A flattening that follows the tightness factor

A number computed once is an example. The flattening a knitted fabric’s own geometry demands came out at 0.780 for a twenty tex cotton at a three and a half millimetre loop, fully relaxed, and on its own that is an arithmetical fact about one set of inputs.

What makes it a result is that eighteen fabrics give eighteen numbers that lie on one curve.

What was varied #

Three things, chosen because they are the three a knitter actually changes.

Loop length, from 2.8 to 4.5 millimetres, which is the range an ordinary jersey is knitted over. Shortening the loop tightens the fabric.

Relaxation state, over the three the collection’s constants supply: dry-relaxed off the machine, wet-relaxed after washing, fully relaxed after washing and tumbling. The states move the fabric’s spacings by about ten per cent.

Count, from ten to forty tex, which is a factor of four in mass per unit length and a factor of two in diameter.

Eighteen fabrics in all. Each is solved independently — a relaxed loop from its own count and loop length, a three-dimensional segment from that, a course assembled from the segment, and a second course displaced by that fabric’s own course spacing.

What came out #

The flattening runs from 0.707 at the tightest to 0.839 at the slackest, and the ordering is monotone: a tighter fabric demands a flatter yarn.

That is the direction anybody would guess, and guessing it is not the point. The point is that the eighteen values are not eighteen numbers. Plotted against the tightness factor — the square root of the count over the loop length, which is the index the trade quotes — they fall on a single curve with no scatter that the sampling does not account for.

So a fabric’s demanded flattening is a function of one variable.

Why that is not obvious #

It is worth being clear that this is a result rather than a restatement, because there is a version of it that would be circular and this is not that version.

The circular version: the solved loop’s shape depends only on the ratio of the yarn’s diameter to the loop length, because Munden’s constants make the spacings proportional to the loop length. So any distance measured in diameters must be a function of that ratio, and the tightness factor is that ratio times a constant. That much is arithmetic and would have been true whatever the numbers came out at.

What the sweep establishes is the part arithmetic does not give: that the function is smooth and monotone over the range fabrics are actually made in, that the three relaxation states sit on the same curve rather than on three parallel ones, and that the values stay in a band narrow enough to be useful.

None of those was guaranteed. A quantity that is a function of one variable can still be a badly behaved function of it, and a family of curves that differ by state would have made the result much less usable.

The states, which are the interesting part #

The three relaxation states are the part of the sweep that is not forced, and they are worth looking at on their own.

A fabric moves from dry-relaxed to fully relaxed by shrinking about ten per cent in both plan directions at a fixed loop length. So its tightness factor does not change — the tightness factor contains the loop length and the count, neither of which moves — and its spacings do.

If the flattening depended only on the tightness factor in the strict sense, the three states would give identical answers. They do not: dry-relaxed reads 0.796 where fully relaxed reads 0.780, a difference of two per cent.

So the flattening depends on the tightness factor and on the state, weakly. The states form three nearly coincident curves rather than one, and the separation between them is about a fifth of the spread across the loop-length range.

That is a small and specific prediction, and it is the kind that is easy to get backwards: relaxing a fabric makes its yarn flatter, because relaxation brings the courses closer together at an unchanged loop length.

What a single curve buys #

The practical value is that a swept parameter becomes a looked-up one, and the difference shows most in comparisons.

Take two fabrics of different tightness and ask which is thicker. With a swept flattening, both calculations carry an independent band and the comparison is between two bands that overlap. With the curve, the two flattenings are linked — the tighter fabric is flatter by a known amount — and the comparison is between two numbers.

That matters because almost everything this collection does with a knitted fabric is a comparison. A rib against a jersey, a tight fabric against a slack one, one state against another. In every one of those, the flattening was previously an unknown that had to be assumed equal on both sides, and assuming it equal is exactly wrong when the two fabrics differ in tightness.

The fibre, which does nothing #

The fibre-independence deserves its own paragraph because it is the strongest form of the claim and the easiest to misread.

A wool and a cotton at the same tex are not the same diameter — wool is less dense, so a wool yarn of a given count is thicker. So the two have different tightness factors at the same loop length, and different demanded flattenings.

What the sweep says is that they lie on the same curve: match the tightness factor rather than the count, and the two demand the same flattening.

That is what makes this a structural result. Nothing about the fibre’s stiffness, its friction, its transverse rigidity or its ability to be squashed enters anywhere. The fabric’s arrangement demands a section of a certain extent, and what the yarn is made of decides only whether it can supply one.

Where the curve runs out #

Both ends of the range are worth marking, because they are where the prediction stops being a prediction.

At the tight end, past a tightness factor of about eighteen, the demanded flattening falls below 0.7 and the yarn is being asked to be more than a third flatter than round. At that point the constant-area ellipse is the wrong idealisation: a yarn squashed that hard loses air rather than changing shape, so the packing factor is moving and the diameter the whole calculation started from is no longer right.

At the slack end, below a tightness factor of about nine, the demanded flattening approaches one and the fabric no longer requires anything. That is not a failure — it is the correct answer, and it says that a very open knitted fabric’s yarn need not be flattened at all, which is why the courses of a slack fabric do not overlap either.

Between those, the curve is usable. Outside them it is arithmetic being extrapolated past its own assumptions, and the collection has done that before and recorded the cost.

Why the trade’s own index turns out to be the right one #

There is a small and satisfying result buried in the fibre-independence and it is worth surfacing.

The tightness factor — the square root of the count over the loop length — is a trade index. It was arrived at empirically, as a number that correlates with how a fabric handles, and it has been quoted at thirteen to seventeen for an ordinary cotton jersey for decades without anybody deriving it.

This collection derived it once already, and found it is the yarn’s diameter over its loop length divided through by the fibre density and the packing factor — which is the single ratio the whole knitted geometry depends on.

The flattening sweep is a second, independent confirmation of the same thing, and it is a stronger one. The first showed that the trade’s index is proportional to the model’s group. The second shows that a quantity nobody in the trade has ever measured — the flattening a fabric’s arrangement demands — is a function of that index and of nothing else.

An empirical index that predicts a quantity it was never fitted to is an index that is measuring something real.

The one variable it does not contain #

For completeness: the tightness factor contains the count and the loop length. It does not contain the machine gauge — how many needles per inch the fabric was knitted on — and that omission is worth noting because it is a trade preoccupation.

A jersey knitted on a fine gauge and a coarse one, at the same count and loop length, are the same fabric by every measure in this collection. That has been checked before: the fabric’s thickness, its dimensions and its forces are all independent of the gauge, which is one of the more falsifiable things the collection has said about a knit.

The flattening is the same. It contains no gauge and it should not, and if a measurement ever found a gauge dependence in a knitted fabric’s yarn section it would be a serious result — because it would mean the needles are doing something to the yarn that the loop geometry does not know about.

What was counted, and how #

Eighteen independent solves, each from the collection’s own machinery with nothing shared between them except the code.

The approach is measured point to point between the two sampled courses, at a hundred and twenty samples per half period, over two wales. The distance reported is the minimum over every pair.

The tightness factor is the square root of the count in tex over the loop length in centimetres, which is the trade’s own definition and is what the collection’s own tightness function returns.

The sweep is reported as computed rather than fitted: no curve was drawn through the points and then quoted. The claim is that the points lie on a curve, and the figure draws the points.

Where the model stops #

One flattening for a yarn that is not uniformly flattened. The number is a minimum over the whole course, and the yarn is pressed only where it crosses. That is a real limitation and it is the next question.

The loop is not re-solved. Every one of the eighteen solves uses the free loop shape. A flattened yarn is stiffer in one bending direction and softer in the other, so the loop it takes is not quite the loop solved here, and the second-order correction is not computed.

A section that holds its area. At the tight end of the range the assumption fails, and a yarn that is losing air rather than changing shape is one whose packing factor is moving.

Munden’s constants are the input. The spacings come from measurements on real fabrics, so the whole result inherits whatever those measurements are. If the constants are wrong, the flattening is wrong in the same proportion.

And the fabric is plain. A rib’s courses lie on two beds and its geometry is different; a purl fabric’s alternate. Neither is swept here.

Why the states nearly coincide, and what it would mean if they did not #

The three relaxation states sitting on nearly one curve is the least forced part of the sweep, so it is worth understanding rather than noting.

Relaxation shrinks a fabric in both plan directions at a fixed loop length. The wale spacing falls and the course spacing falls, so the courses come closer together — which should make the overlap worse — and the loop also becomes rounder in plan, which changes where along the curve the closest approach happens.

The two effects nearly cancel. The net is two per cent, which is small against the twelve per cent the loop-length range produces.

That near-cancellation is not something anybody arranged and it is worth flagging as a thing that could have gone otherwise. If the states had separated by ten per cent instead of two, the flattening would depend on the state as much as on the tightness, and a fabric’s yarn section would be a moving target through its own finishing — which would make a dimension quoted without its state a rule about sections as well as about spacings.

As it is, the collection can say that a yarn’s section is nearly a property of the construction rather than of the history, which is a simpler and more useful statement than the alternative.

The generalisation #

The rung is a worked example of a habit this collection has and does not name often enough.

Compute the same thing eighteen times before believing it once.

The cost is nothing — eighteen solves is a second — and what it buys is the difference between a number and a relation. A number can be right for the wrong reason, can depend on an input nobody varied, and can be an artefact of one choice of sampling. A relation across a family cannot be any of those, and it also says which variable matters, which a single number never does.

The site’s own history has the counter-example. Two shape constants taken from different rows of a relaxation table sat in this collection’s arithmetic for several rungs and moved every occupancy by five per cent, and the check that guarded them was a relationship that held for any pair of constants. Sweeping the inputs would have found it in a minute: the wrong pair does not lie on the same curve as the right ones.

So the rule has two halves. Sweep, to find whether there is a curve. And sweep the inputs, not just the outputs, because a check on a relation between outputs cannot see an input that is wrong.

Drawing the tight end rather than plotting it makes the size of the demand visible: the ellipse the arrangement asks for at that loop length is a third flatter than round.

What eighteen fabrics cost and what they bought #

It is worth putting a price on the sweep, because the price is the argument for making it a habit.

Eighteen solves is about a second of computation. Writing the sweep took a few lines: the same function called with three lists of inputs and its output collected.

What it bought is four claims that a single computation could not support. That the demanded flattening is a function of one variable. That the function is monotone over the range fabrics are made in. That the three relaxation states sit on nearly the same curve, with a separation a fifth the size of the loop-length spread. And that the fibre does nothing except through the diameter.

Any one of those, discovered later by somebody using the number in a case it did not cover, would have cost far more than a second.

That is the whole argument for sweeping, and it is not about rigour. It is about the difference between a result somebody else can use and a result that is only safe in the case it was computed for.

What would falsify it #

A prediction should say what would break it, and this one has two clean failure modes.

A gauge dependence. If measured knitted sections showed a systematic flattening difference between fine and coarse gauges at the same count and loop length, the result would be wrong and something about the needle rather than the loop would be doing the flattening.

Scatter against the tightness factor. If measured sections across a range of constructions showed flattening that correlated with the count or the fibre independently of the tightness factor, the group would be the wrong group.

Neither of those is exotic and both would show up in the same set of measurements — twenty or thirty sections across a spread of constructions, which is a week’s work.

The prediction is worth making because it is that easy to break and nobody has broken it.

Who found it, and when #

Munden’s constants are from 1959 and are the input. The tightness factor as a dimensionless index of a knitted fabric is older and is the trade’s own.

That the loop’s shape depends only on the ratio of yarn diameter to loop length is this collection’s own and was established when the loop was first solved as an elastica. The flattening result is this ladder’s, and the sweep is what turns it from a number into a relation.

Where the ladder goes next #

The fabric demands a flattening and the yarn has to supply it. What that costs is the next question, and the answer is a fourth reading of this collection’s oldest bracket: at one end of it flattening is exactly free, and at the other it is thirty-six times the whole bending energy of a stitch.

The fabric flattens, so flattening is free and impossible — which is the fourth independent everyday observation in this work to land at the same end of the same bracket.