Which knitted fabrics lie flat

Ask a knitter which fabrics curl and the answer comes back immediately: stockinette does, garter does not, rib does not, interlock does not, seed stitch lies flattest of all. Ask about a half-milano, or a rib that tucks on alternate courses, or a fabric with three front wales to every back wale, and the answer is a shrug and an experiment.

That is not because the second list is harder. It is because the first list is the one everybody has already knitted.

What the criterion is #

Each stitch in a knitted fabric contributes a small curling moment, and the sign of that contribution is which face the loop’s head is on.

The size of the contribution is a single constant. It is the same constant for every loop in the fabric, because every loop in the fabric is the same loop — the same yarn, the same length, the same interlacing, the same eccentricity relative to the fabric’s own neutral surface.

So the fabric’s total moment is that constant times a signed count: how much yarn goes to the front bed, minus how much goes to the back. A constant times a signed sum is zero exactly when the sum is, and the constant divides out of the question of whether a fabric curls.

That is the whole criterion. Sum over the grid, divide by the total, and read the answer.

Why the constant can be dropped #

This is the part that makes the criterion possible, and it comes out of asking what a curl is.

A fabric curls to a radius equal to its bending rigidity divided by its residual moment. Both of those depend on the yarn’s stiffness, on the loop length, on the fibre — on everything. Neither is available without a great deal of arithmetic, and the moment in particular needs an asymmetry that this collection’s loop model does not have and cannot supply.

But the question being asked here is not how tightly a fabric curls. It is whether it curls at all. And a sum of terms that are all one constant times ±1 vanishes or does not vanish regardless of what the constant is — even if the constant is unknown, even if it is a bracket a hundred wide, even if nobody has ever measured it.

Dropping a quantity that cannot be computed, because the question does not need it, is worth more here than computing it would have been.

What counts #

A stitch counts on the bed that took its yarn. That is the distinction the counting rungs on this ladder already draw and it is used here unchanged.

A knit takes yarn and pulls it through the old loop: it counts.

A tuck takes yarn and holds it over the old loop without pulling it through. The yarn is on that bed, held there, contributing its own asymmetry to that face. It counts.

A miss takes no yarn at all. The needle does not move, the yarn passes by on the other side, and nothing is added to that face. It does not count.

A needle out of work — one that misses on every course of the repeat — holds no loop, carries no wale and is not in the machine’s action. It contributes nothing, which is how a single-bed fabric is written on a two-bed machine.

The two answers that are not in doubt #

A criterion of this kind is worth exactly nothing unless it gets the known cases right, and there are two that are not arguable.

Single jersey comes out at one — as unbalanced as a fabric can be. Every stitch is on the front bed, nothing opposes anything, and the fabric rolls into a tube at every edge.

A one-by-one rib comes out at exactly zero. Alternate wales go to opposite beds, the counts are equal, and the fabric lies flat at its selvedges however long the piece is.

Those are the two ends. Everything else is between them or at one of them, and the interest is in which.

What it says about the rest #

Of the thirteen structures the collection holds, seven lie flat and six curl.

Flat: the one-by-one rib, interlock, the two-by-two rib, both cardigans, milano, and — the interesting one — a tubular fabric.

Curling: single jersey, single jersey with a float, half-milano, a rib with a miss on the back bed, a rib that both tucks and floats, and a fabric with three front wales to one back wale.

Three of those deserve their own paragraph.

The tube #

A tubular fabric — one course on the front bed, the next on the back — comes out balanced, and the reason is worth stating because it is not the reason a rib is balanced.

A rib balances within a course: the yarn alternates beds as it crosses the fabric. A tube balances between courses: each course goes entirely to one bed and the next goes entirely to the other.

Mechanically the two are quite different. A tube is two separate jerseys, each of which curls hard on its own, and what makes the whole thing lie flat is that the two curl in opposite directions and are attached at the selvedges. Cut the selvedges and both halves roll up immediately.

The criterion cannot see that difference, because it is a sum and a sum does not know where its terms came from. That is a real limitation and it is the clearest one on this rung: a balance of zero means the moments cancel, not that no moment exists.

The cardigans #

A half-cardigan is a rib with the back bed tucking on alternate courses, and it looks front-heavy: the front bed knits on every course and the back bed knits on half of them.

It comes out balanced, and the reason is the tuck. On the courses where the back bed does not knit, it tucks — and a tuck takes yarn. So the back bed takes yarn on every course after all, and the counts are equal.

That is the one place on this rung where the modelling choice does real work, and it is worth being honest that it is a choice. Weight a tuck at zero rather than at one and a half-cardigan becomes unbalanced at a third. Weight it at a half and it comes out at a seventh.

The reason for weighting it at one is that a tuck’s yarn is genuinely on that bed and genuinely held there — it is not passing by, it is caught. The reason for stating the sensitivity is that nobody has measured a half-cardigan’s curl to check.

The front-heavy ribs #

Three structures come out unbalanced without being single-bed fabrics, and all three are front-heavy for the same reason: the front bed knits more often than the back.

A half-milano — a rib course followed by a course on the front bed alone — sits at a third. A rib with a miss on the back bed sits at a third. A three-by-one rib sits at 0.43, which is nearly half way to a jersey.

Those are fabrics that are used precisely because they are one-sided: a half-milano has a face and a back that look different, and a wide rib has a pronounced front. The criterion says they should also curl, mildly, in the direction of the heavier face — and mild curl in a fabric that is otherwise a rib is exactly the thing a knitter notices at a selvedge and blames on the yarn.

That is a prediction, and it has not been checked here.

Why the constant is the same for every loop #

The step that makes the whole criterion possible deserves more than the sentence it got, because it is doing all the work.

Every loop in a knitted fabric is the same loop. It has the same yarn, the same length between interlacings, the same two spacings, the same interlacing at a diameter — and therefore the same eccentricity relative to the fabric’s neutral surface, whatever that eccentricity turns out to be.

So the fabric’s moment is a sum of identical terms with different signs, and identical terms factor out. That is not true of, say, a woven cloth, where a warp end and a weft pick are different objects with different crimps and different contact forces — a signed count over a weave would need two constants and would not factor.

The criterion works because a knitted fabric is made of one part repeated. It is the same property that makes the loop length the fabric’s only dimension, and it is worth noticing that the two are the same fact.

Two sums over one grid #

It is worth putting the two censuses side by side, because a reader could easily take them for the same thing.

The crossing share asks: between one wale and the next, did the yarn change beds? It decides how far the yarn climbs and therefore what the fabric costs in energy and force.

The balance asks: on which bed did each stitch’s yarn end up? It decides whether the fabric’s curling moments cancel.

A one-by-one rib has a crossing share of one and a balance of zero. A tubular fabric has a crossing share of zero and a balance of zero. Two structures at opposite ends of one census, together at one end of the other — which is as clear a demonstration as one could want that the two are measuring different things.

What the count reaches that the earlier one did not #

The rung this follows counts face changes between courses and gets zero for stockinette and one per course for garter. That is exactly right and it is limited in a specific way: it is a count along one direction only.

Garter alternates along the courses. Rib alternates across the wales. Seed and moss alternate in both. A count that walks down the courses sees the first, and can be persuaded to see the second by turning the fabric ninety degrees, and has to be told which way to turn.

A signed sum over the whole grid does not need to be told. It has no direction in it at all — it is a count of yarn on each bed, and yarn does not know which way it was laid down. That is why it handles a milano, whose repeat is three courses long and has different beds active on each of them, without any special case.

What the criterion does not say #

Four things, and every one of them is the kind of claim the ranking above invites.

How tightly the unbalanced ones curl. That needs the constant, and the constant needs an asymmetry the loop model does not carry. What is available is the exchange rate: the eccentricity that a measured radius implies.

Which way each one curls. The sign of the balance says which face is heavier; it takes an extra step, and a convention about which face a heavy count rolls toward, to turn that into an edge rolling forwards or backwards.

Whether the fabric is stable in any other sense. A rib lies flat at its selvedges and is enormously extensible across them. Flat is not stable.

And whether a zero is a cancellation or an absence. A tube’s zero is two large moments opposing; a rib’s is many small ones. Both are zero, and only one of them survives being cut.

The direction each one curls #

The sum’s sign says which face is heavier, and turning that into an edge rolling one way or the other takes one more step and one convention.

The rung this follows establishes the observed directions for stockinette: the top and bottom edges roll toward the back, the sides toward the front. A fabric with a positive balance in this account has more yarn on the front bed, which is the same arrangement as a jersey worked entirely on the front bed, so it should curl in the same senses.

So a half-milano, at a balance of a third, should roll its course-wise edges toward the back like a jersey but with a radius three times as large. That is a prediction with a direction as well as a magnitude in it, and both halves are testable on a swatch.

What the criterion cannot supply is the radius, because a radius needs the constant that just factored out.

What would test it #

Two experiments, and both are a swatch and a ruler.

A half-milano and a two-by-two rib of the same yarn. The criterion says one curls and one does not, and they are structurally close enough that anybody expecting a smooth family would predict them to behave alike.

A half-cardigan against a full cardigan. Both come out flat here, and both would come out flat under any sensible tuck weighting. If either curls, the tuck’s weighting is wrong and the whole family’s numbers move.

Neither has been run. The criterion is offered as a criterion.

Why this is the right shape of answer #

There is a pattern in this collection worth naming, because this rung is the fourth or fifth instance of it.

A question that looks like it needs a hard number often does not. Whether a cloth hangs together is a connectivity question, not a strength question. Whether a satin exists at a given move is an arithmetic question, not a weaving question. Whether a knitted fabric lies flat is a signed sum, not a bending-moment calculation.

In each case the hard quantity is real, is genuinely hard, and cancels out of the question actually being asked. Finding the version of the question the hard quantity cancels out of is most of the work, and it is nearly always more useful than computing the quantity would have been.

What is genuinely new here #

Two things.

A flatness criterion that runs on any structure, computed from the grid in a tenth of a second, for fabrics nobody has published a rule for as well as the six everybody knows.

And a demonstration that the criterion’s constant does not matter. The curl radius is uncomputable here and the flatness is not, and the reason is that one question needs the constant and the other divides by it.

What the pictures cannot show #

The bars are ranked, which invites a reading in which a balance of 0.43 is nearly half as curly as a balance of one. That reading is not supported. The balance is proportional to the moment, and the radius is the rigidity over the moment — so a balance of 0.43 gives a curl radius more than twice a jersey’s, not a curl half as strong.

The bar chart also gives seven structures identical bars of zero length, which reads as seven equally flat fabrics. They are not equally flat in any other sense; they are equally balanced, and a tube’s zero is a very different object from a rib’s.

What is worth taking away #

A signed count over a structure’s own grid says which knitted fabrics lie flat, for every structure a two-bed machine can write, at a tenth of a second each.

It works because the hard quantity — the moment a single loop contributes — is the same for every loop in the fabric and therefore divides out of the question of whether the sum is zero. Finding the version of a question that a hard quantity cancels out of is most of the work, and it is nearly always worth more than computing the quantity.

The one modelling choice, isolated #

Everything here follows from the grid except one decision, and it is worth putting in one place so that anybody who disagrees knows exactly what to change.

A tuck counts on the bed that took its yarn, at full weight.

That decision is why both cardigans come out flat. Weight a tuck at zero and a half-cardigan comes out at a third — the same as a half-milano, and predicted to curl. Weight it at a half and it comes out at a seventh, which is a mild curl rather than none.

The case for full weight is that a tuck’s yarn is caught and held on that bed, contributing its asymmetry to that face exactly as a knitted loop does. The case against is that a tuck is not drawn through and so is not interlaced, and this collection’s own account of what a tuck costs treats it as a different object from a knit in every other respect.

Nobody has measured a cardigan’s curl. The sensitivity is stated so that the measurement, if it is ever made, decides something.

Where the ladder goes next #

Two structures on this list are balanced for reasons that could not be more different, and telling them apart needs the connectivity argument rather than the sum: a tube and an interlock balance for different reasons.

And a fabric whose thickness is set by its bed gap rather than by its yarn is the subject of how thick a knit is, which is where the model’s predictions finally meet what a gauge reads.