A print is as sharp as the hairs are long

Every finishing sequence for a printed cotton begins the same way: singe, desize, scour, bleach, mercerise, print. The singeing is first and its justification in the manuals is that the cloth must present a smooth surface to the screen or the roller.

That is true and it is a description rather than a reason. The reason is an arithmetic one and it produces a number: an unsinged cotton cannot hold a rule finer than about a fifteenth of an inch, whatever the printer does.

The cloth #

A printed edge is not a line; it is a boundary between an inked region and an uninked one, and the ink has ways of crossing it. Diffusion in the paste crosses it by microns; the cloth’s own pore structure carries it further. Capillary flow along the yarn crosses it by however far the wicking takes it before the paste thickens.

And a fibre end lying across the boundary, with one end in the ink and the other outside it, carries colour the whole way along itself in a single stroke. A hair is a wick with no pore in it, and it moves ink further than either of the other two mechanisms.

The claim #

The feather on a printed edge is set by the reach of the hair population: the number of hairs bridging at least a distance x is (n_A λ/2)e^(−x/λ), an exponential with the population’s own decay length, so the visible edge is a quantile rather than a mean. An unsinged sheeting feathers by nearly two millimetres. Singeing caps it at the flame’s own reach of two hundred micrometres, and the cloth goes from six lines to the inch to sixty.

A factor of ten in resolution, bought by burning off under one per cent of the cloth’s mass.

Why the bridging count is an exponential again #

The derivation is one integral and it comes out with the same decay length as the population, which is worth noticing because it did not have to.

A hair of free length ℓ lying at a distance y from the printed edge bridges ℓ − y of it, if it points the right way. Half of them do not. Integrating the population’s exponential over both the position and the length gives

bridging(x) = (n_A λ / 2) · e^(−x/λ)

an exponential in x with the same λ. So the feather’s profile is a scaled copy of the hair layer’s own profile, and everything about it is decided by the same two numbers.

That means the feather is a quantile: the visible edge is where the bridging count falls below what an eye or a densitometer will register, and moving that threshold moves the edge logarithmically. A print judged by eye and one judged by an instrument will disagree about where the edge is, by an amount proportional to λ.

What the flame does, which is not what a raising machine undoes #

Singeing does not scale the population. It truncates it: everything standing clear of the cloth is burnt back to the flame’s reach, and what survives is the short cloud lying within a fibre diameter or two of the yarn.

So the feather after singeing is capped at the flame’s reach — two hundred micrometres — however the arithmetic behaves above that, because there is nothing left up there to carry ink. That is a hard cap rather than a reduction, and it is why the operation is so decisive.

The comparison with raising is instructive. Raising multiplies the population, so the feather grows as λ times the logarithm of the multiplier: sixty-fourfold raising takes the feather from under two millimetres to over four. The two operations are not inverses. One truncates and the other scales, and a truncation is worth far more than a scaling to anything that depends on reach.

What it costs, in the currency a finisher counts in #

The exchange rate is unusually clean.

The mass is under one per cent — the long population is a tenth of a per cent of the cloth’s areal mass and the whole protruding population is under one, which is what a weighbridge records after a singeing.

The strength cost is nothing, which is not true of raising. Nothing structural is burnt; the flame is fast enough that the body of the yarn never heats, and the fibre ends removed were not carrying any load.

The optical cost is eight or nine points of cover. A cloth is more opaque than it is closed, and the hairs are part of why; removing them makes the cloth measurably more transparent, which is a real loss on a fabric sold on its cover.

And the benefit is a factor of ten in resolution. That is the whole of the case, and it explains why singeing is not optional for a printed cloth and is optional for almost everything else.

What a screen ruling actually is, in this arithmetic #

The number a printer works to is a ruling — lines to the inch — and it converts straight into the feather.

A halftone or a line figure needs two adjacent features to stay separate, so the finest usable pitch is twice the feather. At an unsinged feather of nearly two millimetres that is four millimetres of pitch, which is six lines to the inch: coarser than any screen anybody engraves, and coarser than the eye’s own resolution at reading distance.

So an unsinged cotton cannot print a figure at all in the sense a printer means it. It can print a large flat area with a soft edge — which is what a woven outline cannot do and a printed one can, which is what a great deal of historical printed cotton looks like and is usually described as a stylistic choice.

Singed, the same cloth holds sixty lines to the inch, which is in the range of a real engraved roller. The remaining limit is then the paste and the screen rather than the cloth — which is the state a printer wants to be in and the reason the operation is not negotiable.

Where the same argument meets the coating ladder #

A coating fills the crowns before it bridges the holes found that the volume a coating must bury is far larger than a block model predicts — a hundred and fifty-six micrometres against ninety-four for a sheeting — because threads are round and the plan is far more open than the cover factor says over most of the descent.

The hair layer sits above that surface and it is a very small volume with a very large surface. A coating wets surface, and a canopy has π times its own shadow in surface, so a hairy cloth drinks coating into a place where it does nothing.

The effect is small in volume and large in where the volume goes, and this essay cannot compute it because nothing here models a liquid. What it can say is that the error runs toward more coating for the same film, and that it would be attributed to the paste’s rheology rather than to the cloth.

Why this is the sharpest thing the hair layer does #

It is worth saying why the printing case is the clearest of all the applications in this ladder.

Most of what a hair layer does is a modification of something the cloth already did. It adds a little warmth to a cloth that has some; it adds obstruction to a cloth that already blocks light; it adds contact area to a surface that already has some.

A printed edge is different: the cloth’s own contribution to the feather is very small, because a printed edge on a filament fabric — a fibre with no staple length at all — is sharp. So the hair layer is not modifying a number, it is supplying almost the whole of one, and the ratio between the treated and untreated states is a factor of ten rather than a few per cent.

That is what makes the operation worth its place at the head of every finishing route, and it is why the effect was noticed centuries before anybody could compute it.

What was counted, and how #

Three assertions, and one of them is about the cap rather than about a value.

That singeing shortens the feather, and that it buys a finer screen with it. Both are direct.

That raising takes it the other way — which is the check that the model is responding to the population rather than to the singeing flag, because a model that special-cased singeing would pass the first two and fail this.

The reach is capped at the flame’s own height in the arithmetic, explicitly, rather than emerging from the exponential. That is a modelling decision and it is stated: the exponential’s tail above the cap has no members, so a quantile taken on the untruncated curve would report a feather made of hairs that were burnt off.

Where the model stops #

Nothing here is a liquid. Ink is treated as travelling the whole length of any hair that bridges, which is an upper bound: a real hair carries paste only as far as the paste’s own viscosity and the hair’s surface allow, and a thick pigment paste travels much less far than a thin reactive dye liquor.

The threshold is a parameter. The feather is a quantile and the quantile is chosen, at one hair per fifty millimetres of edge. Moving it moves the feather logarithmically, so the number is robust to a factor of two in the threshold and not to a factor of a hundred.

The flame’s reach is a measurement with a wide bracket. Two hundred micrometres is an ordinary figure for a gas singeing and the operation is run at several intensities.

And the short population is not in it. It cannot bridge anything — its members are shorter than a fibre diameter or two — so leaving it out costs nothing here, which is one of the few places in this ladder where that is true.

Which cloths are worst, and it is not the hairiest #

There is a ranking here that goes the opposite way from the one a reader would guess, and it comes from the two laws the population obeys.

The feather is λ times a logarithm of the density. The density goes as the square root of the yarn count and the length does not move with the count at all. So a coarse yarn’s extra hairiness buys only a logarithm of extra feather: doubling the count raises the density by a factor of 1.41 and the feather by λ·ln(1.41), which is two hundred micrometres.

The length is the whole of the leverage and the length belongs to the fibre. A wool’s decay length is nearly twice a cotton’s, because it is fifty-six fibre diameters and a wool fibre is nearly twice as thick — so a wool cloth’s feather is nearly twice a cotton’s at the same density, and no amount of fine spinning changes it.

That is why printing on wool is difficult in a way printing on cotton is not, and why the wools that are printed are the ones with the finest fibre. It is also why singeing is worth so much: it removes the length dependence entirely by replacing it with a cap.

The edge is not displaced, it is smeared #

Calling the feather a quantile settles where the edge appears to be and leaves out something a printer cares about more: how wide the transition is between full colour and none.

For an exponential the answer is a fixed multiple of the decay length. The distance from the ninety per cent bridging level to the ten per cent one is λ ln 9, which for cotton’s six hundred and twenty micrometres is

1.36 millimetres.

That is comparable to the feather itself, so an unsinged print’s edge is not a line in the wrong place. It is a gradient a millimetre and a half wide, and asking where the edge is has no answer better than a threshold somebody chose.

Singeing changes the kind of the answer and not merely its size. The truncation puts every surviving hair inside two hundred micrometres, so the gradient cannot be wider than that — an edge rather than a gradient, and the width is set by the flame rather than by the fibre.

That is a better statement of what the operation buys than a factor of ten in ruling. It converts a soft boundary into a hard one, and the printer’s whole vocabulary of line, outline and register only means anything on the far side of that conversion.

And the flame lands on the eye’s own limit #

There is a coincidence worth noticing in the size of the cap, because it explains why nobody singes harder.

An eye at reading distance resolves about a hundred and twenty micrometres. The singed cap is two hundred — a factor of 1.7 above the limit, which is just visible as a softness if a viewer looks for it and invisible in ordinary use. The unsinged feather of two millimetres is seventeen times the limit and grossly visible.

So a single singeing takes a printed cotton from far outside the eye’s resolution to just inside it, and a second pass would buy something no viewer could see. The operation is calibrated, by accident and by centuries of practice, to human vision — and the flame’s reach, which is a property of a gas burner and a cloth speed, happens to sit where it needs to.

That also says what a finer process would have to be for. Nothing about a printed cotton’s appearance improves past this point; what improves is registration between colours, which is a machine problem, and the sharpness of a photographic halftone, which needs the paste rather than the flame.

A napped cloth feathers unequally #

One further consequence follows from the same integral and it is a defect with a direction in it.

The bridging count carries a factor of a half because only half the hairs point across the boundary the right way. That factor assumes the hairs are randomly oriented, which they are on a bare cloth.

A raised cloth’s hairs are combed. A raising machine works in one direction and leaves the fibres lying preferentially with it, which is what gives a napped fabric its way of the nap. So on such a cloth the half is not a half: it approaches one in the nap direction and zero against it.

A print on a napped cloth should therefore feather up to twice as far downstream of the nap as upstream, and the edge should be visibly asymmetric — soft on one side of a figure and comparatively crisp on the other.

That is a specific, testable prediction, it needs no new arithmetic, and it is consistent with the trade’s blanket rule against printing fine figures on raised cloths. The usual reason given is that the surface is not flat enough to meet the screen; this adds a second reason that survives however flat the cloth is pressed, because a combed hair carries ink along itself whether or not it was touching the screen.

The generalisation #

A boundary’s sharpness is set by the longest thing that crosses it, not by the average thing.

The transferable form is that any interface whose position is determined by transport has a fuzziness equal to the transport’s own reach, and if the reach has a distribution then the fuzziness is a quantile of it. Averages are useless here: a population whose mean reach is a tenth of a millimetre can put a visible edge two millimetres out, because visibility is a threshold and a threshold on an exponential is a logarithm.

The practical consequence is that removing the tail is worth far more than reducing the average, which is the same conclusion the two hairiness instruments reached from the other direction and the reason the trade’s preferred instrument is the wrong one for this question.

The one thing a printer can do that is not singeing #

The exponential has a second lever in it and it is one this collection can price.

The feather is λ ln(bridging density / threshold), so raising the threshold — making the printer’s acceptable level of stray colour higher — shortens the feather logarithmically. That is not a useful lever; a printer does not get to choose what registers as colour.

But the same logarithm says something about ink. Halving the amount of colour a single hair delivers halves the density at every distance, which shortens the feather by λ ln 2 — four hundred and thirty micrometres for cotton, which is a fifth of the untreated feather. A thicker, more pigment-loaded paste that travels less far along a fibre buys exactly that.

So paste rheology is worth a fixed increment and singeing is worth a cap, and the two do not compete: the increment is subtracted from whatever the cap allows. A printer with both has a feather set by the flame’s reach and a printer with neither has one set by a fibre’s length.

That framing also says why the historical progression went the way it did. The flame came first, centuries before anybody could formulate a paste, and it took the problem from insoluble to ordinary in one operation.

Who found it, and when #

Singeing before printing is universal practice and is centuries old — the operation predates gas singeing, having been done over open flames and hot plates. Its justification in the literature is a smooth surface and a clean print, stated as an outcome.

What is added here is the arithmetic: that the bridging count is an exponential with the hair population’s own decay length, that the feather is therefore a quantile, and that a flame’s value is a truncation rather than a reduction — which is why it buys an order of magnitude and a compact spinning frame would not.

Where the ladder goes next #

To the same material doing something useful rather than something to be burnt off. A woven filter beats its own rating finds that the hairs standing in a cloth’s holes catch particles ten times smaller than the holes themselves, which is not enough to filter anything and is exactly enough to start a cake.

And to the far end of the same ladder, where the protruding fibre is put there on purpose and at a length nobody has to guess: hair, nap and pile are one construction.