A pile is the only surface with no crowns

A woven cloth’s outside is the tops of threads that are curving. A knitted cloth’s outside is the tops of loops that are curving. Every fabric in this collection so far has had the same kind of surface — a set of curved crowns — and every consequence traced from it has followed from that curvature.

A cut pile has no curved crowns at all. Its outside is a set of severed ends, cut in one pass by a blade, all of them lying in the plane the blade travelled in.

The claim #

Cutting a pile changes the exponent of its bearing curve from a half to zero, and everything that follows is a consequence of that rather than of the pile’s height, density or fibre.

Three consequences, and they are the reasons a carpet is built this way.

The contact does not vanish with the load. Every other fabric’s contact area goes to zero as the pressure goes to zero, so its area is a function of how hard it is being pressed. A cut pile’s is a constant, and the constant is the tuft area fraction.

The pressure concentration is bounded. On a woven cloth the ratio of contact pressure to nominal pressure is one over the bearing fraction, which grows without limit as the load falls; on a cut pile it is one over the tuft fraction — about nineteen for an ordinary corduroy — at every load.

And the wear is uniform. A rubbing plane meets every tuft at once and takes the same amount off each, so the mass removed and the section removed are the same quantity. A cut pile is the only fabric for which an abrasion mass loss is an honest measure of damage.

Why the exponent is zero #

The bearing curve at small depth is decided by the shape of what is highest, and there are exactly three shapes available.

A doubly curved summit — a crown with no float — opens as an ellipse, linear in depth.

A singly curved ridge — a float’s plateau, or the top of a terry loop — opens as a strip, as the square root of depth.

A flat end opens all at once, because it is already flat. There is no shape to cut through; the whole cross-section is present at zero.

The blade is what supplies the third case. It travels in a plane, it severs every tuft it meets, and it leaves every end in that plane to within the accuracy of the machine. The flatness is not a property of the yarn or of the weave. It is a property of the tool.

What was counted, and how #

The tuft area fraction is the number of tufts per unit area times one tuft’s cross-section, and both come from arithmetic this collection already has.

For a corduroy woven at sixty ends to the inch with six-end floats, the pile geometry gives 8.6 wales to the inch and two legs per wale per cut float. At thirty tex the pile yarn’s diameter follows from the site’s standing route from a count to a diameter. The product is 1.59 tufts per square millimetre, each of 0.033 square millimetres, and 5.2 per cent of the plan.

The comparison is with the same collection’s woven cloth. A two-and-two twill in an ordinary sheeting reaches five per cent of contact at a depth of about eight micrometres, which takes about eight kilopascals — rather more than a resting hand. The pile has it at nothing.

The loop case is computed the same way and comes out with an exponent rather than a value: a terry loop’s top is a curved thread of the pile yarn’s own diameter, so it opens as a strip, and its crown line is the loop density times half a loop’s circumference.

What a bounded concentration is worth #

The pressure concentration is the quantity that does damage, and its behaviour is qualitatively different on the two kinds of surface.

On a woven cloth the concentration is one over the bearing fraction. At a hundred kilopascals it is about six; at five kilopascals about twenty-three; at half a kilopascal about fifty. It rises as the load falls, without limit, because the bearing area is going to zero faster than the load is.

On a cut pile it is one over the tuft fraction, which is nineteen, and it is nineteen at every load.

So the two curves cross. Under a heavy press a woven cloth concentrates less than a pile; under a light touch it concentrates far more. The light touch is the whole of ordinary use — a hand, a sleeve, a foot in a soft shoe, a chair — and it is exactly the regime in which a pile’s constant concentration is the smaller number.

Why a carpet is a pile and not a heavy cloth #

The obvious answer is that a pile is thick and soft, and the obvious answer is incomplete: a heavy woven fabric can be made as thick and nearly as soft.

The bearing argument gives the rest of it. A carpet is walked on, which is a heavy load applied briefly and repeatedly; it is also stood on, which is a light one applied for a long time. Between those two the woven surface changes character completely — its contact area moves by an order of magnitude and its concentration with it — while a cut pile presents the same surface to both.

A pile is a surface with no regimes. That is worth more than any particular number: a fabric that behaves the same at every pressure can be specified, and one whose contact area is a function of the load being applied cannot be, without the load being stated as well.

And the wear result closes it. Because every tuft is cut at once and worn at once, the material removed from a pile is removed evenly, and the section-versus-mass divergence that makes a woven cloth’s abrasion figures so hard to interpret does not arise. A carpet that has lost ten per cent of its pile mass has lost ten per cent of its pile, distributed as it started.

The velvet case, where the tufts are dense #

A corduroy is a sparse pile — 1.6 tufts to the square millimetre, in wales with bare ground between them. A velvet is not, and the arithmetic scales in an interesting way.

Doubling the tuft density doubles the area at zero depth and halves the pressure concentration. Since the area starts at five per cent for a corduroy, a velvet at four times the density is at twenty, and a dense carpet higher again — which means a pile fabric can reach contact fractions at no load that a woven cloth cannot reach at any load a hand can produce.

Above about a third of the plan the argument changes character, because the tufts start to support one another laterally and the surface stops being a forest of independent columns. Nothing here computes that transition; what can be said is that the bearing statement — a constant area from zero depth — holds for as long as the tufts stand, and how long that is is a buckling question.

The reverse limit is the more interesting one. A very sparse pile has a small area at zero and a large concentration, and at some density the pile’s constant concentration exceeds the woven cloth’s at the load in question — at which point the pile is the worse surface. For an ordinary hand’s pressure that crossing is at about a quarter of the corduroy’s density, which is sparser than anything anybody weaves.

Where the surface goes when the pile is crushed #

The flat plane is the fabric’s whole advantage and it is not permanent.

A tuft is a column, and a column under enough load buckles. Once it does, it lies over — and a laid-over tuft presents its side to whatever is above, which is a curved thread, which is a crown. A crushed pile has a woven cloth’s kind of surface: it has recovered its curvature and lost its plane.

This collection has already established that a crushed pile is not held down by its fibres — that the recovery is a matter of the tuft’s own bending and of the friction at its base rather than of anything about the fibres. The bearing curve adds what is lost while it is down: not merely the height, which is what a reader sees, but the exponent, which is what the fabric touches with.

That is why a crushed carpet is worn twice over. It has less pile, and what it has left is presenting the wrong geometry.

Where the model stops #

The blade is perfect and no blade is. A cutting machine leaves tufts at slightly different heights, and the spread of those heights is what the bearing curve would actually be: a step smeared over a few tens of micrometres. Nothing here computes it, and it would come from the machine rather than from the fabric.

A tuft is a bundle of fibres and its cut end is not a disc. The fibres splay, the end is ragged, and what a plane meets is some fraction of the nominal cross-section. That fraction is a spinning and cutting property; taking it as one is an upper bound on the area.

The tufts are rigid. A pile in contact deflects — it is a forest of slender columns — and the load-bearing behaviour of a pile is a buckling problem rather than a compression one. Nothing about the area changes while the tufts stand, which is why the bearing statement survives; everything about the force does.

And this is a cut pile of a stated construction. A velvet, a plush, a moquette and a carpet differ enormously in tuft density, and the five per cent computed here is a corduroy’s. What does not change with construction is the exponent.

What it does to three earlier pile results #

The pile ladder on this site has eight rungs before this one and the bearing curve touches three of them.

Velvet is cut apart established that a double plush woven face to face and cut in the middle is the only way to make a dense pile, and treated the cutting as a production step. It is also the step that creates the surface: everything above is a consequence of the same blade.

How a tuft is held computed the anchorage — the capstan arithmetic that decides whether a tuft stays in the fabric — and found the criterion this site is built on cannot see the difference between a carpet and a fabric that sheds. The bearing curve is the other end of the same tuft: the anchorage decides whether it is there and the cut end decides what it does.

And a crushed pile is not held down by its fibres computed the recovery. What that essay treated as a loss of height is also a loss of exponent, which is a sharper way of saying why a crushed carpet does not merely look worn but behaves differently.

None of the three needed changing. The surface reading sits on top of them, which is what a rung of a ladder should do.

The generalisation #

A surface made by cutting has a different contact law from a surface made by forming.

That is the transferable statement, and it applies well beyond cloth. Anything sliced presents flat faces in the plane of the slice; anything grown, moulded, woven or drawn presents curvature. The first has a contact area at zero load and the second does not, and no amount of care over the material changes which of the two a process produces.

The corollary for design is that flatness at the top of a surface is worth more than smoothness of the surface as a whole, because it is the top that participates. A rough surface with flat summits behaves like a flat one; a smooth surface with curved summits behaves like a rough one.

What the comparison is worth, honestly #

The pile’s five per cent and the woven cloth’s contact at a light touch are close enough that the comparison deserves a careful statement rather than a slogan.

At the depth a standard thickness gauge reaches — about two micrometres on this sheeting — a two-and-two twill is touching 2.6 per cent of its plan and the corduroy is touching 5.2 at no depth at all: a factor of two, not a factor of a hundred. Press harder and the woven cloth catches up; at twenty kilopascals it has passed the pile.

So the pile’s advantage is not the size of its contact, it is the shape of its curve. A woven cloth’s contact is a function of the load, and a pile’s is a constant. That means a pile can be specified and a woven surface cannot, and it means the pile is ahead exactly where the load is small — which is where a surface is judged.

Making the number large as well as constant is a matter of tuft density, and a velvet or a carpet does exactly that. What is being reported here is a corduroy, which is the sparsest pile anybody weaves, so the five per cent is a floor for the construction rather than a typical value.

The crossing is a different pressure for every weave #

The honest comparison above quotes one crossing — the load at which a woven cloth catches a corduroy — and there is not one crossing. There is one per weave, and they span two decades.

Take each draft’s contact fraction at one kilopascal from the contact arithmetic and raise the pressure until it reaches the corduroy’s 5.2 per cent. The bearing area grows as somewhere between the one-third power of pressure, which is what a floated cloth’s geometry predicts, and 0.45, which is what the measured curve gives over a wider range; so each crossing is a band rather than a point.

Against a satin the corduroy has no advantage worth the name. The two are equal at a pressure lighter than a bedsheet, and above that the satin is touching more — so a satin and a sparse cut pile present much the same amount of themselves to anything a person does.

Against a plain weave the pile is ahead over the whole range of ordinary use, up to a firm pinch, which is where the essay’s twenty kilopascals sits and is comfortably inside the band.

So the pile’s advantage is real and it is against the firm weaves rather than against cloth in general. That is a more useful statement than a single crossing, and it says which comparison a designer is actually making: a corduroy against a poplin, where the pile wins over everything short of a press, or a velvet against a satin, where it does not win on area at all and wins on the shape of the curve.

How dense a pile has to be to win outright #

The same arithmetic run backwards says what tuft density it takes to beat every weave at every ordinary load, which is a specification rather than a comparison.

An eight-end satin at a hand’s five kilopascals is touching about eleven per cent of its plan. So a pile that is to be ahead of it there needs a tuft fraction above eleven per cent, which is twice the corduroy’s — and to stay ahead at a firm hundred kilopascals, where the satin reaches perhaps a quarter, it needs four or five times.

A velvet at four times a corduroy’s tuft density is ahead of every weave in this collection at every pressure a person applies, with contact of about a fifth of its plan and a concentration of five rather than nineteen.

That is worth having because it prices the construction. The pile’s structural advantage — a constant rather than a curve — is available at any density; the numerical advantage is not, and it costs tufts. A sparse pile buys predictability and a dense one buys area, and they are separate purchases made with the same yarn.

It also explains the trade’s own division without appealing to appearance. A corduroy is a cheap, sparse, hard-wearing cloth whose pile is there to be seen and felt; a velvet, a plush and a carpet are dense piles whose point is the surface itself. The first is buying the exponent and the second is buying the coefficient, and only the second is expensive.

Who found it, and when #

Nobody, as a statement about bearing curves. Cut pile fabrics are ancient, the reasons given for them in the trade are softness, cover and wear resistance, and all three are correct as far as they go.

What this collection contributes is the observation that the three have a common cause that is geometric and exact — the plane the blade leaves — and that the same argument distinguishes a cut pile from an uncut one in a way that the usual comparison, which is about appearance and about whether the loops snag, does not.

Where the ladder goes next #

Back to the woven surface and its other reading. The same crowns that decide what touches also decide what reflects: a float reflects into a line is the same geometry asked about light instead of contact, and it turns out that the part of a crown that reflects is the part that is flat — which is why a pile, having nothing but flat, behaves so strangely under a light and why which way it is laid decides its tone.