How a tuft is held

There are two classical ways of binding a cut pile into its ground and the trade has clear opinions about both. V-fastening dips the pile end between two ground picks and brings it straight back: seen in section it is the letter, two legs and one turn. W-fastening takes it under one pick, over the next and under the third: three legs of contact.

The trade says W is the one to specify for heavy traffic and V is the one that gives more pile for the money. Both halves of that are correct, both are computable, and neither is visible to the criterion this site decides fabrics with. That last clause is the reason this essay exists: the rule of thumb is right, the machinery that has decided every other fabric on this site returns no opinion about it at all, and what fills the gap has to come from somewhere else and be labelled as such.

Why the criterion cannot help #

It is worth being precise about why, because “the model is too simple” is the wrong diagnosis and would suggest the wrong fix.

The integrity criterion asks whether the strands can be split into an upper set and a lower set such that at every crossing between them the upper strand is on top. That is a question about the existence of a separation, and a separation either exists or it does not. There is no continuum in it, no margin, nothing to be more or less of.

A tuft under one pick and a tuft under three are both connected. Neither can be lifted off without breaking something. The criterion is not failing to notice a difference; it is correctly reporting that on the question it asks, there is no difference. Refining it is not possible, because “how firmly” is not a refinement of “whether” — it is a different question with different units.

So the honest move is a second model, kept visibly separate, with its own assumptions stated.

The capstan equation #

The second model is old, simple and exactly the right shape.

A flexible line wrapped around a fixed cylinder through an angle θ can hold a load exp(μθ) times the tension applied at its free end. The radius of the cylinder does not appear. That last point is what makes the equation usable here: a yarn turning around another yarn is not a rope on a bollard in most respects, and the one thing the model needs — the wrap angle — is a property of the construction rather than of the yarn’s size.

Applying it is then arithmetic. A V tuft turns through one half-turn: θ = π. A W tuft turns through three: θ = 3π. So the holding ratio between them is

exp(μ·3π) / exp(μ·π) = exp(2πμ)

and every number below is that expression at some μ.

Why one number is not the answer #

μ is not a material constant. Yarn-on-yarn friction depends on the fibre, the finish, the twist, the moisture and the angle at which the two threads cross, and reported values for cotton on cotton run from about 0.2 to 0.4 — a spread of the same character as the one between the two thread-section models. Wool on wool differs, and a scaled fibre is famously directional.

That range is not a rounding error in an exponent. At μ = 0.15 the W-to-V ratio is 2.6; at μ = 0.4 it is 12.4. The answer moves by a factor of five across the range somebody might reasonably assume.

So quoting “W holds six times harder” as a fact about pile fabrics is not a useful statement, and this site’s rule about naming the model exists for exactly this case. What is worth quoting is what survives the whole range, and two things do.

The sign survives. W beats V at every coefficient anyone reports, and the assertion in the code is on that rather than on a value.

The order of magnitude survives. The ratio is between about three and about twelve — never 1.1, never 100. A construction decision worth making, and not one that reverses if the yarn is finished differently.

Why the answer is an exponential #

The shape of the capstan relation is worth a paragraph, because it explains why a small change in wrap angle does so much and a large change in load does so little.

Consider a short arc of the wrapped line subtending angle dθ. The tension differs slightly across it; the difference is what friction has to supply, and the friction available is μ times the normal force pressing the line against the cylinder. That normal force is not a constant — it is supplied by the tension itself, turning through the arc, so it is T dθ. Setting the two equal gives dT = μT dθ, whose solution is T growing as exp(μθ).

The essential feature is that the grip at each point is proportional to the tension already there. A wrap does not add a fixed amount of holding; it multiplies. That is why turns compound so violently, why a second turn round a bollard is worth so much more than the first is worth over nothing, and why a W tuft’s three half-turns are worth so much more than three times a V’s one.

It is also why the answer is so sensitive to μ. The coefficient sits in an exponent multiplied by an angle of several radians, so a change from 0.2 to 0.3 is not a fifty per cent change in the answer — at 3π radians it is a factor of 2.6. Any quantity that appears exponentially deserves suspicion, and this essay’s repeated insistence on the range rather than a value is that suspicion acted on.

What W costs #

The other half of the trade’s rule is a count rather than a model, and it is exact.

A V tuft occupies two ground picks — one to dip under and one to move on. A W tuft occupies three. So over a fixed run of ground, the number of tufts a construction fits is the pick count divided by the picks each tuft consumes, and W fits exactly two-thirds as many as V.

That is an unqualified ratio with no coefficient in it, which makes it a different kind of statement from the anchorage figure and worth keeping typographically apart. The site’s habit of naming which model produced a number matters most where two numbers of different status sit side by side, and this is such a place: one of the two numbers below any pile comparison is arithmetic and the other is physics with a measured parameter in it.

The exchange is therefore: give up a third of the pile density, gain between three and twelve times the anchorage. Stated that way it is obvious why heavy-traffic carpet is W-fastened and why upholstery velvet, which is looked at rather than walked on, is not.

The exchange, multiplied out #

The trade’s rule is stated as a trade — a third of the pile density against three to twelve times the anchorage — and left as a choice. The two can be multiplied, and the product says the choice is not as balanced as it sounds.

What a carpet resists is tuft loss per unit area, which is the anchorage per tuft times the tufts there are. V gives 1 × 1; W gives 0.667 × e^(2πμ). At a coefficient of three tenths that is

4.4 against 1 — W wins the product by more than fourfold.

And it wins it over the whole plausible range. The two are equal when e^(2πμ) = 1.5, which is μ = 0.065 — a coefficient far below anything reported for any fibre on any finish. W anchors more total pile per unit area than V at every friction coefficient a real yarn has.

So the trade is not anchorage against anchorage. It is anchorage against appearance: V buys half again as many tufts per unit length, which is pile density, cover and the look of the carpet, and it buys them at a real cost in what stays in. Stated that way the rule stops being a compromise and becomes a straightforward choice between a fabric that is looked at and one that is walked on — which is exactly the division the trade makes.

How many half-turns are enough #

The essay’s better reading of a large anchorage figure — that slipping stops being the mode that decides — can be given a form even without the tail tension it would need to evaluate.

Slipping stops deciding when the capstan’s multiplier exceeds the ratio of the yarn’s breaking load to whatever tension the buried tail carries. Writing that ratio R, the requirement is

number of half-turns ≥ ln R ÷ (μπ),

which has two useful properties and no need for R to be known precisely, since it enters through a logarithm.

The requirement is inversely proportional to the friction coefficient. At the bottom of the reported range a construction needs twice the half-turns it needs at the top, so a fastening that is sufficient on a well-finished wool may be insufficient on a lubricated synthetic of the same construction.

And a finish is worth as much as a leg. Doubling μ halves the half-turns required — so a treatment that raises yarn-on-yarn friction by a factor of two is worth exactly as much as going from a V to a W, and it costs no picks, no loom speed and no pile density.

That is a lever the essay’s list does not contain and it is the cheapest of the three. It also explains why carpet yarns are finished the way they are: a pile yarn is not lubricated for weaving the way a warp is, and the reason is that every unit of lubricant is a unit of anchorage given away in the exponent.

Which sets the order to try things in #

Putting the three together gives a designer a sequence rather than a menu.

Raise the friction first, because it enters as a reciprocal in an exponent and costs nothing structural.

Then add half-turns, because each is worth e^(πμ) and costs a pick — a third of the pile density for the step from V to W, and a further quarter for a five-leg fastening.

And back the fabric last, because adhesion is a different mechanism that makes the whole comparison moot, works better than any of it, and is why the tufted construction displaced the woven one entirely for floors.

The sequence is the reverse of the order the trade’s vocabulary suggests, where the fastening is the named variable and the finish is an afterthought.

What “holding force” means here, and does not #

The capstan gives the tension a wrapped line can sustain before it slips. That is a statement about slipping, and a tuft can be lost three ways.

It can slip out, which is what the capstan models and what wrapping more turns prevents.

It can break, which is a yarn strength question and has nothing to do with the wrap. A tuft anchored so firmly that it breaks before it slips is anchored as firmly as it can usefully be, and adding wraps past that point buys nothing.

The ground can fail around it, which is a question about the ground weave’s own integrity and is back in the criterion’s territory.

So the anchorage figure is an upper bound on one failure mode, and the useful reading of a large number is not “this tuft holds twelve times harder” but “slipping is no longer the mode that decides”. That is how a designer actually uses it, and it is a good deal less impressive and considerably more honest than the multiple by itself.

Two numbers of different status, side by side #

Every figure in this essay prints two numbers about the same tuft, and keeping them apart is more than a presentational nicety.

The layer count is exact. It has no parameters, no tolerance and no model behind it beyond the definition of a separation. Given the contacts, it is what it is, and two people computing it will agree to the last digit for all time.

The anchorage is modelled. It has a named model with a stated idealisation, a parameter that must be measured, and a range of plausible values that moves the answer by a factor of five. Two people computing it will agree only if they agree about μ.

Printing both beside one figure, in the same typeface, with the same air of authority, would be the exact failure this site’s fourth invariant exists to prevent: never claim the machinery went where it did not. So the figures label them — “the criterion” and “the capstan, at μ = 0.3” — and the labels are not decoration.

The habit is worth generalising. A great deal of engineering writing puts an exact count and a modelled estimate in the same table — this site did it once itself, in a tally whose bias row was a slogan — and lets the reader assume they are the same kind of thing. They are not, and the difference matters most exactly where it is least visible: when both numbers happen to be right.

What was counted, and how #

The wrap counts are read off the constructions rather than assumed: a V passes beneath one pick, a W beneath two with one above between them, and those contact patterns are what the graph in the previous rung is built from. The half-turn count and the number of picks a tuft consumes are therefore properties of the same object the criterion ran on, not a separate description of it.

The capstan is then applied to the wrap angle, and the pile density to the pick consumption, and both are returned alongside the friction coefficient that produced the first.

Three assertions run while the figures draw. W anchors harder than V, and W fits fewer tufts — the two halves of the trade’s rule, each asserted so that a sign error in either would stop the build. And across the whole reported friction range, W’s advantage is at least 1.5× and the spread between its best and worst is more than threefold, which is the assertion that the answer depends on μ. That last one is unusual and deliberate: it is an assertion that the model is parameter-sensitive, and its job is to stop a later reader quoting a single number as though it were a measurement.

Where the model stops #

The capstan assumes a fixed cylinder and a flexible line. A ground pick is neither fixed nor rigid; it moves, flattens and is itself under tension. The wrap angle is real and the geometry the equation assumes around it is idealised.

μ is doing an enormous amount of work in an exponent. Any quantity that appears exponentially deserves suspicion, and this one is measured with difficulty and reported over a wide range. Every number here is a number at a stated coefficient.

Nothing here models the finish. A pile fabric is normally sheared, brushed, sometimes heat-set and often backed with adhesive, and a latex or fusible back changes tuft withdrawal entirely by gluing the loop under the ground. Modern carpet’s tuft bind owes more to its backing than to its wrap, which makes this whole comparison a statement about the woven construction alone.

And the criterion has not been improved. It says the same thing it said before: attached. Nothing here has made it sharper, and the second model has been put beside it rather than folded into it, because a topological verdict and a frictional estimate are different kinds of claim and merging them would produce a number that looked exact and was not.

The construction that does not wrap at all #

There is a third way of getting pile into a fabric and it is how most carpet on most floors is now made. A tufted carpet is not woven. A needle punches the pile yarn through a ready-made backing fabric, a loop is formed, the needle withdraws, and nothing holds the loop in but the friction of the backing’s own threads closing round it — which is very little.

The wrap angle in that construction is essentially the small turn the yarn makes passing through the backing, so the capstan gives a holding force barely above one. On the analysis of this essay a tufted carpet should shed its pile immediately, and an unbacked one does.

What holds it is applied afterwards: a latex or polymer coat spread over the back, which bonds the loop to the backing along its whole buried length. That is adhesion rather than friction, it is a different model again, and it works well enough that tufting displaced weaving for floor covering almost completely from the 1950s onward.

The comparison is instructive rather than a digression. Woven pile solves the anchorage problem structurally, by wrapping the yarn round something, and pays for it in loom speed. Tufted pile ignores the structural problem and solves it chemically afterwards, at a fraction of the cost. Both are legitimate and only one of them is a question about cloth, which is why this site can compute the first and only describe the second.

Who found it, and when #

The capstan relation is Euler’s, from 1762, and was rediscovered and popularised by Johann Albert Eytelwein early in the nineteenth century; textile mechanics knows it as the reason a knot holds, a yarn package does not collapse and a belt drives a pulley.

Its application to pile anchorage is not attributable to anybody in particular — it is what any textile engineer reaches for — and the V-and-W distinction is workshop vocabulary considerably older than the physics. Carpet weaving manuals have specified W-fastening for heavy service since well before anyone wrote down why, which is the usual order of events on this site: the rule of thumb is right, and the reason it is right turns out to be a quantity nobody in the workshop was computing.

Where the ladder goes next #

The gap between what the criterion sees and what decides a fabric’s service life has now been named and filled once. It is not peculiar to pile: a leno gauze and an open plain weave differ in exactly the same way and by exactly the same equation, and the essay that puts both together is where this field states its own boundary. Before that, the two pile geometries: corduroy and terry, whose heights are decided by parameters neither construction shares with the other.