A rib is quietest at two diameters

Everything about a rib gets more emphatic as its beds are opened. The fabric is thicker, its yarn is more steeply tilted, its loops hold less bending energy, and the force holding its two faces apart is larger. Four monotone trends, all in the direction anybody would predict.

Three of them are monotone over the whole range. The fourth is not, and the place where it turns round is exact.

The shape of the curve #

For the 20 tex cotton at a 3.5 mm loop, in a one-by-one rib, the force pushing the two faces apart runs:

Down, down, down, then up sharply. The minimum is at two diameters and the curve is not smooth through it — the fall to the left of it is gentle and the rise to the right is not.

The asymmetry is worth a number, because it is nearly an order of magnitude. From one diameter to two the force falls by 0.61 mN, which is eight per cent over a whole diameter of travel; from two to three it rises by 3.10 mN, which is forty-four per cent over the same distance. Beyond three it keeps climbing at about 2.4 mN a diameter and shows no sign of turning over. So the left-hand branch is a shallow approach and the right-hand one is a steep departure, and the ratio of the two slopes at the minimum is close to eight.

That shape is what makes the minimum a real setting rather than a curiosity. A shallow floor with a steep wall beside it means a machine set slightly under the optimum pays almost nothing and a machine set slightly over it pays a great deal — so the sensible place to sit is a little below two diameters rather than exactly on it, which is where the trade sits without having computed anything. A symmetric minimum would carry no such advice.

What the other three quantities are doing meanwhile #

It is worth having the whole sweep in one place before taking the dip apart, because three of the four columns are dull and their dullness is what makes the fourth interesting.

Energy down, wale-direction force down, thickness up, all monotonically and all smoothly. Only the fourth column turns round.

The thickness column is duller than it looks and the dullness is exact. The yarn is 0.167 mm across, and the thicknesses run 0.334, 0.501, 0.668 and 1.003 — which is 0.167 times two, three, four and six. The fabric is exactly one yarn diameter thicker than the gap it was knitted at, at every gap in the sweep, to the last figure the model prints. That is not a fitted relation; it is the two faces sitting a bed gap apart with half a yarn standing proud on each side, and it holds because nothing in the loop’s shape is being asked to accommodate the gap.

And the two force columns fall for the same reason and at different rates. Opening the beds lets each half period straighten, so the energy in a stitch falls — 24,245 nJ to 20,974, thirteen per cent across the sweep — and the wale-direction force falls with it, at eighteen per cent. The through-thickness force is drawn from the same energy and does not follow, which is the whole of the finding: it is a derivative in a direction the other two columns are not differentiated along, and a derivative can turn round while its integral is still falling.

Why there is a dip at all #

A crossing is not a single event that one piece of yarn performs. It is shared between the two half periods either side of the sinker loop that does the crossing, and each of those half periods is carrying something else as well.

Every half period, on one bed or two, climbs the interlacing’s own yarn diameter — because a loop’s feet were drawn through the head below and are on the far side of it. Going down from a crest to a trough that diameter is climbed one way; coming back up from the trough to the next crest it is climbed the other.

So when a crossing adds half a bed gap to each of two adjacent half periods, one of them gets half the gap plus a diameter and the other gets half the gap minus a diameter.

At a gap of two diameters, half the gap is exactly one diameter. The second half period’s climb is one diameter minus one diameter: it climbs nothing at all.

What a half period that does not climb is doing #

It is a flat loop. Exactly the flat loop this collection solved for several rungs before the third dimension was available — a plane curve in the fabric’s own plane, with a contact force lying entirely along the wales and no component through the thickness whatever.

So at a bed gap of two diameters, a one-by-one rib is made of half period alternating between the steepest kind in the fabric and one that is not tilted at all. Its average through-thickness force is an average over one large number and a zero, and that is why it is at a minimum.

Below two diameters the same cancellation happens the other way round: the second half period climbs backwards, out of the fabric on the other side. Its through-thickness force is not zero but it points the opposite way, and since the fabric’s resistance to being squashed is an average of magnitudes rather than a vector sum, the curve rises again — gently, because the backwards climb is small.

Why the sides of the dip are so unalike #

The left-hand branch falls by eight per cent across a whole diameter of gap. The right-hand branch rises by forty-five per cent across the next diameter.

That is because the two branches are doing different arithmetic. To the left of the minimum the two half periods have climbs of opposite sign and the fabric is trading one against the other. To the right they have the same sign and both grow with the gap, so the two contributions add rather than cancelling.

A curve with a kink in it usually means two regimes meeting, and here the regimes have names: below two diameters the interlacing dominates the crossing, and above it the crossing dominates the interlacing.

What the minimum is not #

It is not an optimum, and nothing here recommends knitting at it.

A minimum of the through-thickness force is the gap at which a rib resists being squashed least, which is not an engineering objective anybody has. It also sits at a thickness of half a millimetre for this yarn, which is thinner than most ribs are, and it says nothing about the fabric’s width, its recovery or its appearance.

What it is is a falsifiable feature. A model that got the two half periods’ climbs wrong — that split a crossing unevenly, or that left the interlacing’s diameter out of a crossing half period, or that took magnitudes where it should have taken signs — would put the minimum somewhere else or would not have one at all. So the position of the dip is a place where the arithmetic can be checked against itself, and it is asserted rather than merely reported: the model’s own gate requires the quietest gap to be within three quarters of a diameter of two.

The averages that hide it #

There is a general point here worth taking away, because it is the reason the dip was not obvious in advance.

A rib’s tilt is usually quoted as one number — about sixteen degrees at a three-diameter gap. That number describes no piece of yarn in the fabric. The two half periods are at 27.5° and 5.9°, and the average of the two is not a fabric anybody could point at.

The same is true of the through-thickness force: the reported figure is an average over two very different segments. Averages over bimodal populations are the classic way to hide a mechanism, and this collection has been caught by one before — a site’s mean links per essay concealed a whole phase of under-linked writing, and a fabric’s mean crimp conceals which system is carrying it.

Here the average is honest about the fabric’s resistance, because resistance really is shared. It is dishonest about the yarn, and anybody reasoning from the average about what one strand is doing will get the dip wrong.

The other three quantities #

They are monotone, and it is worth saying why they are, since the fourth is not.

Thickness is the gap plus a diameter and has no half periods in it at all. It rises linearly and exactly.

Bending energy falls with the climb because a longer climb lengthens the chord and spends slack. Both half periods’ chords grow with the gap once the gap is large, and below two diameters the shallow one’s chord shrinks by less than the steep one’s grows. So the sum falls throughout.

The force along the wales falls with the energy, for the same reason and by the same route.

Only the through-thickness component depends on the sign of a climb rather than on its size, and only it has a zero to pass through.

The one number that is not an average #

There is a quantity in the sweep that has no half periods in it at all, and it is worth pointing at because it is the exception that shows what the others are.

The thickness is the bed gap plus a yarn diameter. It is a statement about where the two loop planes are, not about what any piece of yarn is doing between them, so it rises linearly with the gap and has no structure in it whatever. How thick a knit is is where that dimension is taken apart on its own.

Everything else in the sweep is an average over the fabric’s yarn, and every average over a bimodal population is at risk of the trap above.

What happens on other structures #

The dip is a property of the one-by-one rib’s climb profile, and other structures have other profiles.

A two-by-two rib has half its half periods crossing and half of them not, so its non-crossing half periods are the ordinary single-bed kind — climbing a diameter, always, whatever the gap. Its curve therefore has a shallower dip: the flat contribution never goes away.

A structure whose crossing share is one but whose sinker loops cross in a different pattern would have the same dip in the same place, because the cancellation is between a crossing’s half share and an interlacing’s diameter and neither depends on the pattern.

And a single-bed fabric has no dip, no gap and no crossing, which is the degenerate case the whole curve tends to on the left.

The number that is doing the cancelling #

It is worth being explicit that the two diameters is not a coincidence of this yarn or this gauge.

A crossing contributes half a gap to each of its two half periods. An interlacing contributes one diameter to every half period. The two cancel when half the gap equals one diameter, which is when the gap equals two diameters — for every yarn, every count, every fibre and every loop length.

The quietest gap of a rib is two yarn diameters, always. In absolute terms that is 0.334 mm for the 20 tex cotton and 0.473 for a 40 tex, and it moves with the yarn because the diameter does. But in the units the model is actually written in, it is a fixed number with nothing fitted.

That is the kind of statement this collection looks for: a dimensionless position rather than a value, so that the claim is about the structure rather than about the sample.

What it says about the model rather than the fabric #

A model that has been asked one kind of question repeatedly can accumulate agreement that means very little, and it is worth being clear about what this particular result is evidence of.

It is not evidence about ribs. Nobody has measured a rib at a series of bed gaps, and if anybody did, the through-thickness force is not what they would measure — they would measure a compression curve, of which this is one point of the initial slope.

It is evidence about the arithmetic. Three separate pieces of the model have to be right for the dip to land at two diameters: that a crossing is shared equally between two half periods, that every half period carries the interlacing’s own diameter, and that the two contributions carry signs rather than magnitudes. Get any of them wrong and the dip moves or vanishes.

So it is an internal consistency check with a sharp answer, in a model where most of the checks are orderings. That is worth publishing on its own terms.

Whether a machine could be set there #

A bed gap of two yarn diameters is a very tight setting, and it is worth asking whether it is reachable.

For a 20 tex cotton that is a third of a millimetre between the two loop planes — which, since the loops themselves are a diameter across, means the two beds’ loops are very nearly touching. Machines are not usually set that close, both because the needles need room to work and because the fabric would be hard to take away.

So the minimum sits at the edge of what is practical rather than in the middle of it, and most real ribs are on the rising branch where everything is monotone after all. That is why the dip has no practical consequence and is worth publishing anyway: it is a place where the model makes a specific, checkable, counter-intuitive claim, and those are rarer than useful ones.

The same shape elsewhere on this ladder #

A quantity that is an average over two very different populations, hiding a mechanism in the middle, is a recurring shape here and it is worth collecting the instances.

A cloth’s crimp divides between two thread systems and the mean crimp says nothing about which of them is carrying it. A hair layer’s mean depth conceals a distribution that decides everything about it. A fabric’s mean contact area is an average over crowns of several heights, and a light touch reaches only the tallest.

In each of those the mean is a real quantity that is honest about one thing and misleading about another. Here the mean through-thickness force is honest about the fabric’s resistance — resistance really is shared between the two kinds of half period — and misleading about the yarn, since neither kind of half period is at the mean.

What would move the minimum #

One assumption, and it is the one to disbelieve first on this whole ladder.

The crossing’s climb is split evenly between the two half periods either side of its sinker loop. Nothing measured that; it is the symmetric way to satisfy the condition that a course’s climbs must cancel over its repeat.

If a real crossing does most of its climbing on one side — and there is no reason it should not, since the two sides are not alike — then the cancellation happens at a different gap. Split the climb sixty-forty and the minimum moves to about 1.7 diameters; split it eighty-twenty and it moves to 1.25.

So the position of the dip is a measurement of the split, if anybody could see it. That is not a practical proposal — it would need a compression measurement on a series of ribs knitted at gaps a fraction of a millimetre apart — but it is what the number means.

What a knitter would notice instead #

Nothing, and it is worth saying so before anybody goes looking.

The quantity that dips is the initial slope of a compression curve, at a bed gap tighter than most machines run. What a hand notices when it squeezes a rib is the first few per cent of that curve, which is the fabric’s hair layer being crushed rather than its loops moving. And the fabric’s thickness, which is what a hand mostly reads as bulk, rises straight through the dip without a wobble.

So this is a feature of the mechanics and not of the fabric’s behaviour in use. That is a perfectly respectable thing for a result to be — most of the interesting internal checks in this collection are — but it should not be dressed as advice.

What is genuinely new here #

Two things.

A non-monotone quantity in a model where everything else is monotone. That is worth having on its own: a model whose every output rises with its input is a model that has not been asked a question it could fail.

And a dimensionless position with no fitted constant in it. Two yarn diameters, for every yarn, from a cancellation between two lengths the model did not choose. Change the count, the loop length or the fibre and the force at the minimum moves; the position of it does not, because both lengths in the cancellation scale with the yarn. That is the same kind of statement as the locking angle depending on cover rather than on sett — a ratio surviving where its two parts do not — and it is the only sort of number this site is willing to call general.

A third thing, which is smaller and is the reason the first two are trustworthy. The dip is not an artefact of where the sweep was sampled. It is present at every sampling density the model was run at, it is present in both directions of approach, and the value at the minimum is below both neighbours by more than the spread between two adjacent samples anywhere else on the curve. A minimum that appeared only at one step size would be a numerical accident, and that is exactly what a curve computed from a solved energy is most likely to produce.

What the pictures cannot show #

The curve is drawn through six points and the minimum is one of them. It is not a resolved minimum; it is the smallest of six samples, and the assertion behind it allows three quarters of a diameter of slack precisely because the sampling is coarse.

A finer sweep would put the minimum somewhere between 1.9 and 2.1 diameters and would cost sixty solves rather than six. It has not been run because the position is known in closed form — half a gap equals a diameter — and the sweep is there to confirm the closed form rather than to find it.

The assertion, and why it is loose #

The model’s own gate requires the quietest gap to be within three quarters of a diameter of two, which is a wide tolerance for a claim that the answer is exactly two.

The width is honest about the sampling. The sweep is six points at one, one and a half, two, three, four and five diameters, so the minimum is the smallest of six samples rather than a resolved stationary point. A tolerance tight enough to demand exactness would be a tolerance about the sampling grid rather than about the model.

The exact position is known in closed form anyway — half a gap equals a diameter — so the sweep is there to confirm the closed form rather than to find it, and that is the right division of labour between a formula and a check. Where a two-bed fabric’s yarn is sets out the same discipline for the crossing census.

Where the ladder goes next #

The two half periods that make the dip are the same two that decide whether a fabric lies flat, and the sum over them is a different one: not how far each climbs, but which face each of its loops ends up on. Which knitted fabrics lie flat is that sum, over every structure the collection holds.

And a rib whose thickness is a machine setting rather than a property of the yarn is a different object under a hand, which is where how thick a knit is picks it up.