A hole with nothing crossing

Hold a piece of mock leno up to the light and it has holes in it — square, regular, a millimetre or so across, in a cloth of ordinary threads at an ordinary sett. Nothing in it crosses anything. Every warp end runs straight down the cloth and every pick runs straight across, exactly as in a plain weave, and the holes are there anyway.

The trade calls it mock leno because it imitates the appearance of a real leno, which makes its holes by crossing warp ends over one another with a doup. The imitation is close enough to have earned the name and the mechanism is not related at all.

The threading does it #

Two adjacent ends that interlace identically are lifted together on every single pick. No weft ever passes between them: the shed opens with both of them above it or both below, and the pick goes past on one side.

So nothing holds them apart. In a finished cloth they lie touching, at one yarn diameter, and the reed’s average spacing is not a constraint on them individually — it is a constraint on the group.

That is the whole construction. A mock leno draft is a plain weave of bundles rather than of threads: three ends threaded to lift together, three picks inserted to bind together, and the resulting cloth is a plain weave whose unit is a bundle. The bundle’s members close up, the space they vacate collects at the bundle’s edge, and the collected space is the hole.

What was counted, and how #

The arithmetic is a conservation and a subtraction, in that order.

The reed sets the ends at a fixed average spacing p, so a group of m ends occupies m spacings of the cloth’s width whatever happens inside it. The threads within the group sit at one diameter d, so they occupy m·d. The difference is m(p − d), and it has to be somewhere: it is the gap at the group’s edge.

Written the other way, the gap is m·p(1 − k) where k is the cover factor of one system. The hole is exactly the cover the sett leaves over, collected into one place instead of being distributed between every pair of threads.

Nothing is created and nothing is destroyed, and the generator asserts that: each end must sit exactly one dent pitch from its opposite number in the next group, or the layout is wrong. That check is worth having because the tempting one — measuring from the first thread to the last and comparing with the flat spacing — is short by exactly one gap, since the grouping pulls both outer threads inwards. It is a check that fails on the correct answer.

At twenty threads per centimetre and a 0.3 mm yarn:

The hole grows linearly with the bundle and the hole count falls as its square, so the open area is constant: a mock leno of any bundle size has exactly the same fraction of open surface, because the cover factor has not changed. Bundling redistributes the openness and does not create any.

Which is the answer to the obvious question — why not use six to a bundle and have larger holes — and the answer is the float. A bundle of six is a float of six, and the cloth loses firmness exactly as fast: 0.50 interlacings per intersection at two to a bundle, 0.167 at six. The trade’s usual choice of three is where the hole becomes visible and the cloth is still a cloth.

The connection is worth making explicit because the trade treats the two as different subjects. A float is a run of one thread on one face; a bundle is a run of adjacent threads doing the same thing. Both are counted off the matrix, both lower the interlacing count, and both are bought for a surface property — one for lustre, the other for openness.

The two refusals #

The arithmetic refuses in two places and both refusals say something.

A draft with no grouped threads leaves no hole. Ask for the mock-leno geometry of a plain weave and it declines: plain weave’s ends are all distinct, its groups are of one, and its gaps are the ordinary spaces between neighbouring threads. Calling those holes would be calling every cloth ever woven a mock leno. The refusal is not a guard against a silly input; it is the statement that the effect requires the threading and nothing else supplies it.

A cloth already at its jam cannot open a gap. At thirty-two threads per centimetre and a 0.3 mm yarn the cover factor is 0.96 and the hole is thirty-eight microns — which is to say the threads were nearly touching before they were grouped and there was no space to collect. Push a little further and the arithmetic refuses outright: there is no surplus, so there is no hole, and a mock leno woven at a close sett is a plain weave with a texture.

A mock leno is an open cloth pretending to be an open cloth in a different way. Its whole hole budget is the openness the sett already had.

The difference from a real leno is not the hole #

Both fabrics have holes of about the same size in cloth of about the same weight. The difference is what happens to the hole when the cloth is used.

A real leno’s crossing is a topological arrangement: the doup end passes to the other side of its partner between picks, so the two are linked, and the pick is trapped between them by a crossing rather than held by a bend. Nothing short of breaking a thread undoes it, and the hole cannot close because the threads on either side of it are wound round one another.

A mock leno’s bundle is held together by nothing. The threads within it lie touching because the weave gives them no reason to be anywhere else, and the gap stays open because the threads on either side of it are, at that moment, where the last finishing operation left them. Rub the cloth, wash it, put it under a tension across the holes, and the bundles will spread and the gaps will close. Some of it recovers and some does not.

The mock leno’s hole is metastable and the real leno’s is topological, and that difference is invisible in both the draft and the photograph. It is the reason a leno is used for a filter or a scrim where the aperture is a specification, and a mock leno is used for a summer shirting where it is an appearance.

The picks do it too, and that is why the holes are square #

Everything above is written across the width, and the same argument runs down the length: picks that bind identically have no end passing between them, so they close up and the space collects at the group’s edge.

That is why a mock leno’s holes are square rather than slotted. The bundle in the warp gives a horizontal gap, the bundle in the weft a vertical one, and the hole is where the two coincide — so a fabric with three to a bundle in the warp and one in the weft would have vertical slits, and the trade’s fabrics do not, because the construction is symmetrical.

There is a second reason to expect square holes and it is not the same reason. A cloth of one yarn at one sett in both directions is balanced, so p − d is the same in both directions before any grouping happens; an unbalanced mock leno — more ends than picks, which is the commoner cloth — has rectangular holes taller than they are wide, and the ratio is exactly the ratio of the two setts. Nothing in the construction makes a square; the balance of the base cloth does.

The pick side has one difference worth naming. The warp’s spacing is set by a reed, which is a physical comb with a fixed pitch; the weft’s is set by the take-up motion, which is a rate. So the warp bundles are held to their average by hardware and the weft bundles only by how fast the cloth is drawn forward, which is one more reason the fabric is less stable in the length than across it.

Grouping is not the only way to leave a gap #

It is worth putting the mock leno beside the other two constructions on this site that leave holes, because the three make their openings by three unrelated mechanisms and the trade’s names do not distinguish them.

A plain weave at a low sett has openings between every pair of threads, and the opening size is p − d at every one of them. That is the baseline the mock leno redistributes: the same total open area in fewer, larger holes.

A leno links its ends, so the opening between two crossings cannot close whatever the cloth is asked to do.

A mock leno groups its ends, so the openings between them close and the one at the edge of the group opens by as much.

The three sit in a clear order by how much of the answer is geometry. The open cloth’s holes are decided entirely by the sett; the mock leno’s are decided by the sett and the threading; the leno’s are decided by the threading and are held there by a linkage. Only the last of those survives a cloth being used.

That last figure is the practical conclusion and it is the opposite of the obvious one. Faced with a specification for a larger hole, the temptation is to bundle more threads, and bundling costs float length quadratically in what it does to the fabric’s firmness while gaining hole size only linearly. Opening the sett costs cover — which is the same budget, spent honestly — and costs no float at all.

Why the open area cannot change and the specification still does #

The open area is constant across every bundle size, which sounds as though bundling achieves nothing measurable. It changes two things a specification actually names, and neither of them is an area.

The largest hole grows linearly with the bundle, and every requirement about what passes through a fabric is a requirement about the largest hole rather than about the total. A grain, a feather, a finger, a mesh’s rated aperture: each of them meets one hole at a time. So a mock leno at three to a bundle passes things a plain weave of identical open area stops, and the specification that would separate them is the one the filtration ladder recommends and nobody takes.

And the hole count falls as the square, which decides how the openness looks. Forty-four holes to the square centimetre reads as a texture and eleven reads as a mesh, at the same open area and the same yarn — so the whole visual effect the construction is bought for is a redistribution that no measurement of openness records.

That gives the construction a clean description. A mock leno converts a fabric’s open area from many small holes into few large ones, at no cost in area and at a cost in float. Everything it buys and everything it spends is in the redistribution.

It also says which specifications the construction is safe against and which it is not. A requirement on air permeability is nearly indifferent to bundling, because permeability reads a mean over the holes and the mean has barely moved. A requirement on opening size is entirely sensitive to it. So a mock leno substituted for a plain weave of the same cover will pass an air test and fail a bead test, and the two tests were both being used as proxies for openness.

The air test is not quite indifferent, and the direction it moves in is the one the slot arithmetic gives: a fixed open area in fewer, larger holes passes more, because what a channel carries rises faster than its area. So bundling raises the permeability slightly while leaving the open area exactly alone — which means a mock leno reads as marginally more open than its cover says on the one test, and enormously more open on the other, and neither reading is the area anybody thought was being measured.

Which is a good deal more useful than it sounds, because a construction whose three openness measures move by three different amounts is a calibration object: it separates instruments that would otherwise agree on every ordinary cloth.

All three numbers describe the same fabric and only one of them is unchanged. That is as clean a demonstration as this collection has that openness is not a single quantity, and it is available on a construction anybody can weave on a plain-weave loom.

Where the model stops #

Nothing here is a force. The threads within a bundle are assumed to close completely and the gap to open completely, and what actually decides how far they move is the friction between yarns and the residual tension in the cloth. This site has a capstan model for exactly that quantity and it is not used here, because the mock leno’s geometry has no wrap angle in it — the threads are parallel, not crossing, which is the whole point.

The bundle is treated as rigid. In a real cloth the members of a bundle bow apart slightly in the middle of a float and touch only near the interlacings, so a hole is a rounded square rather than a square and is a little smaller than the arithmetic says.

And the finishing is missing. A mock leno is usually finished with some treatment that sets the threads where they are — a resin, a heat set on a synthetic, or simply enough shrinkage to lock them — and how far that fixes the metastable arrangement is not geometry. The essay’s central claim about stability is qualitative for exactly that reason: a resin-finished mock leno is more stable than an unfinished one and it is still not a leno.

Who found it, and when #

Mock leno is a nineteenth-century mill construction and appears in the pattern books as a group of “imitation gauze” weaves, given as drafts with the note that they produce an open effect on plain-weave machinery. The economic point is the whole of it: a leno needs a doup harness, which is an extra mechanism, extra breakages and a slower loom, and a mock leno needs a threading.

That the imitation works at all is a fact about the eye rather than about cloth. At arm’s length an opening is an opening, and the two fabrics have openings of a similar size and a similar count. The difference the eye cannot see is the one the arithmetic above makes visible: one of them holds its openings by geometry and the other by nothing in particular.

There is one place the trade has always known the difference. Filter cloths and reinforcing scrims are specified by aperture, and neither is ever mock leno. Nobody wrote down why; the fabrics that closed up in service settled it.

Where the ladder goes next #

Three rungs of this anchor have taken the same object — the float — and asked it for three different things: a gradient that buckles, a scatter that hides a line, and a group that leaves a gap. The fourth thing a float does is the one the loom cares about and no rung of this anchor has reached: a long float is a long unsupported length in the shed, and the limit past which a weft cannot be inserted cleanly is a machine number rather than a geometric one. It is recorded here as not done.