A bouclé loop is an elastica, not a semicircle

The account of a bouclé that put its crimp in the wrong thread needed a size for the loops, and took the simplest loop there is. An effect thread fed eighty per cent faster than its core has surplus to lose; a semicircle of radius r holds r(π − 2) more thread than the base it stands on; so the radius is the overfeed times the loop spacing over π − 2. On loops every three millimetres that is 2.10 millimetres at eighty per cent and 0.79 at thirty. The account of how a bouclé cloth is set built its whole jamming bracket on that radius.

Both essays said the semicircle was wrong, and both guessed which way. A real loop is a teardrop, the reasoning ran, a teardrop spends more thread per loop, and so the semicircle overstates the loop. The guess is half right, and the half that is wrong is at the small overfeeds, where it is wrong by more than a factor of two.

A loop’s shape is not a choice anybody makes. It is a length of thread held at two binder points, leaving the core along the core and arriving at the next binder point along it again, and between two such points a thread of a given length takes one shape: the elastica, the curve of least bending energy with those ends. Solved, that loop is taller than the semicircle at every overfeed below 66 per cent and shorter at every overfeed above. At thirty per cent it stands 1.10 millimetres, not 0.79; at eighty, 1.95, not 2.10; at three hundred, 4.83, not 7.9. Its sides first stand vertical at an overfeed of 119 per cent, its neck closes at 559, and nothing in any of those numbers comes from the fibre.

A semicircle meets its core at a right angle #

Two things are wrong with the semicircle before any mechanics is done, and both can be seen by drawing it on its core.

The first is its feet. A semicircle leaves its base at a right angle and arrives at a right angle. A binder presses the effect thread against the core, so at each binder point the effect thread lies along the core, and to turn from lying along the core to pointing straight up in no length at all is a corner. A corner in a thread with any bending stiffness at all costs an unbounded energy, so no thread makes one. The semicircle is an elastica — a thread of length πR between two points 2R apart comes out as exactly that semicircle when its ends are required to point straight up — but it is the elastica for ends held perpendicular to their base, and a binder does not hold anything that way.

The second is its base. The semicircle’s feet are 2r apart, and the only places on the yarn that hold the effect thread down are the binder points a whole spacing apart. Between a semicircle’s foot and the next binder point the effect thread would have to lie flat along the core with nothing pressing it there. And once the overfeed passes (π − 2)/2, which is 57 per cent, the semicircle’s base is longer than the spacing it stands in. At eighty per cent each loop is 4.20 millimetres across on a three-millimetre pitch.

One number sets the shape, and the fibre is not it #

Take the effect thread between two neighbouring binder points. It is the spacing times one plus the overfeed long — 5.4 millimetres of thread across a three-millimetre gap at eighty per cent — and it leaves one binder point and reaches the next pointing along the core. That is a complete elastica problem, and it contains only three quantities: the thread’s length, the gap, and the thread’s bending stiffness.

The stiffness drops out of the shape. An elastica’s energy is the stiffness times the sum of the curvature squared along it, so multiplying the stiffness multiplies every candidate shape’s energy by the same factor and leaves the cheapest one cheapest. Five times the stiffness gives the same loop to nine decimal places. The gap drops out too, because curvature is one over a length: a loop on a five-millimetre spacing is the loop on a three-millimetre spacing enlarged. The eighty-per-cent loop stands 0.65 of its spacing whatever that spacing is — 1.30 millimetres on two, 1.95 on three, 3.25 on five.

So a bouclé’s free loop is fixed by one ratio, the thread over the gap, and scaled by one length, the gap. A stiff effect thread and a soft one, a wool, a mohair and a polyester, make the same loop at the same overfeed on the same spacing. That a yarn’s bending stiffness is a bracket five hundred times wide is no obstacle to knowing the shape of its loop, because the shape never asks.

The loop is also unlike the other curved threads whose shapes can be solved. A woven thread has no room to bend: its whole crimp is spent wrapping the thread it crosses. A knitted loop is nine tenths free run, its contacts confined to the rest. A bouclé loop is all free run — nothing inside it but air, nothing touching it between its binder points — which makes it the purest elastica any yarn contains.

That any two of loop size, loop spacing and overfeed fix the third survives; the relation between them is simply not the straight line the semicircle drew.

A shallow loop stands up faster than its surplus #

A loop with little surplus is a buckled strut: a thread clamped along the core at both ends and made a little longer than its gap, rising in one smooth wave. For a shallow wave the extra length goes as the square of the height, so the height goes as the square root of the extra length — $\tfrac{2}{\pi}\sqrt{v}$ of the spacing, for an overfeed $v$. It is the same square root that ties a gently crimped thread’s wave height to its crimp, which is a fitting thing to find in a yarn whose loops are crimp in the wrong thread.

The semicircle’s radius goes as v itself, and a square root beats a straight line near zero. The smaller the overfeed, the worse the semicircle does. At ten per cent it gives 0.26 millimetres and the loop stands 0.62, 2.3 times as tall. At twenty per cent the loop is 1.7 times the semicircle, at thirty 1.4, at fifty 1.1. The strut’s formula is itself within two per cent of the solved loop at ten per cent and about five per cent low at thirty, and falls away above that as the loop stops being shallow; at eighty per cent it is twelve per cent low.

A small overfeed buys much more loop than the semicircle allowed. The thirty-per-cent yarn, whose loops were computed at a quarter of their spacing and called a gimp, has loops over a third of their spacing — 1.10 millimetres on three — and they are bells, their feet running out along the core and their tops round. A designer choosing a small overfeed to keep a yarn quiet gets a louder yarn than the semicircle promised.

The semicircle is right once, at two thirds #

The two heights cross at an overfeed of 65.9 per cent, where both stand 1.73 millimetres on a three-millimetre spacing. It is an agreement of one number and nothing else. The semicircle there is 3.46 millimetres across on a three-millimetre pitch, already past fitting, and meets the core at right angles; the loop is a bell whose feet leave along the core and whose sides steepen only partway up.

Above the crossing the semicircle overstates the loop, and by more the larger the overfeed. At eighty per cent it is eight per cent too tall, at 150 per cent a third too tall, at three hundred 63 per cent and at five hundred 81 per cent. The reason is where each shape puts its extra thread. A semicircle has one dimension, so all its extra length goes into its radius and therefore into its height. An elastica past a certain surplus stops rising efficiently and starts to fatten: the extra length goes into the width of a bulb that swells sideways over its own feet, and height grows only in proportion.

That is why the solved loop’s height, which rises as a square root at small overfeeds, settles into a straight line at large ones with less than half the semicircle’s slope. The semicircle is too short for shallow loops because it cannot flare, and too tall for deep ones because it cannot bulge.

At 119 per cent a loop starts to overhang #

At eighty per cent the loop’s steepest point leans at 80 degrees from the core: a bell with nearly upright sides. At an overfeed of 118.8 per cent the steepest point reaches 90 degrees, and past that it goes beyond — the loop’s sides lean back over its own feet and the loop becomes a teardrop, a bulb wider than its waist. The thread in the loop is then 2.19 times its gap, and like everything about the loop’s shape that ratio holds on any spacing and for any fibre.

That teardrop is the shape both earlier essays reached for, and it arrives only well above the overfeed they were describing. The eighty-per-cent bouclé they priced has bell loops, taller than a semicircle’s base is wide but not yet hooked. The teardrop belongs to overfeeds well beyond eighty per cent, where the loops stand 2.92 millimetres on three at 150 per cent, 4.83 at three hundred and 7.25 at five hundred.

An overhang matters for more than looks. A bell’s whole outline is visible from above, and anything pressing on it meets its top first. A teardrop has thread hidden under its own bulb, a space beneath the overhang where a neighbouring yarn’s loop or a crossing pick can lie, and a surface of teardrops catches on what is drawn across it in a way a surface of bells does not. Where a loop sits on that line — bell or hook — is a question the overfeed and the binder settle together, and the semicircle could not ask it.

At 559 per cent the neck closes and the neighbours meet #

As the surplus keeps growing the bulb widens and the neck under it narrows. At three hundred per cent the neck is a third of the spacing wide and the bulb two thirds; at four hundred the neck is a fifth and the bulb four fifths; at five hundred the neck is under a thirteenth and the bulb over nine tenths. At 559 per cent the two sides of the neck touch.

At the same overfeed, to within a thousandth of the spacing, the bulb becomes exactly as wide as the spacing. A loop closes its own neck and touches both its neighbours at the same moment. A row of loops in one plane therefore has a hard ceiling: past 559 per cent the loops cannot lie side by side in that plane without passing through themselves and each other, and a real yarn fed faster than that must turn its loops out of the plane, lay them over one another or wrap them round the core.

The line has no thickness in that account, and a thread has. A thirty-tex effect thread is about a fifth of a millimetre across, and on three-millimetre spacing the five-hundred-per-cent loop’s neck is already 0.23 millimetres, so a real thread’s neck closes near five hundred per cent rather than 559. The ceiling is a property of the geometry; the thread’s own diameter brings it lower.

A binder that pulls the feet in makes a shorter loop overhang #

Everything so far stands a loop on the whole spacing: the binder holds the effect thread at one point and at the next, and all the thread between is loop. A binder wound tightly may instead pull each loop’s feet closer together, leaving part of every spacing with the effect thread lying along the core. The loop then holds its own base plus the whole surplus, and its thread over its base is one plus the overfeed divided by the base’s share of the spacing. Pulling the base in takes thread out of the loop and raises that ratio at the same time.

On the eighty-per-cent yarn on three millimetres: with its base the whole spacing the loop is a bell 1.95 millimetres tall; at four fifths of the spacing, a bell 1.79 tall; at two thirds, 1.68 tall and overhanging; at two fifths, 1.43; at a quarter, a narrow teardrop 1.27 tall. The loop starts to overhang once its base is under 67 per cent of the spacing and closes its neck once its base is under 14 per cent.

The same overfeed makes a tall bell or a short teardrop according to how the binder holds it, and the loops grow shorter as they grow more hooked. The semicircle cannot see this at all: its radius is the overfeed times the spacing over π − 2, whatever the binder does. A specification that names a bouclé’s overfeed and loop spacing has fixed the thread in each loop and left the loop’s shape and height to a setting it does not name.

The bracket widens at low overfeed and narrows at high #

A bouclé cloth’s jamming bracket runs from the sett at which its loops touch to the sett at which its counts touch, and its width is the loop envelope over the count’s diameter. The envelope is the core plus a loop on either side, so the semicircle’s radius entered it directly, and the solved height replaces it.

At eighty per cent the change is small. The envelope is 4.12 millimetres rather than 4.43, the bracket is 11.7 wide rather than 12.6, and the open end of the bracket is 1.21 ends a centimetre rather than 1.13, at 21.6 grams a square metre rather than 20.1 and with 4.3 per cent of the cloth’s area under yarn by count rather than 4.0. A seven per cent correction at the one overfeed that was drawn is why nothing about that yarn looked wrong.

At thirty per cent it is not small. The envelope is 2.43 millimetres rather than 1.80, the bracket 7.6 wide rather than 5.6, and the open end 2.06 ends a centimetre rather than 2.78, at 30.5 grams a square metre rather than 41. At 160 per cent it runs the other way: an envelope of 6.33 rather than 8.63, a bracket of 16.0 rather than 21.7, an open end at 17.8 grams a square metre rather than 13.1.

The bracket still widens with every step of overfeed, but not in proportion to it. Doubling the overfeed from eighty to 160 per cent widens it by 37 per cent rather than the 73 the semicircle gave, and lightens the open end by eighteen per cent rather than by a third. A designer reaching for a much more open cloth by doubling the loops’ surplus gets a little over a third more room, because the loop spends its extra thread on its bulb.

The reading from weight hardly moves #

The bracket’s ends move; the reading of a real cloth from its weight barely does. An eighty-per-cent bouclé cloth of 100 grams a square metre has 5.62 ends a centimetre and its yarns touch at 0.89 millimetres, as before — that part comes from the count and the weight, and the loop’s shape does not enter it. On the logarithmic scale from the count end to the envelope end the cloth now sits 37.7 per cent of the way up rather than 37, and its loops are pressed to 17 per cent of their free height rather than 16. Both still round to the same words: about a third of the way up the bracket, loops pressed to a sixth.

The reason is that the envelope enters the position only through a logarithm, and a seven per cent change in a number twelve times the count’s diameter moves its logarithm by three per cent. At thirty per cent overfeed, where the envelope moved by a third, the same weight places the cloth 41 per cent up with its loops at 24 per cent of their free height; at 160 per cent, 38 per cent up with its loops at fifteen.

What the loop’s shape does change is the reference: the free height a cloth’s pressing is measured against, and the open end of the bracket, the lightest cloth the yarn makes while still jammed.

The stiffness is in the push, not the shape #

A loop pushes on its binder points. A thread clamped at both ends and barely buckled pushes with $4\pi^2$ times its stiffness over its gap squared, 39.5 in those units; the loop at ten per cent pushes with 34.2, at thirty 26.5, at eighty 15.8, at two hundred 6.75 and at five hundred 1.97. A taller loop is a softer spring. And the push is the one place the fibre’s stiffness appears: an effect thread five times as stiff makes the identical loop and pushes five times as hard, to six decimal places.

On a yarn of equal loops each binder point is pushed equally from both sides, and the two pushes cancel. The push is a compression locked into the effect thread, carried from loop to loop through every binder point, and it loads neither the core nor the binder. A free bouclé therefore carries no mark of its effect fibre anywhere in its geometry. The stiffness shows only when the loops are disturbed unevenly — a cut end that releases a row, a loop pressed by its neighbour and not by the next — and that is exactly what weaving the yarn into a cloth does.

The size a snarl coils to makes the same move: the bending rigidity cancels and leaves the twist and a ratio of stiffnesses. Here it leaves the thread’s length over its gap. In both, the stiffness hardest to measure is not needed for the shape and is still needed for the force.

What was solved, and how #

Each loop is the elastica between two points one gap apart, with the thread’s tangent along the line joining them at both ends and a length a stated multiple of the gap. The solver is the one used for a thread between two crossings, which describes the curve’s angle by sixteen modes and minimises the bending energy under the two end conditions and the length. A loop is started from a shallow sine wave whose chord matches its length, solved at every length ratio from 1.02 to 7.0 in steps of 0.04 with each solve started from its neighbour, and any other ratio is then solved from the nearest of those. Started from a straight line the solver has nothing to push against; started carelessly it can land on a shape with two humps, which no overfed yarn makes.

The loop’s height is its highest point; its overhang is its steepest tangent passing 90 degrees; its neck is the closest approach of its two sides away from the feet and the top, with a crossing test so that a loop that has passed through itself is not read as open. The overhang and neck thresholds were found by bisection. With a base pulled in, the loop holds its base plus the whole surplus.

Confirmed: the buckled strut’s height for a shallow loop, within two per cent; height rising and push positive at six ratios from 1.1 to 6; the stiffness absent from the shape to nine places and exactly proportional in the push; both thresholds inside stated bands; the semicircle below the loop at thirty per cent and above it at eighty; and an overhang when a binder halves the base. Loops with no surplus, no overfeed, no spacing or a base wider than the spacing are refused.

Where the model stops #

The loops lie in one plane, and all in the same plane. A binder is wound helically, so a real bouclé’s loops stand at different angles round the core and neighbours rarely share a plane. The ceiling at 559 per cent, where neighbours meet, is a ceiling for loops in a row; loops fanned round the core meet later, and only their own necks close on schedule.

The feet are held exactly along the core. A binder wrap has thickness and some give, so a real foot can tilt a little off the core, and a loop with tilting feet is a different elastica. A shallow loop’s height comes out the same to first order whether its feet are clamped or free to turn; the overhang and the neck closure are not computed for the second case, and nothing says they sit at the same overfeeds.

The thread is a line. Its own diameter brings the neck closure lower, to about five hundred per cent for a thirty-tex effect on three-millimetre spacing, and at every overfeed it makes the envelope a thread’s diameter wider than the centreline’s.

The thread has no set and no twist. A loop formed and then steamed takes its shape as its natural curvature, which is what a loop that is set and not sprung does in a knit, and then its shape is whatever it was set in rather than the elastica. A twisted effect thread carries torque, and torque turns a loop out of its plane toward a coil.

The binder’s grip on a loop’s base is given, not solved. How far a binder pulls each loop’s feet in depends on its tension and its wrap, and the base fraction here is a parameter the model accepts rather than a number it predicts.

Still open: how hard a loop resists its neighbour #

The free loop’s shape is now known, and it carries none of the fibre. The fibre enters when a loop is pressed — and a bouclé cloth sits somewhere in its bracket precisely because its loops have been pressed by their neighbours, to about a sixth of their free height in a hundred-gram cloth of the eighty-per-cent yarn.

How hard a loop resists that is the elastica with a contact on it: a loop pushed down or sideways by a neighbouring yarn, its height reduced, its push on its binder points rising toward the flat limit. A knitted loop’s pressing force was solved that way from its own bending, and came to tens of millinewtons a stitch. The same solve for a bouclé loop would give a force per loop against a loop’s height, and with it a bracket in which a cloth’s position is predicted from its beat-up rather than read from its weight — the first place in this account where the fibre’s stiffness would finally matter.

Who worked it out #

The elastica is Euler’s, from 1744, and his drawings of the elastic curve’s forms already include the bell, the overhanging loop and the loop that closes on itself; the elastic-rod literature since has treated loops of this kind, with and without contact, many times over. The semicircle is the natural first estimate of a loop’s size, because it needs nothing but the surplus and the number π. Treating a fancy yarn’s loop as a clamped elastica, the threshold overfeeds, the effect of a binder pulling the base in and the correction to the jamming bracket were computed directly.