Twist is not torsion

Two quantities in this subject are called by the same word, and they are not the same quantity. One belongs to a curve and one belongs to the stuff the curve is made of, and a collection that has just acquired a torsional rigidity is exactly the kind of place where confusing them would do damage.

The damage is available and specific. This collection’s account of a knitted loop ends by proving that the solved loop is a plane curve — the fabric’s own plane, rotated a dozen degrees out of it. A plane curve has zero torsion everywhere, and that is a theorem rather than an approximation. It would be very easy to read that sentence as saying the loop carries no twist, and to conclude that a torsional rigidity has nothing to act on. It says nothing of the kind.

What a curve’s torsion is #

A smooth space curve carries a frame at every point that is decided entirely by its own shape. The tangent points along it. The normal points the way it is turning. The binormal is perpendicular to both.

The curvature is how fast the tangent turns as one moves along the curve, and the torsion is how fast the plane containing the tangent and the normal turns about the tangent. A curve with no torsion stays in one plane; a curve with torsion is leaving the plane it was momentarily in.

Everything about that is a statement about the centre line. Nothing about it says what the curve is made of, or whether it is made of anything.

Two consequences follow that the word “torsion” makes hard to believe on first reading. A straight rod has no torsion, because a straight line has no normal direction — the tangent never turns, so there is no plane to turn about. And a plane curve, however violently it bends, has torsion zero at every point, because its osculating plane is the plane it is drawn in and that plane never moves.

What a material’s twist is #

Now paint a line down the length of the rod, or — what is the same thing and is what a spun yarn actually has — lay its fibres in a helix about its axis.

The twist is how fast that painted line rotates about the tangent as one moves along the rod. It is a property of the material and of nothing else. The centre line can be straight, bent, coiled or tied in a knot; the painted line either winds round it or does not, and how fast it winds is a number that can be measured on the object without knowing anything about its shape.

That is the quantity a torsional rigidity resists. The energy is one half the rigidity times the square of the twist rate, integrated along the rod, and it does not contain the curve’s torsion anywhere.

The two are independent #

The clean way to say it is that a rod’s configuration needs two things: where its centre line goes, and how its material is rotated about that centre line. The first supplies the curvature and the torsion. The second supplies the twist. Fixing either one leaves the other completely free.

A straight rod can carry any twist. A plane curve can carry any twist. A curve with enormous torsion can carry none at all — a helix whose fibres run exactly parallel to its own axis has a large geometric torsion and no material twist whatever.

That last case is worth holding onto, because it is not a curiosity. It is a description of one of the two ways a yarn can be laid into a helix, and the difference between the two ways turns out to be what the folding rules of the trade are about.

Where the two get coupled #

If they are independent, why does the word get shared at all?

Because they are coupled by a condition rather than by a definition, and the condition is important. Take a rod whose ends are clamped so that the material cannot rotate at either end. The total amount of winding — the number of times the material goes round the centre line, added to the number of times the centre line goes round itself — is then fixed. It cannot change while the rod stays unbroken.

That total is the linking number, the winding of the material about the axis is the twist, and the winding of the axis about itself is the writhe. The three are related by an identity that is exact and is one of the more beautiful things in the subject, and it is what a closed thread cannot choose.

The coupling is therefore between the twist and the shape — not between the twist and the torsion. A curve’s torsion is a local quantity and the writhe is a global one, and they are related by an integral rather than by an equality. It is entirely possible for a curve to have torsion everywhere and no writhe at all, which is what happens whenever the curve has a mirror symmetry.

What the distinction saves here #

This collection is unusually exposed to the confusion, for two reasons that are both its own doing.

The first is the theorem about the loop. The solved half period leaves and arrives at extreme points of its wave, both end tangents lie on one axis, and rotating the problem about that axis maps the constraint set to itself — so the solved shape is the flat one turned, exactly, and the loop is a plane curve. If torsion and twist were the same quantity, that theorem would say a knitted loop stores no torsional energy and the whole of this ladder would be unnecessary.

They are not the same quantity, so the theorem says only that the loop’s shape is planar. It says nothing at all about what the yarn’s fibres are doing about that shape, and a spun yarn arrives at the needle with several hundred turns a metre already in it.

The second is that this site’s own word for the helix angle of a spun yarn is twist, and that essay establishes — correctly — that a yarn’s twist is one angle and that everything else about twist follows from it. That essay is about the material quantity throughout. Nothing in it is about a curve’s torsion, and nothing in it needed to be, because the yarn it describes is straight.

The quantity a fabric has to satisfy #

Once the two are separated, the question a fabric raises can be asked properly.

A yarn is spun with a twist. That twist is a property of the material and it does not go away when the yarn is bent into a loop. If the yarn’s ends are held — and in a fabric they are, by every interlacing between here and the selvedge — the sum of the twist and the writhe of the path is fixed. So a fabric can relieve a yarn of some of its twist, but only by giving its path a writhe.

That is a real mechanism, it is the accepted explanation for why a jersey knitted from a hard-twisted yarn leans, and it is the reason the third dimension mattered enough to build. Whether this collection’s own model of a course actually has any writhe to offer is a separate question with a surprising answer, and it is the next rung.

The one case where they do coincide #

There is a case where the geometric torsion and the material twist are equal, and knowing which case it is prevents the confusion from returning.

Lay a rod along a curve so that its painted line always points along the curve’s own normal — so that the material frame is the Frenet frame. Then the material rotates exactly as fast as the osculating plane does, and the twist rate equals the torsion. That is a particular way of laying the rod and there is nothing natural about it.

The natural way, and the one a rod with no torque in it actually takes, is the opposite: the material frame rotates as little as possible while staying perpendicular to the tangent. A rod laid that way carries no twist at all, whatever its centre line does, and the frame that describes it turns relative to the Frenet frame at exactly minus the torsion.

So the geometric torsion is not the twist. It is the amount of twist that would appear if somebody insisted on using the curve’s own frame to measure the material’s rotation, which is a choice about bookkeeping rather than a fact about the rod.

A yarn arrives with its twist already in it #

The reason this matters for cloth rather than for rods is that a spun yarn is not a rod somebody twisted. It is a rod that was made by twisting, and the twist is what holds it together.

A staple yarn is a bundle of fibres a few centimetres long in a thread that is kilometres long, and nothing joins one fibre to the next. What stops the bundle pulling apart is that the twist presses the fibres against one another, so that a fibre being pulled out has to slide against a normal force the twist supplies. That is why a bundle is weaker than its threads and why the strength rises with twist to a maximum and then falls again: more twist grips harder and also runs the fibres further from the yarn’s own axis, so less of each fibre’s strength points the way the yarn is being pulled.

So a yarn’s twist is not an accident of handling. It is a construction parameter with an optimum, quoted as a twist factor, and an ordinary cotton at twenty tex carries somewhere between six hundred and a thousand turns a metre.

That number is what makes the distinction on this rung expensive to get wrong. Eight hundred turns a metre is eight tenths of a turn per millimetre, and a knitted stitch is about three and a half millimetres of yarn — so a single stitch contains nearly three full turns of material twist, all of it stored against the torsional rigidity, all of it invisible to a solve whose configuration space is centre lines.

What the collection has been doing instead #

It is worth being clear that nothing computed so far is wrong because of this.

Every force this collection reports for a knitted loop is a derivative of a bending energy with respect to a position, and the twist does not appear in either. A twist energy that is constant as the fabric deforms contributes nothing to any force; only a twist that changes with the deformation does. So the omission costs nothing at all for a comparison at fixed twist, which is most of what has been computed.

Where it costs something is exactly where the earlier ladder said it would: in the shape of a curve rather than its height. A fabric being pulled open changes the path its yarn takes, and a changed path is a changed writhe, and a changed writhe is a changed twist. So the torsional energy is not constant along a load–extension curve even though it may be nearly constant at any one point on it, and a discrepancy that changes with extension is the signature to look for.

What was counted, and how #

Nothing on this rung is a computation, and that is deliberate: it is a distinction rather than a result, and a distinction is checked by producing the cases that separate it.

Four cases separate them, and all four are drawn above. A straight rod with twist: torsion undefined, twist nonzero. A plane curve with twist: torsion zero, twist nonzero. A helix with its fibres along its own axis: torsion large, twist zero. And a rod laid along the Frenet frame: the two equal, by construction and by nothing else.

The check that matters for this collection’s own machinery is the second, because it is the case the loop is in. The solve that produces a knitted loop’s shape reports its energy, its forces and its curvature, and the curvature is the only material constant that enters. There is no twist in the solve at all — the configuration space is centre lines, and a centre line has no material to rotate. So the solve is not silently wrong about torsion; it is silent about it, which is a different and much more repairable condition.

Where the model stops #

Nothing here computes a twist. The distinction says what quantity would have to be computed and it does not compute one. How much twist a knitted loop’s yarn actually carries depends on how the yarn was fed, what the machine did to it, and how much the fabric has been allowed to writhe, and only the last of those is a question about geometry.

And the coupling is stated for a rod with clamped ends. A yarn in a fabric is not clamped; it is held by friction at every interlacing, and friction lets a yarn rotate slowly and stops it rotating quickly. So the sum of twist and writhe is fixed on the timescale of a pull and not on the timescale of a wash, which is one of several reasons a fabric’s behaviour depends on how long it has been left alone.

The rod is treated as having a circular section throughout. A section that is not circular has a preferred bending direction, and a rod with a preferred bending direction couples its bending to its twisting: bend it and it wants to rotate. That coupling is absent here and is not absent from a yarn in cloth.

The generalisation #

The lesson is about vocabulary rather than about rods, and it recurs across this collection.

Three times now a word shared between a geometric quantity and a material one has hidden a real distinction. Crimp is both a length ratio and a wave height, and this collection had to separate them before the interchange arithmetic would close. Set is both a state of the yarn and a fraction of the natural curvature, and separating those is what made a loop set rather than sprung a statement with content. And now torsion, which is a curve’s and a material’s and only ever one of them at a time.

The general shape is that the geometric member of the pair is computable from the drawing and the material member is not, so a collection that draws things will tend to reach for the geometric one and quietly answer a different question. The defence is to ask, of any such word, whether the quantity would still exist if the object were made of nothing.

A curve’s torsion would. A yarn’s twist would not.

Who found it, and when #

The Frenet frame dates from the 1840s and the distinction between it and a rod’s material frame is as old as Kirchhoff’s rod equations of 1859, which are written in the material frame precisely because the geometric one is unusable for the purpose: it is undefined wherever the curve is momentarily straight, and a rod is often momentarily straight.

The frame that rotates as little as possible was named and studied by Bishop in 1975, which is late for such an elementary object and is a good indication of how easy the confusion is to make.

The identity that couples the twist to the shape is Călugăreanu’s, from 1959 and 1961, and it arrived in this subject through molecular biology rather than through textiles.

Where the ladder goes next #

The distinction opens a question the collection can now answer with its own machinery: how much writhe a knitted fabric’s own path actually has, and therefore how much twist the fabric can relieve its yarn of.

The answer is none, and the reason is worth more than the number. A course of the solved fabric is carried to itself by a reflection, writhe changes sign under a reflection, and so a jersey’s course has no writhe — not approximately, but as an identity, and the identity is a statement about what the model left out rather than about knitting.