Seven friezes

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.

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.

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.

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 nine are worth separating from the eight, because the two numbers are easy to run together and this page did for a while. Eight candidates land on p2mm — every subset containing two of {H, V, R}, and every subset of size three or four. Nine candidates collapse, which is the eight that land on p2mm minus the one of them that got there first, plus HG landing on p11m and RG landing on p2mg. Sixteen candidates, seven groups, nine collapses; eight of the sixteen ending at the same place is a different count and a smaller one.

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.

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 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.

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.

Where the exactness stops #

Everything above is exact, and none of it applies to a real border without a decision being made first.

A frieze group is defined for a pattern that is infinite in one direction. A band round a vase is not: it closes on itself after a finite number of repeats, and its symmetry group is finite — a cyclic or dihedral group acting on a circle, not a frieze group at all. The seven are recovered only by agreeing to treat the band as a sample of an infinite strip, and that agreement is a modelling choice made before any classification happens rather than a consequence of one.

The choice has a cost that is easy to state. A finite band of $N$ repeats admits translations by $1, 2, \ldots, N-1$ steps if it is read as a circle, and by none of them if it is read as a segment. Neither reading is wrong; they answer different questions. The frieze reading is the useful one because it is the one that survives cutting the pot, and that is a claim about which description is stable under the accidents of preservation rather than about which is true.

The second condition is placement tolerance. The three forcing rules above compose operations exactly: two perpendicular mirrors give a half turn about their crossing point, and if the mirrors miss each other by a millimetre they give a glide instead. A hand-painted border misses by more than a millimetre. So an archaeologist applying the classification is deciding, for every pattern, how much slippage counts as intent — and the seven are sharp while the assignment of a real pot to one of them is not.

This is worth saying plainly because it is the opposite of the situation on the rest of this site. Where a pattern is generated, the group is decidable and no tolerance appears anywhere. Where a pattern is measured, a tolerance is unavoidable, and the classification’s exactness says nothing about the reliability of the measurement that feeds it.

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.

One dimension further down #

The strip is the plane’s classification with one translation instead of two, and running the same question one step further down gives an answer short enough to state in a sentence — and a useful check on the method.

A pattern on a line, living in one dimension, has translations along it and exactly one other candidate operation: a point reflection, which reverses the line about some point. There is no strip to be flipped over and no direction across which to reflect, so the four extras of this essay reduce to one.

So there are two. Translations alone, and translations together with a point reflection — the one-dimensional line groups, and no others.

And the forcing rule survives. A point reflection composed with the translation is another point reflection, half a repeat along, exactly as the horizontal mirror composed with the translation gives a glide here. The mechanism is the same; there is simply less to force.

Two, seven, seventeen, two hundred and thirty. The jumps are not a matter of degree, and this essay is the one place where the whole argument fits on a page.