip, and marks the elements. The plate that draws all seven asserts that there are seven; if the table gained or lost an entry, the figure would throw and the build would stop.

The frieze machinery is deliberately simpler than the wallpaper machinery. A frieze group is small enough to write out completely, and running it through the full integer-matrix detector would be using a hammer on a drawing pin. That is a considered choice rather than an oversight, and it is the honest description of what these particular figures check.

The frieze group p2mgEvery way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.p2mgspinning sidlesolid line: a mirror · dashed: a glide · lens: a half-turn centre
Fig. 7 The spinning sidle, where a half turn and a vertical mirror force a glide between them. This is the frieze classification’s one case in which the position of an element, rather than its presence, decides which group the pattern belongs to.

Who classified them

The frieze groups have no single discoverer, which is unsurprising for a result this small. They fall out of Fedorov’s 1891 derivation as a degenerate case, and they were certainly known earlier to anybody who thought about it carefully.

Their modern prominence is largely pedagogical, and largely down to two things. George Pólya’s 1924 paper on the plane groups made the frieze case the natural warm-up. And the informal names — hop, step, sidle, jump — come from Conway, who has done more than anybody to make this subject nameable.

Archaeologists use the classification directly. The distribution of frieze groups on decorated pottery varies systematically between cultures and periods, and the classification gives a description of a border pattern that survives translation between languages and does not depend on what the motif depicts. A ceramic tradition can be characterised by which of the seven it prefers, which is a genuinely useful thing to be able to say.

Where the ladder goes next

The obvious sequel is the same argument with a second translation direction: the seventeen, where five lattices replace the frieze’s one and the branches multiply accordingly.

The obvious companion is the four motions, since the frieze case is where the glide first becomes unavoidable and where its awkwardness is easiest to see.

What the pictures here cannot show. Each strip is a few repeats of an infinite band, and no drawing shows the absence of a symmetry. That p11g has no mirror is a claim about a completed search; the figure can mark the glide axis and cannot mark the mirrors that are not there.

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. flip and slide — neither motion alone is a symmetryglide The four motions of the plane Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these. first: mirrorthen: mirrorlands on: rotation of order 4closure is what makes it a group, not a listp4m Why it is a group and not a list The symmetries of a pattern cannot be chosen independently. Do two of them in succession and the result is forced to be a third, which is why there is no eighteenth wallpaper for anybody to invent.