A honeycomb gets its cells in the wash

A honeycomb towel has cells in it. They are perhaps half a millimetre deep, they are what the towel is bought for, and they are not in the draft.

That last part is worth sitting with. The draft is a matrix of ones and zeros; a cell is a place where the cloth stands away from its own plane. No arrangement of ones and zeros contains a height. So the cells are a consequence of the draft, computed by some route the point paper does not show, and the first job is to find out which route.

The construction is a diamond, and the gradient is what it is for #

A herringbone reverses the threading and leaves the treadling running straight, so the twill line zigzags across the width. Reverse both and the zigzag closes into a diamond — and the interesting thing is not the shape.

Point a 1/3 twill at a reversal of five and the longest float goes from three to six. That is not an accident of this weave: reflecting a twill makes the run at the point twice as long, because a thread that was rising through a run now reaches the reversal and comes back down through the same run on the other side. Pointing a twill lengthens its longest float, always, and the generator asserts it rather than assuming it, because a diamond with no float gradient would have no relief and every amplitude computed from it would be zero for a reason nothing in the figure would show.

The gradient is what makes a relief weave. A region of long floats interlaces rarely; a region at the diamond’s boundary interlaces at nearly every crossing. Those are two different fabrics sharing a set of threads.

The obvious mechanism, and why it is wrong #

Every end of a warp comes off one beam. A beam is a barrel and it unwinds at one rate, so every end receives the same length of yarn per unit of cloth woven. An end that interlaces often consumes more of that length than an end that floats — so a draft whose float lengths vary across the width hands some of its ends a surplus, and the surplus has nowhere to go in the plane.

That is the first model this site tried and it is wrong for a diamond. The arithmetic says so plainly.

Take the honeycomb above and average the float length along each end over one repeat. The local floats run from one to six, a factor of six. The per-end averages run from 3.20 to 3.80 — a spread of nineteen per cent, and after the crimp arithmetic the take-up difference between the heaviest and lightest end is smaller still.

The reason is the diamond’s own geometry. Every end passes through the diamond’s centre and its boundary alike as the repeat goes down the cloth, so the gradient runs diagonally and averaging along an end runs across it. There is nothing much for a beam to notice.

Which is a claim with a consequence anybody in a mill can check, and it is the right way round: a honeycomb weaves from a single beam. A fabric whose take-up really does vary across the width cannot — a terry needs two, one for the ground and one for the pile, because the pile warp is fed several times as fast. The existence of the second beam is the trade’s own measurement of when take-up differs, and honeycomb does not get one.

The diamond is what hides the gradient from the beam #

The reason the take-up model fails is worth stating as a positive claim rather than as an arithmetic accident, because it says which relief weaves would have needed a second beam.

The gradient in a pointed twill runs along the diagonal, because the construction points in both directions at once. An end walks from the diamond’s boundary to its centre and back as the repeat goes down the cloth, so it meets every float length in the design and its average is very nearly everyone else’s. Pointing in both directions is precisely the operation that projects the gradient onto a direction the beam cannot see.

Point in the threading only and the picture changes completely. The long floats then lie in a band running the length of the piece, every end in that band floats and no end outside it does, and the per-end averages separate as far as the local float lengths do. That fabric genuinely would take up unevenly and genuinely would need two beams — and it is a warp stripe rather than a relief weave, which is what the trade calls it.

So the family divides on a structural criterion rather than on appearance. A relief effect whose gradient is warpwise is a two-beam fabric; a relief effect whose gradient is diagonal is a one-beam fabric that develops in the finishing. A terry is the first kind and says so with its second beam. A seersucker is the first kind and gets its second beam too. A honeycomb, a waffle and a huckaback are all the second kind, and all of them are pointed both ways — which nobody appears to have chosen for this reason, and which is the only arrangement that makes a single-beam relief weave possible.

Why the float does more by being wide than by being long #

The buckle amplitude is a half-wavelength times the square root of an excess, and the two factors are not equally powerful.

A longer float raises the excess, because a region that interlaces rarely wants to contract much less than the cloth around it does. It also widens the region, because the region is the float. The first enters under a square root and the second enters linearly, so most of what a longer float buys is width rather than surplus — doubling the float roughly doubles the half-wavelength and multiplies the amplitude by a good deal more than the extra shrinkage differential does on its own.

That has a design consequence which the table above obscures by moving both quantities together. A construction that widened the low-interlacing region without lengthening its floats — a longer twill repeat pointed at a larger reversal, say — would buy nearly the same relief and would keep the float short, which is the quantity everything else about the cloth is paying for. It would also be a much larger cell, which may or may not be wanted.

And it explains why the shallow end of the family is so shallow. A 2/2 diamond’s floats run from two to four, so it has both a small surplus and a narrow region, and the two shortfalls multiply. Eighty-five microns is not a sixth of the honeycomb’s relief because its gradient is a sixth as steep; it is a sixth because two independent factors are each down by rather less than that.

What is left is finishing #

Each region of the cloth has its own float length, so its own crimp, so its own natural relaxed size — and they are all held at one reed pitch and one pick rate on the loom.

Off the loom the cloth contracts once, by one amount. The regions that want to contract more than that are held open; the regions that want to contract less are compressed. Compressed cloth in the plane is cloth out of it, and that is the cell.

The claim has a consequence a reader can check without any arithmetic at all: a honeycomb comes off the loom nearly flat and gets its cells in the wash. So does a seersucker, so does a cloqué, and so does every blister fabric in the trade. Relief weaves are sold as woven structures and made in the finishing.

What was counted, and how #

The contraction of a region comes from the site’s own finishing model, run in an unfamiliar direction.

relaxed works forwards: hand it the relaxed spacings, because Peirce’s closure condition describes a cloth in equilibrium with itself and a cloth under load on a loom is not one, and it works out what the loom state must have been. Here the question comes the other way round — the loom set every region at the same pitch and the regions want different relaxed states — so the relaxed spacing is solved for by bisection until the loom state it implies is the one the loom actually had. The equations are unchanged and the residuals are the same; only the unknown has moved.

The float enters through one step, and it is the only modelling decision in the whole calculation. A warp end floating over four picks passes three of them without turning, so the distance between its bends is four weft spacings — and Peirce’s geometry, which knows nothing about weaves and everything about a thread turning round another at a stated spacing, answers that by being handed the longer spacing.

At sixteen threads per centimetre and a quarter-millimetre yarn, that gives a contraction per region of:

A factor of fourteen from the shortest float to the longest. The cloth as a whole contracts by 2.68 per cent, which is the area-weighted mean over the repeat, so the float-of-six regions are compressed by about two per cent of their own area and the float-of-one regions are held open by nearly six.

The buckle follows from the standard small-amplitude result: an excess length fraction ε over a half-wavelength L shows as an amplitude of (2L/π)√ε. Taking the half-wavelength as the region’s own float — which is how wide the region is — gives a peak relief of 171 microns on a cloth 500 microns thick, or about a third of a thickness.

For the waffle it is 242 microns, near half a thickness, from floats running to ten. For a huckaback of a 3/3 twill it is 115. For a 2/2 diamond it is 85 and the fabric is a pattern rather than a relief.

What the gradient costs, and what it buys #

A relief weave is bought for its surface and paid for in the two quantities the float has always decided.

The paying is firmness. Interlacings per intersection is the number behind how closely a cloth can be set and how well it resists a thread sliding, and pointing a twill lowers it: the 1/3 twill interlaces half its crossings, the honeycomb made from it 0.400 of them, and the waffle 0.278. A relief weave is a loose cloth by construction, and it is loose in exactly the regions that stand proud — which are also the regions a finger touches.

The buying is surface. A cloth with cells has more area than its own footprint and holds air in the hollows, which is why the honeycomb is a towel weave and not a shirting weave: what a towel does is hold water against a large area of yarn, and a flat cloth of the same threads holds less of both. The same long floats also make the fabric raisable, because a raising wire can only catch a thread that is not held down — so the relief weaves sit next to the pile weaves in a mill for a structural reason rather than a commercial one.

The float limit is the other half of the trade-off and it is the usual one. A honeycomb’s cell depth rises with its longest float; so does its tendency to snag and its rate of abrasion loss, because both of those are the same length of unsupported thread measured for a different purpose. A designer choosing between a honeycomb and a waffle is choosing where on that line to stand, and the arithmetic above gives both ends of it in the same units.

The construction can fall apart, and it does #

A diamond is not a safe operation. Point a 4/4 twill at a reversal of four and the result is a perfectly reasonable-looking draft that describes sixteen separate cloths.

The reason is the same reason it makes a good relief: pointing lengthens the floats, and floats long enough that whole groups of threads never interlace with whole other groups are floats long enough to break the cloth into pieces. The connectivity criterion catches it, and it is the only thing that does — nothing about the drawing betrays it, and the interlacing filter passes it because every end and every pick does interlace somewhere.

So the relief generator asserts the layer count before it computes anything, and asking it for a honeycomb of a 4/4 twill at that reversal is refused with the count it found. That refusal is not a courtesy. A relief weave is pushed towards long floats by the thing it is for, so it walks towards the failure by design, and a construction that walks towards a failure needs the check on rather than nearby.

Where the model stops #

The amplitude is an upper bound, not a prediction. The regions with a deficit cannot buckle inwards. They come under more tension and straighten, losing some crimp, and how much they straighten is a question about the yarn’s bending and transverse stiffness that this site has no model for. So the surplus computed here is larger than the surplus that actually reaches the surface.

One buckle per region is an assumption. The arithmetic puts the whole surplus into a single half-wave across the region’s own width, which is the lowest mode and therefore the cheapest — but which mode a real cloth takes is decided by bending stiffness against the in-plane stress, and that is the wrinkle calculation rather than this one.

Nothing here is anisotropic. The surplus is treated as an areal quantity and the buckle as one-dimensional, and a honeycomb cell is a two-dimensional dimple with a rim. The number computed is the depth a section through the middle would show.

And the finishing is a single stated relaxation. How much crimp the loom took out is a tension the model is given rather than one it computes, and a cloth finished harder has more relief. Every figure here says which tensions it used.

Who found it, and when #

Honeycomb weaves are old and anonymous, like most structures in the trade: they appear in nineteenth-century pattern books as a class of “cellular” or “spot” weaves with the construction given as a recipe — point the threading, point the treadling, use a long-float twill — and the mechanism given not at all.

The mechanism, when it is named, is usually named wrongly. The common explanation is that the long floats “stand proud” because they are unsupported, which has the surface behaviour right and the cause backwards: a float is not pushed up by anything, it is a region of cloth that failed to shrink as much as its neighbours and had to go somewhere.

The finishing half is better documented in the fabrics where it is impossible to miss. Seersucker is made by feeding two warps at different rates, so the slack warp puckers, and nobody calls that a weave effect. Cloqué and blister fabrics are made by shrinking one system of threads differentially, and the trade calls the process what it is. Honeycomb is the same phenomenon achieved without a second beam and without a shrinking yarn — by float length alone — which is why it gets described as a weave and why the finishing half goes unmentioned.

Where the ladder goes next #

The relief weaves use a float gradient to make a surface do something. The next rung uses the opposite: a draft designed so that no arrangement of its floats produces a visible line anywhere. It turns out to be impossible in an exact sense, and the amount of structure that cannot be removed is fixed before any design decision is made.