Two drafts of twenty-two thousand

The surface of a cloth is decided by its draft, and a draft is a small binary matrix. So the question of which weaves have which surfaces is not a matter of judgement about pictures — it is a finite enumeration, and the catalogue this collection runs every enumeration over is the same one here: all 22,874 four-by-four drafts in which every end and every pick interlaces at least once.

The claim #

Over the whole catalogue, the crown line a surface carries is a narrow and unimodal quantity with one exact exception at zero, and it is smallest for the balanced drafts rather than largest.

Three findings, and the third is the one that is not obvious in advance.

Exactly two drafts touch at points. A draft with no plateau on either face is a draft with no run of two anywhere in either direction, which is an alternation both ways, which is the plain weave. There are two of them in the catalogue because a plain weave can be written starting on either thread.

The range is narrower than a float count suggests. From 0.33 to 1.40 millimetres of crown line per square millimetre, with eighty per cent of the catalogue between 0.99 and 1.36. Most drafts have very similar surfaces and a few are unusual.

And crown line is a U-shaped function of balance. A draft with twelve of its sixteen intersections warp-up carries 1.400; a draft with eight carries 1.136 on average; a draft with four carries 1.300. The most one-sided cloths have the most crown line and the most balanced have the least — which reverses the intuition that a floated cloth is a floated cloth whichever way up it is.

Why balance decides it #

The mechanism is arithmetic and takes one line.

A crown line is contributed by a face run of two crossings or more, and a run of length n contributes n − 1 spacings. So a system whose face is one long run contributes a great deal and a system whose face is chopped into singles contributes nothing. The two systems share the intersections: warp-up and weft-up sum to sixteen in every draft, always.

A balanced draft splits those sixteen evenly, which means both systems have their face broken into short runs, which means both contribute little. A one-sided draft gives one system nearly the whole face — long runs, much crown line — and gives the other almost none, but the other was contributing almost nothing anyway. The trade is not symmetric, because the crown line rewards concentration: three runs of one contribute nothing at all, and one run of three contributes two.

The twenty-four drafts at the maximum are exactly the drafts with twelve warp-up intersections arranged so that every end and every pick has one down — the three-and-one family, of which the twill is the familiar member. Every one of them lands on 1.400 to the last digit, because the quantity depends only on how the runs are distributed and they all distribute them the same way.

What was counted, and how #

The census does not sample any surfaces. It cannot: sampling a height field on a grid takes a fraction of a second and there are twenty-two thousand of them.

It does not need to. The crown line comes off the float map in closed form — for each thread, the maximal runs of face are found cyclically, and each run of length n contributes (n − 1) spacings — and the crown height comes from one Peirce solution that is the same for every draft because the construction is the same for every draft. So the whole catalogue is answered by a loop over sixteen-bit integers, and it takes under half a second.

The two point-contact drafts are found by the property that defines them rather than by being recognised: a draft whose total crown line is exactly zero is asserted to have a longest float of one, which is the statement that the census and the float count agree about what a plateau is. Both come out at 1, and there are two of them.

The two ends of the catalogue #

The maximum is a family of twenty-four and the minimum is a family too, and the two are worth setting beside each other because they are not opposites in the way one would guess.

At the top, twelve warp-up intersections with one down in every end and every pick: the three-and-one family. Every end carries one run of three, every pick carries none, and the total is 1.400 millimetres of crown line per square millimetre.

At the bottom of the non-zero drafts, 0.325 — a quarter of the median. Those are drafts in which both systems are chopped nearly to alternation, with one long run somewhere to keep them off zero. A representative is a cloth that is a plain weave over most of its repeat with a single doubled crossing in it: almost all of its intersections are isolated, its crown is almost all summits, and the one run of three it contains is the whole of its crown line.

The distance from that draft to a plain weave is one intersection, and the distance in crown line is the difference between a little and none. What that says is that the point-contact property is exact but fragile: it is the property of having no run, and one wrong lift anywhere in the repeat destroys it. A plain weave with a single missing end or a single wrong shed is no longer a point-contact cloth, and the cloth’s whole contact regime has changed for a fault that is invisible.

The system that touches is not always the system on the face #

Here is the finding that took the census to produce, and it is a caution as much as a result.

A one-and-three twill has its weft on the face — three quarters of what a reader sees is weft float, and every one of those floats carries a plateau. Its warp is on the face at one intersection in four, in isolated points with no plateau anywhere.

And in the model as constructed, a one-and-three twill touches at points.

The reason is the step between the crown heights. The warp crowns at 190 micrometres above the mid-plane and the weft at 183, so for the first seven micrometres the cloth’s outside is entirely the warp’s isolated summits, and the weft’s ribbons — which are what the cloth looks like it is made of — are out of play.

Whether that survives contact with a real fabric is a different question, and the answer is that it depends on a number nobody measures. The step comes from the division of crimp between the two systems, and this collection’s standing position is that the crimp ratio is not a measurement but a convention. Change the convention and the step changes sign, and with it which of the two regimes sits at the top of the curve.

So the honest statement is a conditional one: given a division of the crimp, the surface follows exactly; the division itself is an input, it is small, and it decides something surprisingly large.

The narrowness, which is the other result #

Most of the catalogue is alike. The interquartile range of the crown line is small, the median draft carries 1.175 millimetres per square millimetre, and a draft picked at random is overwhelmingly likely to be within twenty per cent of that.

That is worth saying because it is the opposite of what the float count does. The longest float over the catalogue takes only three values — one, two or three — and 22,784 of the drafts have a longest float of three, so the float count classifies almost nothing. The crown line is a continuous quantity that distinguishes drafts the float count cannot, and it still finds most of them similar.

What that says about designing a surface is unflattering. Within a four-by-four repeat there is very little room: a designer choosing among twenty-two thousand drafts is choosing among surfaces that mostly differ by twenty per cent in the quantity that decides what the cloth touches with. Getting a large change means leaving the repeat — going to an eight-end satin, which carries 2.1 and is off this scale entirely — or leaving the draft and going to the finish, which is where a calender multiplies the answer by twenty and the draft cannot. Leaving the repeat is the other lever, and how many shafts that costs is the price.

What a designer can actually reach #

Putting the numbers in one place makes the practical range visible, and it is not large.

A plain weave carries no crown line at all. A two-and-two twill carries 1.350 counting both systems, and 0.700 counting only the one that bears. A hopsack of the same order carries the same 0.700, which is worth noticing — two drafts that look nothing alike and behave differently in almost every other respect have identical surfaces by this measure, because both give every thread runs of exactly two. A three-and-one twill carries 1.400. And an eight-end satin, outside this catalogue, carries 2.100.

So the whole four-by-four catalogue spans a factor of about four in a quantity that spans a factor of infinity if the plain weave is included and a factor of three if it is not. The plain weave is not at one end of a range. It is off the scale, because zero is not a small number in a ratio, and every statement of the form “a plain weave has less crown line than a twill” understates by an amount that has no upper bound.

That is the sense in which the exception is worth a census to find. Nobody would have doubted that a plain weave has the least float; what the enumeration establishes is that it has none, that nothing else in the catalogue shares the property, and that the property is exact rather than approximate.

Where the model stops #

The census is a four-by-four census and four is small. A satin of eight ends carries half as much crown line again as anything in this catalogue, because its plateaux are longer and its turns fewer. The catalogue’s narrowness is a fact about small repeats rather than about weaving.

The crimp division is an input. It is held equal for every draft in the census, which is defensible as a convention — the same convention for every draft makes the comparison a comparison of drafts — and is certainly wrong in detail for the one-sided ones, where a real cloth’s two systems do not crimp alike.

The crown line is a length and not an area. What bears is the length times a width, and the width is the same for every draft in the census because they share a yarn. That is why the census can be a length census at all, and it stops being one the moment two cloths are compared across different counts or different finishes.

And nothing here is the plateau’s own straightness. Every float is taken as lying flat on its supports. A real float sags between them, which rounds the ends of every ribbon slightly and shortens none of them.

The census has a closed form #

The crown line is computed here by walking every draft’s float map, and it does not need to be. A run of n contributes n − 1 spacings, so summing over a system’s threads gives its face intersections less its number of maximal runs — and the whole census collapses to two counts per draft.

Writing w for the warp-up intersections, R₁ and R₂ for the number of maximal warp runs and weft runs, and p₁, p₂ for the two thread spacings,

crown line per unit area = (w − R₁) ÷ 16p₁ + (16 − w − R₂) ÷ 16p₂.

Two integers and two spacings, and no float map at all.

It reproduces the census’s own landmarks exactly. The maximum: twelve warp-up intersections in four unbroken runs of three, and four weft-up intersections in four isolated runs, gives (12 − 4)/16p₁ + 0, which at twenty-eight ends to the centimetre is 1.400 — the census’s figure to the last digit. And the zero: a plain weave has eight warp-up intersections in eight runs and eight weft-up in eight, so both numerators vanish and the crown line is exactly nought, which is the exception the whole essay is about, arriving as an arithmetic identity rather than as an enumeration’s output.

Which explains why the U is lopsided #

The essay reports the U-shape as asymmetric — 1.400 at twelve warp-up intersections against 1.300 at four — and offers no reason. The closed form gives one immediately.

Complementing a draft exchanges w with 16 − w and R₁ with R₂, so it exchanges the two terms and their denominators. The total is therefore invariant only if p₁ = p₂, and the sheeting the census runs on is set at twenty-eight ends and twenty-six picks.

The ratio of those is 1.077, and 1.400 ÷ 1.300 is 1.077.

So the whole asymmetry of the U is the sett ratio and nothing else. On a square-set cloth the two arms would be exactly equal and the census’s list would be a census of complementary pairs, each pair sharing one value.

That has a consequence for how the finding should be read. The most one-sided cloths have the most crown line is symmetric in the draft; which of the two one-sided families wins is decided by the construction, and turning the cloth’s setts round swaps them. A weft-dense sheeting would put the one-and-three family at the top and the three-and-one family below it, with the same twenty-four drafts in each place.

And it makes the census a census of pairs #

The invariance also halves the catalogue in the way the orbit count does, and for the same reason.

At equal setts, a draft and its complement carry identical crown line — and a complement is the same cloth turned over. So the surface census is really a census over cloths seen from either face, and the 22,874 drafts collapse onto roughly half as many distinct values before any other identification is applied.

That is worth knowing before a search. A designer looking through the catalogue for a particular surface is looking through half of what the list suggests, because every draft’s complement is already there carrying the same number — and picking between them is a decision about which face is up, which the surface arithmetic cannot make and the trade cares about a great deal.

The generalisation #

An enumeration over a small combinatorial space usually finds it narrower than the extreme cases suggest.

That is the transferable lesson and it recurs across this collection. The catalogue of four-by-four drafts is famous for containing a plain weave, a twill, a hopsack and a mat; the census says those four are unremarkable members of a population that is tightly clustered, and the interesting drafts are the twenty-six at the two ends of it.

The second half of the lesson is that a property can order a catalogue that a classification cannot. Longest float is a classification and takes three values here. Crown line is a property and takes hundreds. Anything that needs the catalogue ordered — a search, an optimisation, a claim that a particular draft is best at something — needs a property, and the surface supplies one that nothing else on this site had.

What it shares with the hole census #

The plain weave has now been singled out twice by two enumerations that have nothing in common, and the coincidence is not one.

Only a plain weave has one size of hole: every aperture in it is the same aperture, and every other draft in the catalogue has at least two sizes. Only a plain weave touches at points: every crown in it is a summit, and every other draft has at least one ridge.

Both are the same statement about the matrix wearing different clothes. A run of two is a second scale, and the plain weave is the only draft in the catalogue with no run of two anywhere. A cloth with no second scale has one kind of hole and one kind of crown; a cloth with a second scale has two kinds of each.

That is worth stating as a property of the plain weave rather than as two separate observations, because it explains why the plain weave keeps turning up as the exceptional case in enumerations that appear to be about quite different things. It is not that the plain weave is special about holes and separately special about contact. It is that it is the only alternating matrix, and everything that follows from alternation follows twice.

Who found it, and when #

The four-by-four catalogue is this collection’s own, enumerated at the outset and used since for layer counts, colour collapses, fault responses and raisability. The surface reading of it is new here.

Nothing in the textile literature counts crown line, as far as this collection can find, because nothing in the textile literature computes a woven surface as a height field. The nearest published relatives are the bearing curves of surface metrology, which are measured off real specimens with a stylus, and the “float ratio” figures quoted in fabric design texts, which count intersections rather than lengths and therefore cannot see the difference between three runs of one and one run of three.

Where the ladder goes next #

Out of geometry and into force: how much of a cloth is touching turns a depth into a pressure and finds that at any load a hand can apply the answer is well under one per cent.

Sideways, the U-shape has a companion. The same census run for reflection rather than contact — how much every four-by-four draft can shine — uses both systems rather than only the higher one, and comes out with a different ordering from the same list, because a bearing curve cares which crown is higher and a reflection does not.