A hole is a channel, not an opening

Hold a scrim up to a window and the light comes through in a grid of small bright squares. Count them, measure one, and the arithmetic is immediate: the squares are the spacing between the threads less the threads’ own width, they are all the same size because a weave is a repeat, and the fraction of the surface they occupy is one minus the cover. Every one of those statements is true, and this collection has been making them since the hole between four threads was first drawn. The last of them is the one that has done the most work: it is why a filter cloth can be specified by two numbers, and why holding a sheet to the light measures its cover and not its thread count.

Every one of them is also a statement about a projection. What a person at the window is looking at is the cloth’s shadow: the region of the plane that no thread covers, seen from one direction, with the third dimension flattened out of it.

Cloth has a third dimension. A muslin is a third of a millimetre thick and its holes are a quarter of a millimetre across, so the passage through one of them is very nearly as long as it is wide — a short tube rather than a window. And a tube has a profile: a clear cross-section at every height, which need not be the same at every height, and whose smallest value is what a thing going through has to fit.

The claim #

The opening a cloth shows and the passage a cloth has are two different quantities, the second is never smaller than the first, and how far apart they are is decided by the weave.

The first is a property of the plan: the set of points no thread covers when the cloth is seen from directly above. The second is a property of the solid: the narrowest cross-section anywhere along the channel between the four threads that bound a hole.

They cannot be the wrong way round. The straight-through region is the intersection of the clear region at every height, and an intersection is contained in each of the things intersected, so whatever can be seen through can certainly be passed. The interesting direction is the other one, and there the gap can be considerable.

The argument #

A thread of diameter d whose axis sits at height z₀ occupies, at height z, a half-width of √((d/2)² − (z − z₀)²) on each side of its axis — the chord of a circle at that height — and nothing at all once |z − z₀| passes d/2. So the clear width across a gap of spacing p bounded by two such threads is p less the two half-widths, and that comes to p − d only where both threads are cut through their own axes.

Everything follows from asking where the axes are, and the answer comes out of the matrix.

A hole in the repeat sits between two adjacent ends and two adjacent picks. Ask one of those ends where it is at that point. If it passes over both of the picks bounding the hole, it is floating on the face and its axis is at the top of its own swing. If it passes under both, it is at the bottom. If it passes over one and under the other, it is on its way between them, and at the midpoint — which is where the hole is — it is at the middle.

That is three states, it is read straight off the weave matrix, and it is the whole of what the weave contributes. It says nothing about how far the top of the swing is, which is Peirce’s crimp height and comes from the geometry that gives a cloth its thickness.

A plain weave alternates at every pick. So every end, at every gap, is transiting: it is at the middle. So is every pick. So the four threads round every hole in a plain weave are level in pairs, they are all cut through their axes at the mid-plane at once, and the narrowest section is there and equals the projection. Not approximately — exactly, and for every hole in the cloth.

Any weave with a float does not. A float lays an end on the face across two or more picks, so at a gap inside a float that end’s axis is at the top of the cloth and not the middle. Its neighbour may be at the bottom. At the height where one of them presents its full diameter the other presents very little, and the sum of the two intrusions is less than d. The hole is wider than the projection says, at every height, and its waist — still at the mid-plane, by symmetry — is wider too.

Where the waist is, and why the answer is not obvious #

The clear cross-section is a rectangle: a width across the ends and a length along the picks, each governed by its own pair of threads. What has to be minimised is the product, and the product’s minimum need not be at either factor’s minimum.

Where the ends are level and the picks are level, both factors bottom out at the mid-plane and so does the product. Where the ends are level and the picks are not — which happens in a twill wherever an end transits between two picks that are themselves both floating — one factor is narrowest at the middle and the other at a height a little off it, and the waist lands between the two.

So the waist is computed rather than assumed: the profile is sampled through the whole depth the threads occupy, and the narrowest section is found by looking. The sampling is honest about itself. A sample lands within half a step of an axis rather than on it, and a chord half a step from the centre of a circle is shorter than the diameter by at most Δz²/8r, so every equality asserted about these numbers is asserted to that tolerance and not to zero. The tolerance travels with the answer instead of being chosen by whoever reads it.

What was counted, and how #

A repeat has ends × picks holes and those are all the holes there are — the sentence that used to justify computing one hole, used here to justify a finite census instead. Every one of them is built, its four threads asked their levels, its profile sampled, and its waist recorded.

At a muslin’s construction — 24 ends and 22 picks per centimetre in 20 tex cotton, threads 167 µm across — the census gives:

Every row has the same cover, the same open area and the same quoted opening — that equality is asserted to twelve decimal places while the table is built, because a table in which the constructions had drifted would be comparing nothing. And no two rows have the same answer.

The excess saturates. Every twill and every satin gives the same 274.3 µm, which is 9.89 per cent over, and the basket gives more. That is not an accident of the sample: the excess is set by how far apart the levels of two adjacent threads can be, and once a weave has any pair of neighbours at opposite extremes it has all the excess that geometry allows. A twill’s step of one moves each end by one pick, so at most one of the two ends round a gap is at an extreme and the other is transiting. A basket takes its threads in pairs, so both can be at extremes at once, and it is the basket that gets the larger number.

The count of distinct sizes is not ordered by float length. A 4/4 twill has four sizes and an eight-end satin has two, though the satin’s float is twice as long. What decides it is how many distinct level patterns a repeat presents, which is a question about the arrangement of the turns and not about their spacing — the same distinction the fold census met when it found that where in a repeat a fold lands is a question about a matrix, and the same one that separates a satin’s scattered turns from a twill’s aligned ones.

The two numbers a specification confuses #

A woven filter cloth is sold on an opening size, and the opening size is measured by the projection — optically, or by the largest glass bead that reliably passes, which are two ways of asking the plan the same question.

The bead is the interesting one, because a bead is not a line of sight. A rigid sphere entering the top of a channel can move sideways as it descends, so what stops it is the narrowest place along its path rather than the narrowest place seen from outside. In every weave but the plain weave, those are different, and the difference is in the direction that lets the larger bead through.

That is not an argument that opening-size measurements are wrong. It is an argument that two measurements which agree on a plain weave will disagree on a twill, and that the disagreement is calculable in advance from the draft.

Why the basket is the widest, and which weaves join it #

The census gives every twill and every satin the same 9.89 per cent excess and gives the basket fifteen. That difference has a structural cause, and once it is named the weaves that share it can be listed without running the census on them.

The excess comes from the two ends bounding a gap not presenting their full diameters at the same height. In a shift-rule weave with a step of one, adjacent ends are one pick out of phase, so at any gap at most one of them is at an extreme and the other is transiting — one full diameter, one reduced. That is the 9.89 per cent, and it is the same for every twill and every satin because every one of them has a step of one.

The basket does better because its adjacent ends are identical. A 2/2 basket takes its ends in pairs, so the two ends bounding a gap inside a pair are at the same level — both floating over, or both under. Neither is cut through its axis at the mid-plane, both present a reduced chord, and the passage is wider still.

So the condition is exact and it is one this collection already computes for another purpose:

the widest passages belong to gaps between two ends that share a shaft.

A shaft is a distinct column, so two ends share one exactly when their columns are identical — and identical adjacent columns are precisely what produces the maximum excess. The list of weaves with that property is therefore the list of weaves whose threading has duplicated adjacent ends, and this collection has met it twice before under other names.

Baskets, hopsacks and every doubled weave, by construction: taking the ends in pairs is what a basket is.

Ribs and cords, which double the ends to make a raised line.

And a point-reversal herringbone, which duplicates exactly two ends per repeat at its seams — so a cracked herringbone has two gaps per repeat with the basket’s wide passage in them and the rest at the twill’s.

That last is a genuinely new statement about a familiar defect. A cracked line is a line of wide holes as well as a line of doubled thread, and in a filter or a screen cloth it is a row of oversize passages running down the piece — which is a much more serious fault than a visible line and is not what anybody inspects for.

The design rule is short and it is checkable off a threading draft rather than off a cloth: a fabric whose passage must match its rated opening needs no two adjacent ends on one shaft. A plain weave satisfies it trivially; a twill and a satin satisfy it; a basket, a rib and a point herringbone do not.

Where the model stops #

The level rule is this collection’s own and is the weakest thing here. An end’s axis is placed at the top of its swing where it floats over, at the bottom where it floats under, and at the middle where it transits, with Peirce’s plain-weave amplitude used throughout. That is the approximation this site has recorded against itself before, under the heading that the wrap angle is a plain weave’s in every weave: a satin’s crimp is genuinely smaller than a plain weave’s at the same sett, so its threads do not swing as far and its excess should be smaller than the table gives. The ordering is what the census claims and the ordering does not depend on the amplitude; the individual numbers do.

The threads are round and uniform along their length. Peirce’s circular section is what supplies every diameter here, and a real thread is flatter where it crosses than where it does not — which this collection computes separately and which would change the chord at every height. A flattened thread intrudes further at the mid-plane and less at the extremes, so it should widen the excess rather than close it.

Nothing here has moved. The whole census assumes the ends are where the reed put them, which is a large assumption in an open cloth and is taken up separately.

And the sphere is rigid. A flexible particle, a fibre, an aggregate of fine ones — none of them is a bead, and for those the channel’s narrowest section is a poor description of what stops them. The claim is about the geometry, and the geometry is the part a specification is written in.

The generalisation #

The shape of this is not about textiles at all.

A silhouette is an intersection and a passage is a path, and a structure with any depth to it distinguishes them. Whenever the thing being characterised is a solid rather than a plane, a measurement made by looking through it returns the intersection over all depths, and a measurement made by pushing something through returns the minimum over one path. The two agree exactly when the obstructions all lie in one plane — which is what a plain weave arranges, by alternating, and what almost nothing else arranges by accident. It is worth setting beside the other place on this site where the plain weave turns out to be the extreme rather than the average: it is the firmest interlacing there is, for the same reason, that it turns at every opportunity.

The everyday version is a sieve. A wire mesh is a plain weave of wire and its aperture is honestly its projection, which is why sieve analysis works and why it is done with woven wire rather than with a knitted or twilled one. The choice looks like tradition and is a theorem.

The second half generalises differently. A quantity computed once and reused because “every one of them is the same” carries a hidden assumption about symmetry, and the assumption is usually true of the simplest member of a family and false of the rest. It fails quietly, because the computation still returns a number.

Who found it, and when #

Peirce’s geometry, from 1937, supplies the heights and the diameters, and the clear opening as spacing-less-diameter is older than that — it is how a reed is set out and how a bolting cloth has been specified since silk was used for it.

Extending the section model past a plain weave is standard in the textile-mechanics literature, and the usual extension is the same one used here: keep the arc-and-straight construction and let the thread stay at its level through a float. Nothing in that is new.

What appears to belong to this collection is the question, which nobody seems to have had a reason to ask. An opening size is a specification and a specification is a single number, so the machinery for computing one has never been asked what the distribution looks like, and the distribution turns out to be where the weave lives.

Where the ladder goes next #

The next rung takes the last column of the table seriously and asks which weaves have holes of one size. The answer is a short list with one entry: only a plain weave has one size of hole, and the reason is exactly the alternation that makes its threads level in pairs.

Sideways, the two statistics of the distribution turn out to belong to different questions — a filter is rated by the hole it does not show, while what a cloth passes in bulk is governed by the mean — and the same channel, asked about light instead of about beads, gives up the covering rule entirely: one minus the cover is the openness of a cloth with no thickness.