The seventeen groups a draft can have

A draft is a periodic pattern of filled and empty squares in the plane, so it has a plane symmetry group, and there are exactly seventeen of those. The obvious question is which ones a weave can have.

Enumerate every four-by-four draft in which each end and each pick interlaces at least once — 22,874 of them — classify each, and count.

Twelve. And the five that never occur are p3, p3m1, p31m, p6 and p6m — precisely the five with three-fold or six-fold rotational symmetry.

Why the five are missing, at any size #

The census is finite and the claim is not, so the reason has to be an argument rather than a count.

A symmetry of a draft is an isometry of the plane that maps the pattern to itself. The pattern lives on the intersections of warp and weft, which form a square grid, so any symmetry has to permute the intersections — it maps the grid to itself.

The isometries of the plane that map a square grid to itself have linear parts drawn from the eight symmetries of the square: the identity, three rotations by multiples of a right angle, and four reflections. None of them has order three.

So no draft, of any repeat size, can have three-fold rotational symmetry, and the five hexagonal groups are unreachable. The census is consistent with that; it is not evidence for it.

This is the crystallographic restriction, met in a place where it has an unusually concrete form. In crystallography the restriction says a lattice can only have two-, three-, four- or six-fold symmetry. Here the lattice is fixed in advance — it is the warp and weft — and the restriction bites harder: only two- and four-fold survive, because the grid is square rather than merely periodic.

What “classifying” actually requires #

Naming a group is harder than it looks, and the distinctions that matter are exactly the ones invisible in a drawing. The classification here is done in four steps.

The translation lattice first. A four-by-four repeat may really be a two-by-two one, and it may contain a centring translation that no edge of the drawn repeat reveals. Plain weave written on four ends has period two and also a centring shift; everything downstream has to be stated relative to the lattice the pattern actually has, not the box it was drawn in.

Then the point group. Which of the eight square operations are symmetries at all, with any translation whatever. That alone separates p1 from p2 from p4 and so on.

Then, for each reflection, whether a mirror exists or only a glide. An element mapping $x$ to $Ax + t$ with $A$ a reflection is a pure reflection when some lattice translation cancels the component of $t$ along the mirror axis, and a glide when none does. This is the step that separates pm from pg, and no amount of looking at a picture settles it.

And whether the lattice is centred. That is the pm-against-cm and pmm-against-cmm question, and it cannot be answered from the reflections at all: a group with a mirror always contains glides too, because composing a mirror with any lattice vector running along it produces one. The distinction is a property of the lattice — whether the shortest lattice vectors along and across the mirror span the whole lattice, or only half of it with a vector at the centre of that rectangle making up the rest.

Getting that last point wrong is easy and the error is silent. A classifier that used the presence of glides as the test for centring would report cmm where pmm is correct, on patterns that look entirely ordinary.

Checking the classifier #

A classification is a claim, so it needs to be able to fail. Three checks are run.

Known answers. Plain weave is p4m; a two-two basket is p4m; a two-two twill is pmg; a five-end satin is p4; an eight-end satin with move three is cmm. Every one of those can be worked out by hand, and the classifier is required to agree.

Invariance. Rotating, reflecting, translating or colour-inverting a draft gives the same cloth seen differently, so the group name must not change. Running that over thousands of drafts and four transformations apiece is a property test with real teeth: a classifier with a coordinate-order bug fails it immediately.

Point-group order. Each of the seventeen groups has a known point-group order — one for p1, two for p2 and the pm family, four for the pmm family and p4, eight for p4m and p4g. The classifier’s own count of operations has to match the name it assigned.

None of those is decorative. The second one in particular caught a real error while this was being written: a first version searched for cancelling lattice vectors only among the representatives inside one repeat rather than over the whole infinite lattice, which misclassified a family of drafts as glide-only when a mirror existed a repeat away.

What the census actually shows #

The distribution is lopsided in a way worth noticing.

Most drafts have no symmetry at all. 14,848 of the 22,874 — nearly two thirds — are p1. That should not be surprising: a random binary matrix is unlikely to be symmetric, and the interlacing filter does not select for symmetry.

The next largest groups are the ones with a single reflection. cm and pm together account for another 5,728. A single mirror is the cheapest symmetry to have, and it is what a draft acquires by being drawn symmetrically about one axis.

The highly symmetric groups are rare. p4m has 186 drafts, p4 and p4g have 32 apiece, pgg has 32. Together the eight-element groups account for under one per cent of the census.

And the named weaves are all in that last one per cent. Plain weave, basket, the twills and the satins are all highly symmetric, because they are all generated by rules — and a rule is a statement of symmetry written out. That is the reason the weaves anybody uses are so unrepresentative of the drafts that exist.

The rule-generated weaves and their groups #

Working through the site’s own weaves is the best way to see what the classification is sensitive to.

Plain weave: p4m. Four-fold rotation, mirrors in all four directions, every one a genuine mirror. It is the chessboard, which is the most symmetric two-colour pattern on a square lattice.

Basket: p4m. The same, at a larger scale — a basket is plain weave with the threads taken in groups, and grouping preserves every symmetry.

Two-two twill: pmg. This one is instructive. The twill has a half turn and both diagonal mirrors as operations, but only one of them is a genuine mirror: reflecting along the twill line requires a shift of half a lattice vector, so it is a glide. The twill’s step is exactly that glide, which is a satisfying way of saying what a twill is.

Five-end satin: p4. Four-fold rotation and no mirrors at all — the satin is chiral. Its move of two satisfies $m^2 \equiv -1 \pmod 5$, which is precisely the condition for a quarter turn to be a symmetry, and the absence of mirrors is what gives a satin its handedness.

Eight-end satin, move three: cmm. Diagonal mirrors, a half turn, no quarter turn — because $3^2 \equiv 1$ rather than $-1$ modulo eight. So two satins of different orders can have different symmetry groups, decided by a congruence.

That last point is worth dwelling on, because there are two congruences in it and not one. Turning a satin’s draft a quarter turn carries the satin on move m to the satin on minus the inverse of m, and reflecting it across the diagonal carries it to the satin on the inverse of m. So a quarter turn is a symmetry exactly when $m^2 \equiv -1$, a mirror exactly when $m^2 \equiv 1$, and a satin with neither keeps only its half turn, which is the group p2. Each is a question in elementary number theory with a clean answer, and each shows up as a visible property of a fabric.

A quarter turn needs minus one, and a mirror needs plus one #

The five-end satin comes out p4 because its move squared is congruent to minus one, and the eight-end satin comes out cmm because its move squared is congruent to plus one. That is stated above as a fact about two weaves. The quarter turn, at least, is a condition on the order, and the condition has a clean answer.

A quarter turn is a symmetry of an order-n satin exactly when some admissible move satisfies m² ≡ −1 (mod n). That congruence is solvable precisely when n is one, two, or a product of primes congruent to one modulo four, optionally times a single factor of two — the classical condition for minus one to be a quadratic residue.

Running it over the satin orders a weaver reaches:

Only three orders between five and sixteen admit a quarter turn, and two of them are in common use. Everything divisible by four is excluded outright, which takes out the eight-, twelve- and sixteen-end satins together; and every order with a prime factor congruent to three modulo four is excluded, which takes out seven, eleven and fourteen.

A mirror is no commoner. It needs a square root of one other than one and minus one, which only some orders have — eight, twelve, fifteen and sixteen in this range — and even there only two of the admissible moves have it. Every other satin is chiral: its mirror image is the satin on the complementary move, and no turn of the cloth carries one to the other.

Two things are worth taking from that.

The two commonest satins in the trade fall on opposite sides. The eight-end satin is the standard lining and furnishing weave, and its group is cmm: it has diagonal mirrors and no quarter turn, so it looks the same in a mirror. The five-end satin, which is the standard dress-weight one, is p4: it has a quarter turn and no mirror, so its mirror image is a different cloth. Two fabrics a weaver would put in one family differ in whether they have a handedness, and the difference is a congruence.

And the handedness is not the twill’s. A twill’s diagonal runs one way or the other and reversing it gives a different draft — that is a chirality of the drawing, present in every twill. A satin’s is a chirality of the symmetry group: no reflection carries the draft to itself, so its two mirror images are genuinely distinct cloths rather than one cloth turned round. The five-end satin adds a four-fold rotation to that, which is the rarer property; the seven-, nine- and eleven-end satins are just as handed with a half turn alone, and what carries their handedness is the diagonal their lattice keeps.

So the classification earns something the census alone could not say. Which satins have a handedness is decided by elementary number theory on the move: the ones without are a short list, the moves whose square is one, and it explains why the question never comes up for the two most-woven satins: one of them is on the list and one is not, and nobody has had occasion to notice.

Reading a group name #

The names look like a code and are one, and decoding them makes the census readable.

The leading letter is the lattice: p for a primitive cell, c for a centred one. The first digit is the highest order of rotation: 1, 2, 4 — and, in the five groups a draft cannot have, 3 or 6. The letters after that describe reflections and glides in the principal directions, m for a mirror and g for a glide.

So p4m is a primitive lattice with four-fold rotation and mirrors; p4g is the same rotation with glides where p4m has mirrors; pmg has mirrors in one direction and glides in the perpendicular one; pgg has glides in both and no mirror anywhere; cmm is a centred lattice with mirrors in two directions.

The distinctions between them are exactly the ones this essay’s classifier has to compute rather than see. p4m and p4g look alike at a glance and differ in whether a mirror passes through a four-fold centre. pm and cm differ only in whether the lattice is centred, which is a property of translations rather than of anything drawn. And pgg — thirty-two drafts in the whole census — has no mirror at all despite having two reflection directions, so a reader looking for an axis of symmetry in one will not find one.

That is worth saying plainly: five of the twelve groups a draft can have are distinguished by properties with no visual signature, which is the same complaint this site makes about layer count and about float length, arriving from a third direction.

Two-colour, and what is being postponed #

One more thing needs saying before the limits, because the classification above quietly took a decision.

A draft is not really a two-colour pattern. Warp-up and weft-up are two states of an intersection, and the operations that act on a weave divide into those which preserve them and those which exchange them. A half turn about the centre of a plain weave leaves it alone. The transpose — which is physically the cloth seen with warp and weft interchanged — turns every warp-up into a weft-up, and is just as much a symmetry of the fabric.

Counting those separately is the theory of two-colour or dichromatic groups, and it turns the seventeen into forty-six. This essay has counted the first kind only, treating the pattern as filled and empty squares.

That is a legitimate simplification and it is not the whole story, and which one is wanted depends on the question. For “what does the surface look like”, the uncoloured group is right. For “what does the cloth do when turned over”, the two-colour group is the one with the content — and a weave whose warp-weft exchange is a symmetry is a weave that looks the same from both sides, which is a real and useful property.

What the group leaves undecided #

Three limits, and the third is the important one.

It says nothing about integrity. Two drafts with the same plane group can differ in layer count, and the classification has no term that could detect it. Symmetry and connectivity are independent properties of the same matrix.

It says nothing about how the cloth is woven. A group is a property of the repeat, and two drafts with the same group can need quite different numbers of shafts — which is the constraint a weaver actually works under. Symmetry is cheap on paper and not on a loom.

It says nothing about float length. p1 drafts include both the tightest and the loosest weaves in the census. A symmetry group is a statement about how the pattern repeats, not about what is in it.

And it is about the uncoloured pattern. A weave is not really a pattern of filled and empty squares; it is a pattern of warp-up and weft-up, which are two states rather than two colours, and some operations preserve them while others exchange them. Counting those separately turns the seventeen groups into forty-six, and that is the next essay’s subject. Everything in this one has quietly ignored the distinction, which is legitimate and is a simplification.

Where the seventeen came from #

The classification of the plane groups is one of the more curious results in the history of mathematics, because it was arrived at three times.

By artisans, over centuries. The Alhambra’s tilework is the standard example, and the question of whether it contains all seventeen has been argued over for decades — the honest answer is that it contains most of them, and the argument is about how strictly to read a pattern’s symmetry when its colours differ.

By crystallographers, in the 1890s. Fedorov classified the plane groups in 1891 and the two hundred and thirty space groups shortly after; Schoenflies and Barlow arrived at the three-dimensional classification independently and almost simultaneously. Their motivation was crystal structure, not decoration.

And by mathematicians, as part of Hilbert’s eighteenth problem and its resolution by Bieberbach in 1911, which generalised the result to every dimension.

Weaving is missing from that list, and the omission is worth noticing. Woven cloth is the oldest large body of periodic two-dimensional pattern there is, it was made in vast quantity for millennia, and nobody classified it. Grünbaum and Shephard’s work in the 1980s is the first sustained treatment of fabrics as geometric objects — nearly a century after crystallography got there, on a subject far older and far more accessible.

The reason is probably that weavers had no need of it. A classification answers “which patterns are possible in principle”, and a weaver’s question is “which patterns can this loom make”, which is a completely different and much more restrictive question. The five missing groups were never missed.

Where the ladder goes next #

The rung below is weaves as plane patterns, which introduces the classification and the two-coloured question together. The next rung is colour and weave as a two-colour problem, where the surface stops reporting the structure at all.

The nearest result elsewhere on the site is the satin theorem, because a satin’s symmetry and its existence are both decided by congruences on its move — one asks whether the move is coprime with the order, the others whether its square is minus one or one.