No weave draws an unbroken line one thread wide

The narrowest stripe there is, on paper, is one thread wide. A designer wanting a fine line down a dark suiting draws a single light column on point paper, and a warper puts a single light end into the warp at that place. The drawing is a line; the warp has a line in it.

The cloth does not. A warp end is on the face only at the picks where it passes over the weft, and at every other pick the weft covers it. So what a reader sees of a single light end in a dark cloth is the end where it is up and the dark weft where it is down — and in any weave that holds its threads in, every end goes down somewhere.

A single end shows only where it is up #

What decides whether an end is on the face at a pick is its own column of the draft, and the draft is a matrix whose every column, in a cloth that hangs together, has both states in it. An end that was up at every pick would lie on the surface for the length of the piece, crossing nothing, and the integrity criterion would report it as a thread held by nothing. So every end in every weave that interlaces goes under at least one pick of its repeat.

That is the whole of the first result. A light end one thread wide is a line with gaps in it, in every weave there is, and the gaps are exactly the runs of picks over which that end is down — the weft floats lying across it, which are the same floats that decide a weave’s lustre and its snagging.

In a 2/2 twill the end is up at two picks and down at two, so its line is half light and half dark, in dashes two picks long. In a 3/1 twill it is up at three and down at one, so the dashes are longer and the gaps shorter. In an eight-end sateen it is up at seven picks in eight and down at one. The dash pattern of a single-end pinstripe is the column of the draft read down the cloth, and it is as different between weaves as the weaves are.

Neighbours fill one another’s gaps #

Put a second light end beside the first and the line is light at every pick where either of the two is up. The gaps that remain are the picks at which both are down together. A third end fills more, and so on, until at some number of neighbours there is no pick at which all of them are down at once. At that point the line has no gap.

That number is a property of the weave, and it is found by trying every run of k adjacent ends round the repeat and asking whether, at every pick, at least one of them is on the face. The smallest k for which some run succeeds is the narrowest unbroken line the weave can draw in the warp.

For the 2/2 twill in the figure it is three. Two adjacent ends of a twill are the same column shifted by one pick, so they are down together at one pick in four, and the line has a gap one pick long there; a third end, shifted by one more, is up exactly at that pick.

A plain weave needs two, and can never need more #

Plain weave is the easiest case and it has a clean reason.

In a plain weave adjacent ends are complements: each is up exactly where the other is down. One end draws a line of dots, and two adjacent ends draw a line that is unbroken and exactly half light at every pick, because one of the two is always up and the other always down. Two is the narrowest unbroken line a plain weave can draw, and it is the narrowest any weave can draw, since one is never enough.

The same is true of a basket, with a condition on where the stripe goes. A 2/2 basket’s ends move in pairs, so two ends from the same pair are the same column and draw the same broken line; two ends that straddle a pair boundary are complements and draw an unbroken one. The narrowest unbroken line is two ends, but only two ends in the right place, and a warper who puts the stripe one end to the side has woven a dashed line with the same number of threads.

Every four-by-four draft #

The search is cheap, so it can be run on the whole catalogue rather than on the named weaves alone.

Of the 22,874 four-by-four drafts that interlace, not one draws an unbroken line one thread wide in either direction, which is the interlacing condition checked rather than restated. For a warp line, 18,530 need two ends, 4,320 need three and 24 need four.

Four is the most a four-end repeat can need, because all four ends together are up somewhere at every pick — a pick with every end down would be a weft thread lying on the face across the whole repeat, crossing nothing. The interesting count is the twenty-four that reach it.

The drafts that need every end are the permutations #

A draft needs all four ends for an unbroken warp line exactly when every run of three adjacent ends leaves some pick uncovered. That happens when every pick has exactly one end up: then whichever three ends are chosen, the pick whose one mark is in the fourth end shows no light.

A draft with exactly one warp mark in every pick and every end is a permutation matrix, there are four factorial of them, and the census finds precisely those twenty-four among the drafts needing four. The drafts with the widest unbroken warp line are the ones that put the fewest warp marks on the face — the weft-faced single-mark weaves of which a satin is the regular member — and at four ends there is no satin, so all twenty-four are twills and irregular arrangements of one mark per thread. Their four-end line, when it comes, is a quarter light at every pick.

The same twenty-four, turned over, are the warp-faced drafts with one weft mark in every end and pick, and they need four picks for an unbroken weft line and two ends for an unbroken warp one. That is the first sign that the two directions are not the same question.

Unbroken is not solid #

The census asks whether a line has a gap. It does not ask how light the line is where it has none, and the answer is almost never all of it.

No line in any weave is solid, in the sense of being light at every thread across its width at every pick, because that would need every stripe end up at every pick, which is an end that does not interlace. What an unbroken line has instead is a share of its width that is light, pick by pick, and that share varies.

A 2/2 twill’s three-end line is light at a third of its width at some picks and two thirds at others — beaded, in a rhythm four picks long. Widen it to four ends, a whole repeat, and every pick has exactly two of the four up: the line is evenly half light for the first time. A 3/1 twill’s two-end warp line runs from half light to fully light and back. An eight-end sateen’s two-end warp line spends six picks in eight fully light and two half light, so it averages seven eighths.

A line becomes even exactly when it is a whole number of repeats wide, because then every pick sees each of the weave’s columns the same number of times, and its light share is the weave’s own face fraction — a half for a balanced weave, three quarters for a 3/1 twill, seven eighths for an eight-end sateen. That is the fraction a damask’s figure and ground are built from, arriving as the brightness of a stripe.

The beading matters more than its size suggests. A variation that repeats every four picks is periodic, and a periodic variation shows where a random one of the same size hides — so an unbroken but beaded line can read as a row of beads at a distance where a line of random unevenness would read as smooth.

A stripe and a bar of the same width need different threads #

Ask the same question across the cloth — how many adjacent picks of a light weft draw a bar with no gap — and the answer changes with the weave in a different way.

A 3/1 twill is warp-faced: each end is up at three picks in four, and each pick is on the face at one end in four. So a warp line needs two ends, since two adjacent ends are down at different picks, and a weft bar needs four picks, since fewer leave some end at which every one of them is covered. The same cloth draws an unbroken stripe two threads wide and an unbroken bar four threads wide — and the stripe averages three quarters light while the bar is a quarter.

Over the catalogue this is common rather than exceptional. The two directions give the same answer for 14,234 drafts and different answers for 8,640 — more than a third. On every one of those a check drawn with lines of equal thread counts in both systems is a check with an unbroken line one way and a broken one the other, and the difference is invisible on point paper, where a column and a row of the same colour look exactly alike.

The named weaves #

The weaves a pinstripe is actually woven in fall into a short table, and the table sorts them by how faced they are.

The balanced weaves need the same number both ways: two for plain and basket, three for the 2/2 twill. The warp-faced weaves need two ends for a warp line and as many picks as their repeat for a weft bar. An eight-end sateen draws the narrowest possible warp stripe and the widest possible weft bar in the same cloth, and a single light end in it shows at seven picks in eight — so close to unbroken that the gap is one pick in eight, and never actually closed.

That makes the sateen a special case for anybody designing a check in it. A warp stripe of two light ends has no gap and is seven eighths light on average; a weft bar of two light picks is two sparse rows of dots, an eighth of the bar light, and nothing a reader would call a line. Even the eight-pick bar with no gap in it is an eighth light: an even haze across the cloth rather than a bar.

What a broken line looks like from across a room #

A dashed line and an unbroken one are different on the drawing and may not be different to an eye, and the arithmetic of that decides whether any of this shows.

The gaps in a single-end pinstripe are a few picks long. In a 2/2 twill worsted of thirty picks to the centimetre, a gap of two picks is about two thirds of a millimetre. An eye resolves roughly a sixtieth of a degree, which is about a tenth of a millimetre at reading distance and nearly a millimetre across a room. In the hand, the line is visibly dashed; from three metres it reads as a continuous line of half the contrast, because the eye has averaged the dashes it can no longer separate.

So a one-end pinstripe is a real design choice rather than a mistake: it is a finer, fainter line at a distance and a textured one close to. The same averaging turns a beaded line into its mean — which is why, from across a room, what matters about a pinstripe is its light share and not its gaps, and the light share of any line that is a whole repeat wide is simply the weave’s face fraction.

A pinstripe is a colour order with one odd thread #

Every result here is a special case of colour and weave, in which an order of coloured threads laid on a weave makes a pattern that is in neither. A pinstripe is the simplest colour order there is — one or a few light threads in a dark ground — and the pattern it makes is the dash pattern above.

What the colour-and-weave arithmetic adds is where the contrast goes. Where a light end is down, the thread on the face is the ground’s weft, and if the weft is dark the gap is dark; if the weft were light, the intersection would be blind and the gap would not show at all. A pinstripe has no gap in any weave if its weft is the colour of the stripe — but only because the whole cloth then shows that colour wherever the ground’s ends go down, so it is no longer a fine line in a dark cloth but a light cloth with dark dashes in it. That is the construction a tartan makes at its crossings, and a pinstripe in a dark cloth is dashed for the same reason a tartan’s crossing squares are mixtures.

A one-pick bar, and the loom that throws it #

The weft direction has one more constraint that the warp does not, and it bears directly on the narrowest bar.

A weft bar one pick wide is a run of one pick in the weft’s colour order, and a loom with shuttle boxes at one side throws its weft in pairs: every coloured run is an even number of picks. So the narrowest weft bar such a loom can weave at all is two picks — which in a plain weave is exactly the narrowest unbroken one, and in a 3/1 twill is half of it.

The two constraints come from completely different places — one from the draft’s columns, one from where the shuttles are — and they happen to agree on a plain weave. On every other weave the loom’s minimum and the draft’s minimum are separate numbers, and a designer wanting a fine bar with no gap has to satisfy both.

How the widths were found #

For each draft and each direction, every run of k adjacent threads round the repeat is tested for whether, at every crossing, at least one of the k is on the face — a warp end up, for a stripe, or a weft pick not covered, for a bar. The width is the smallest k for which any run succeeds, starting from one. Each draft is first checked to interlace, so that every thread has both states and a width of one is excluded by the definition rather than by the search.

The census applies that to all 22,874 interlacing four-by-four drafts in both directions and counts the pairs of widths. The permutation result is a separate check: the drafts with one warp mark in every pick and every end are listed, and the set is compared with the set of drafts whose warp width is four. The light shares are counted pick by pick along the narrowest unbroken line and along a line a whole repeat wide, for each named weave.

Where the model stops #

A float is not a sharp rectangle. A light end lying over two dark picks spreads a little sideways, flattened by the picks it crosses, and a dark weft float over a light end spreads over it in turn. Both blur a gap at thread scale and neither is modelled, so the widths are those of the draft read literally.

The ground is one colour. A pinstripe in a striped or checked ground meets a different colour at its gaps in different places, and the question becomes a full colour-and-weave problem rather than a width.

And the eye is a sentence, not a model. Where a dashed line starts reading as continuous depends on the sett, the contrast, the light and the viewer, and the figure of a tenth of a millimetre at reading distance is a rule of thumb rather than a measurement of anybody’s vision.

Still open: whether an unbroken line has a straight edge #

A 2/2 twill’s three-end line has no gap, and the drawing of it shows something neither its width nor its light share says: the light cells step down and across with the twill, so the edges of the line are not straight but stepped, with a period of four picks. An unbroken line is not the same as a clean one. Whether the stepped edge reads as a rope-like texture or averages to a straight edge at a distance, and which weaves give the straightest edge for a given width, is a third question about the same columns — and a cord, whose threads move in groups, may be the construction that answers it.