Why sixteen become seven

A frieze is a pattern that repeats along a line. Beyond the repetition there are four further operations available — a half turn, a mirror across the strip, a mirror along it, and a glide — so there are sixteen combinations to consider and exactly seven of them are different groups. The nine that disappear are the interesting ones, and the standard treatment disposes of them in a sentence about compositions being forced.

That sentence is true and it hides two different mechanisms. Ten of the sixteen candidates come back from the closure holding an operation nobody gave them, which is forcing in the ordinary sense. Five of them do something else and stranger: they change the lattice they are measured against, and one group turns out to be another group seen at half the spacing.

Why sixteen, and not some other number #

The count of candidates is a count of subsets, and it deserves a moment because everything downstream depends on the list being complete rather than customary.

A frieze group contains translations along one direction, and every other operation in it must map that direction onto itself — otherwise it would produce a translation in a second direction, and the pattern would be a wallpaper pattern rather than a strip. A rotation that maps a line to itself is a half turn. A reflection either has its axis along the line or across it. A glide reflects across the line and slides along it. There is nothing else: four operations, and any subset of them is a candidate.

Sixteen subsets, then, and the classification is the statement that they give seven groups. Neither number is written into the figure above. The candidate count is 2⁴ computed from the list of extras, and the group count is the size of the set of names that come out.

What closure actually produces #

Composition is not optional. If a pattern has two symmetries it has their composition, so a candidate that is handed a horizontal mirror and a vertical one is also handed their product whether or not anybody listed it.

The three simplest cases are the ones a reader can check by eye. A horizontal mirror and a vertical mirror compose to a half turn about the point where they cross. A horizontal mirror and a half turn compose to a vertical mirror. A vertical mirror and a half turn compose to a horizontal one. Any two of the three produce the third, so the three of them are never independent and the candidate holding two of them is the candidate holding all three.

That accounts for three collapses. The remaining six of the ten acquire operations that are harder to name, because they are the same operations displaced along the strip: a half turn about a centre half a repeat over, or a vertical mirror on an axis half a repeat over. Those are not new kinds of symmetry, and the figure names them with a prime rather than a new letter — a distinction that matters for counting elements and not at all for classifying the group.

The eight classes, and why the algebra is finite #

The closure above is exhaustive rather than clever, and it can be because the object it closes over is tiny.

Fix the translation lattice and every operation of a frieze group is described by three things: whether it preserves or reverses the direction along the strip, whether it preserves or reverses the direction across it, and whether it slides by nothing or by half a repeat. Anything sliding by a whole repeat is a lattice translation and is already accounted for; anything sliding by a third would generate translations the lattice does not have.

Three binary choices give eight classes, and composition is coordinate-wise: the signs multiply and the offsets add modulo one. The offsets add without a sign because reversing a direction sends a half to minus a half, and minus a half and a half are the same class.

So the whole classification is a computation inside a group of order eight — the elementary abelian group of three binary digits — and every question this essay asks is a finite closure in it. That is why the argument can be checked rather than believed, and it is the reason a strip is where the classification should be met first: the plane’s version of the same computation runs in a quotient that is not abelian and is not eight elements long.

The half translation, which is a different kind of event #

Five candidates acquire something that is not a mirror, a rotation or a glide. They acquire a translation by half the repeat, and that is a fact about the lattice rather than about the ornament sitting on it.

A horizontal mirror composed with a glide along the same axis is a pure translation, because both reverse the across-strip direction and the reversals cancel. The glide slides by half a repeat, so what is left is a slide by half a repeat and nothing else. The group therefore contains a translation the original lattice did not have — and a group whose translations are finer than the lattice it was described on is being described on the wrong lattice.

Rescale, and the picture changes its description without changing a single point. The glide slides by half of the old repeat, which is exactly one of the new repeats — so it is the mirror composed with a translation the group already contains, and it is not an extra operation at all.

The two candidates that are the same group #

This is what makes {H} and {H, G} identical rather than merely similar.

A reader looking at the two candidates as lists sees one operation’s difference and expects two groups. A reader looking at them as patterns sees the same pattern twice. The list is misleading because it names operations relative to a lattice that the second candidate does not have; once each group is measured against its own translations, the difference evaporates.

The same move settles four other rows in the same way, and it is the reason the classification cannot be done by counting subsets. Subsets of a fixed set are a finite and obvious thing to enumerate. Groups are not subsets — they are subsets together with the lattice they are described against, and closure can change the second.

What the check checked, and how #

The enumeration behind every figure on this page is the same computation, and it is worth stating what would happen if the renormalisation were left out.

Each of the sixteen subsets is closed under composition in the eight-class group of strip isometries — a sign along the strip, a sign across it, and an offset of nothing or a half — which is small enough to close exhaustively rather than cleverly. Each closed set is then rescaled onto the lattice it actually generates, and named from the elements that survive.

Omit the rescaling and the sixteen candidates give ten groups, not seven. That is the negative test this collection keeps beside the positive one: the same enumeration, with the one step removed, has to produce the wrong answer, because an assertion that cannot fail is not evidence of anything. Ten is a specific and checkable wrong answer, and it is the answer a careful enumeration gets if it treats a group as its list of operations.

There is a second, independent check. The seven names the enumeration arrives at are compared against the seven the plate of friezes draws, which come from a table on the other side of the same file and share no code with the closure. Nothing connects the two but the classification being right.

Nine collapses, and eight rows at one group #

Two numbers on this page look as though they should be the same and are not.

Nine of the sixteen candidates collapse — they name a group that a row above them has already named. Eight of the sixteen land on p2mm, the group with everything at once. The two counts are close and they count different things, and the arithmetic that separates them is worth doing once. Of the eight rows arriving at p2mm, one is the first to get there and the other seven are collapses. The remaining two collapses land elsewhere: one on p11m, which is the {H, G} case above, and one on p2mg. Seven plus one plus one is the nine, and the full tally over the seven groups is 8, 2, 2, 1, 1, 1, 1.

This site’s earlier essay on the friezes ran those numbers together and said p2mm was “where nine of the sixteen candidates end up”. It was a plausible sentence and it was wrong by one, and the error was found by making the figure compute what the caption claimed. The figure now computes both counts and does not appear unless they differ, which is a slightly unusual thing to require and is exactly the shape of the mistake it prevents.

Dropping an ingredient #

The enumeration takes the list of extras as an argument, which makes it possible to ask what each one contributes rather than only what the four give together.

Remove the glide and eight candidates give five groups: p1, p2, p1m1, p11m and p2mm. The two that vanish are p11g and p2mg, the only groups whose defining operation the glide is — which is the concrete version of the observation that a glide is not a composition of the other three. It has to be given, and a classification that forgets it is short by exactly two.

Keep only the vertical mirror and the glide and four candidates give four groups, all of them distinct: p1, p1m1, p11g and p2mg. Nothing collapses at all, because nothing here composes into anything that was already available — the half turn that {V, G} acquires is at a shifted centre, and no other candidate in that sub-list has one.

The contrast between those two runs is the argument for the glide’s independence in a form nobody has to take on trust.

Seven arrangements, four groups #

The collapse from sixteen to seven is a collapse of candidates. There is a second collapse waiting one level further in, and running it says exactly what the seven are a classification of.

Ask which of the seven are the same group abstractly — ignoring where the operations sit, ignoring what they do to the strip, keeping only the multiplication table. The answer is four.

Two of them are infinite cyclic. The group with translations and nothing else is generated by one translation, so it is a copy of the integers. The group whose only extra is a glide is also generated by one element — the glide — because the glide squared is the translation. Every element is a power of it, so that group is a copy of the integers too. The two friezes look nothing alike and are indistinguishable as groups.

One is the integers with a two-element factor. The horizontal mirror commutes with the translation and squares to the identity, so the group it generates alongside the translations is the direct product of the integers with a group of order two.

Three are the infinite dihedral group. Vertical mirrors and a translation; half-turns and a translation; vertical mirrors and glides. Each is generated by two elements of order two whose product has infinite order, which is the presentation of the infinite dihedral group, so the three are one group written three ways.

And one is that group with a two-element factor. The most symmetric frieze has everything, and its horizontal mirror commutes with all of it, so it splits off as a direct factor exactly as it did above.

Two, one, three, one. Seven arrangements, four groups, and the arithmetic is the same arithmetic in each case.

What the extra three are, then #

If three of the seven are abstractly indistinguishable from one another, the classification is not classifying abstract groups, and it is worth saying what it is classifying.

It classifies actions. A frieze group is not a group in isolation; it is a group together with the way it moves the strip. The infinite dihedral group acts on a strip in three inequivalent ways — its reflections can be vertical mirrors, or half-turns, or a mix of mirrors and glides — and no change of coordinates carries one of those actions onto another, because a change of coordinates preserves which operations reverse the strip’s two sides and which do not.

That is the same distinction the eight-class group above encodes. The three binary digits are the sign along the strip, the sign across it, and the half-offset — and an abstract multiplication table records only the third of those indirectly and the first two not at all. The digits are geometry that the group structure has thrown away, which is why the closure computation has to be done on the digits rather than on the abstract elements.

The presentation makes the point checkable. The infinite dihedral group is $\langle a, b \mid a^2 = b^2 = 1 \rangle$, and its three appearances differ only in what $a$ and $b$ are taken to be: two vertical mirrors half a repeat apart; two half-turns half a repeat apart; a vertical mirror and a half-turn a quarter repeat apart. The relations are identical in all three, and the strips are visibly different.

So the same phenomenon appears at every scale of this subject. Seventeen plane groups have fewer abstract types than seventeen; two hundred and thirty space groups have fewer still. The number a classification reports is always a number of actions, and a reader who takes it as a number of groups is counting something the classification never claimed to count — which is the same confusion, one level up, as reading a list of subsets as a list of groups.

Where the exactness stops #

The enumeration is exact, and its scope is one dimension.

It settles strips, not the plane. The plane’s classification has five lattices instead of one and its collapses are more numerous, more varied, and not checkable by hand — which is why the plane’s proof is organised by branching rather than by enumeration. The frieze case earns its place as a rehearsal, not as a miniature of the argument.

It works modulo the translations. Everything above happens in a group of eight classes, which is the quotient of the frieze group by its own translations. That is the object the classification is about, and it is not the pattern: two patterns with the same class group can look nothing alike, and the site’s round trip exists because the reverse can also happen.

It does not decide where the elements sit. A frieze group says a vertical mirror is present; it does not say where the mirror lines fall, and the choice of origin moves them. The classification is insensitive to that choice by construction, which is a strength when naming a pattern and a nuisance when drawing one.

Who worked it out, and when #

The seven friezes are old and their attribution is diffuse. They appear as a complete list in the crystallographic literature of the late nineteenth century, alongside the classification of the plane groups that Fedorov, Schoenflies and Barlow arrived at independently between 1891 and 1894, and the strip case was never the hard part of that work — it is the exercise the harder theorem is built on.

What is more recent is the framing used here. Treating the classification as an enumeration over subsets, closed under composition and renormalised, is how a computer algebra system would be asked to do it, and it makes the argument’s shape visible: the theorem is not a list of seven with reasons attached, it is a finite search with a rescaling step in the middle.

The rescaling is the part that survives into the harder cases. In the plane it reappears as the choice between a primitive and a centred cell, where a group’s translations are again finer than the cell chosen to describe it, and in three dimensions it reappears as the centrings that turn seven crystal systems into fourteen lattices.

What the pictures here cannot show. Every figure on this page is a table or a strip, and the collapse itself has no picture: the claim that two candidates are the same group is a claim that one set of operations equals another after a change of lattice, and a drawing of either shows one pattern rather than an equality. What the drawings do show is the half translation arriving — the finer set of rules in the lower strip — and that is the fact the equality rests on rather than the equality itself.

Where the ladder goes next #

The classification this rehearses is the seventeen, where the same forcing arguments run with two translation directions and five lattices, and the proof organised one branch at a time.

The place the friezes turn up again is inside those seventeen: every plane group contains frieze groups along its lattice rows, and which of the seven appear where is a fact about the plane groups that their symbols do not state.

The notation that makes the strip’s three positions readable, and explains why p1m1 keeps a trailing 1 that p4mm is allowed to drop, is Hermann–Mauguin — and the notation index derives all seven symbols from the operations rather than listing them.