A nap is paid for by the taper of the pattern

Every commercial pattern for a garment prints two fabric requirements, one headed without nap and one with nap, and the second is always the larger. A velveteen, a brushed cotton, a corduroy, a melton or a satin with a directional sheen all take the larger figure, and the instruction behind it is simple: every piece is laid with its top pointing the same way along the cloth.

Raising moves the surface onto the hairs, and the raising wire and the brush after it do not leave those hairs standing straight. They lean them, all one way. The instruction follows from that lean, and the extra cloth follows from the instruction — but the size of the extra is not a fact about the nap at all. It is a fact about the shape of the pattern pieces, and for the commonest shape it has an exact value.

A leaning fibre shows more of itself to one side #

Take a nap whose fibres lie at a lay angle β to the cloth, all leaning the same way, and look at it from an elevation φ in the plane of the lean. A straight fibre covers an amount of the view in proportion to the sine of the angle between it and the line of sight.

From the side the fibres lean toward, that angle is φ − β. From the side they lean away from, it is φ + β. The two differ everywhere except straight overhead.

At a lay angle of 20 degrees and a viewer at 45, one side sees 2.14 times the fibre the other does. At 30 degrees of elevation it is 4.41 times, at 60 it is 1.53, and at the lay angle itself the side the fibres lean toward sees them exactly end-on, covering nothing.

So two panels sewn side by side, one turned end for end, present different amounts of fibre to the same eye. What brightness that makes depends on how the fibre and the ground beneath it reflect, which this geometry does not decide and which is why a velvet looks richer one way and a brushed cotton only paler — but a difference is guaranteed at every viewing angle but one, and the eye is good at seams.

The lay angle of any particular raised cloth is not measured here, and nothing below depends on its value. It depends only on there being one. The same hairs veil a highlight when they stand without order; laid one way by a brush, they carry a direction of their own instead.

The instruction is a restriction to translation #

A cutter lays pattern pieces on a cloth by moving them and turning them. For a cloth without a nap a piece may be turned end for end — rotated a half turn in the plane of the cloth — and a cutter uses that freedom constantly, because a tapered piece turned end for end nests against its neighbour.

A napped cloth takes that freedom away. Pieces can still be moved anywhere, but every one must keep its top pointing the same way along the bolt. In the vocabulary of geometry the allowed placements are translations only, and the question of what a nap costs becomes the question of how densely a shape packs when it may not be turned.

For a rectangle the answer is immediate: a rectangle turned end for end is the same rectangle, so nothing was ever gained by turning it and nothing is lost by forbidding it. A straight scarf, a rectangular curtain or a plain tube skirt costs nothing to cut from a napped cloth. Whatever a pattern envelope’s with-nap figure is paying for, it is not paying for those.

The same holds for the other freedom a cutter has, which is to lay a piece at an angle to the grain. A bias cut turns a piece an eighth of a turn and a nap does not forbid that, because the nap still runs one way along the piece; what the nap forbids is only the half turn, and the half turn is the move that nests tapers.

A gore, the commonest tapered piece #

The piece to analyse is a trapezoid: a panel wider at one end than the other, with straight sides. It is a skirt gore, a trouser leg to a first approximation, a flared sleeve, the panel of a bell tent. Its taper is there because a body is not a cylinder, which is why clothes need darts at all, and a gore is the dart’s answer spread along a seam. Call its wide end b, its narrow end a, its length H, and its taper r = a/b — 1 for a rectangle, nought for a triangle.

The worked panel throughout is a six-gore skirt’s: 14 centimetres at the waist and 32 at the hem including seam allowances, and 75 centimetres long, cut across a cloth 150 centimetres wide. Its taper is 0.44.

Laid two ways, a gore and its neighbour turned end for end make a parallelogram, and parallelograms tile. Each gore then takes (a + b)/2 of width on average, 23 centimetres, and six of them fit across 150 centimetres with room to spare.

Laid one way in a lane, every gore needs its full wide width, because its wide end sits beside its neighbour’s wide end. Four fit across 150 centimetres, and the skirt needs a second lane: 75 centimetres of cloth becomes 150.

The price in lanes is (1 − r)/(1 + r) #

In a long marker, where the edges of the cloth stop mattering, the lane arithmetic becomes a ratio. Two ways, a gore takes (a + b)/2; one way, b. The extra cloth one way is therefore

(1 − r)/(1 + r)

of what two ways would need. For the skirt gore that is 39.1 per cent. For a triangle it is a hundred per cent — the cloth doubles — and for a rectangle nought.

That is an upper bound on what a sensible cutter pays, because a lane is the least imaginative way to lay one-way pieces. It is also a price paid in exactly the currency a cutter pays a fault in: panels, not square metres. The interesting question is how much of it imagination can recover.

Staggering recovers exactly half #

The obvious improvement is to shift neighbouring columns of one-way gores half a length along the cloth, so that the narrow end of one sits beside the wide part of the next. Columns can then close up, and for the skirt gore they sit 27.5 centimetres apart instead of 32.

The general lattice is columns of gores head to toe, one length apart, with each neighbouring column shifted half a length and set (3b + a)/4 across. It covers a fraction 2(1 + r)/(3 + r) of the cloth, and the extra cloth it needs is

(1 − r)/(2(1 + r))

— exactly half the lane’s price, at every taper. The skirt gore costs 19.6 per cent instead of 39.1; a triangle 50 per cent instead of 100.

The exactness is not a coincidence of the arithmetic, and it is worth seeing why, because it also settles whether any cleverer one-way arrangement could do better.

Why nothing laid one way does better #

Minkowski proved the relevant fact about lattices more than a century ago. A lattice of translations packs a convex shape K — no two copies overlapping — exactly when it packs the shape ½(K − K), the set of half-differences of two points of K. That second shape is always centrally symmetric, and for a trapezoid it is a hexagon.

A centrally symmetric hexagon tiles the plane. So the lattice that tiles the hexagon packs the trapezoid, and it packs it at the largest density any lattice can: the trapezoid’s area over the hexagon’s, which is 4·area(K)/area(K − K).

The staggered columns are that lattice, and for the skirt gore they cover 83.6 per cent of the cloth against 71.9 in lanes. And a theorem of C. A. Rogers, published in 1951, closes the last gap: for convex shapes in the plane, no packing by translations — lattice or not — is denser than the best lattice. So half the lane’s price is not a good arrangement; it is the best there is, for a panel of that shape laid one way on an unbounded cloth.

That is the surprising connection this essay turns on. A with-nap allowance on a pattern envelope is, for a gored skirt, a quantity from the geometry of numbers: Minkowski’s difference body decides it, and the hexagon’s tiling is why it comes out as exactly one half.

On a real width the price comes in whole lengths #

A cloth is not unbounded, and on a finite width both lays lose to the edges. The losses do not average away for a single garment; they arrive as whole panel lengths.

Across 150 centimetres, two ways, six gores fit a width. One way, four fit a lane and five staggered columns fit, the stagger giving back one panel in five. Ten gores take 150 centimetres two ways, 225 in lanes and 187.5 staggered, which is half a length saved by the half-length shift.

The staircase shows the two things a single garment meets. The extra swings between nothing and a hundred per cent as pieces are added: one to three gores fit a single band either way; four fit one way only if they are in a lane; six are the worst case, a full second band. And over many pieces the extra settles toward what the widths allow, which on 150 centimetres is 20 per cent — five pieces per band against six — close to the 19.6 per cent of the unbounded lattice.

So the with-nap figure on an envelope is a finite-width number for a particular set of pieces, and a single pattern can be anywhere on the staircase. A skirt that happens to need six gores pays the whole of a second band.

What the envelope’s two figures are really measuring #

Put together, the difference between the two yardages on a pattern envelope is a sum over its pieces of something that depends on each piece’s taper and on how the pieces happen to share widths. It is not a percentage of the cloth, and a fixed percentage added to a fabric order for a napped cloth is right on average over many garments and wrong for any one.

Three pieces make the point. A rectangle costs nothing. A panel with the skirt gore’s taper costs between a fifth and two fifths of its own area, depending on whether it can be staggered. A triangle costs between half and all of it. A pattern made mostly of rectangles — a straight skirt, a shift dress, a tote — should carry a with-nap figure close to its without-nap one, and a pattern made of flared gores should carry one a third larger or more.

The same logic runs through the allowance a repeat demands, where every panel of a patterned cloth must start at the same phase and the price is a whole repeat per panel. Both are restrictions on where a piece may go, and both are charged in whole units of something the cloth has — a repeat there, a panel length here — rather than as a smooth share.

A one-way print pays the same price #

Nothing in the packing arithmetic knows why the pieces must lie one way. A print with a direction — flowers that grow upward, a figure that must not stand on its head — imposes the same restriction to translation, and its with-nap allowance is exactly the nap’s for the same pieces.

So is a satin whose lustre depends on the direction of its floats — a float reflects into a line and a line has an end that points somewhere — and so is a pile — corduroy is a cut float and velvet is cut apart, and both are laid with their pile running one way for exactly the reason the lean gives. The cost is a property of the pieces and the restriction, and the cloth’s contribution ends at imposing the restriction.

Where a cutter can escape it #

The arithmetic assumes the restriction is absolute, and in practice it bends in three places.

Small pieces can be turned. A pocket bag, a facing or an inside yoke that no one sees side by side with a main panel can be laid either way, and a cutter uses them to fill the gaps a one-way lay leaves — the same gaps a fault map lets a cutter place condemned cloth in. Those gaps are real cloth: in the stagger, the triangles between columns are 16.4 per cent of the marker for the skirt gore.

A piece can sometimes lie across the grain. A panel cut with its length across the cloth still has its nap running one way, and a cutter who turns a whole garment’s worth of pieces a quarter turn together keeps the nap consistent. That changes which dimension is limited by the width, and so where the staircase steps fall, but not the unbounded price.

And the eye tolerates some panels. A difference in the shade of two panels matters where they meet at a visible seam and hardly at all between a sleeve and a side panel seen at different angles. A cutter who knows which seams show can turn the pieces that do not, and the lean arithmetic says why the risk is real on exactly the seams that are seen head-on.

What was computed, and how #

The lean is a straight fibre at a lay angle β in the plane of the viewer’s line of sight at elevation φ; each side’s coverage is the absolute sine of the angle between fibre and line of sight, φ − β toward and φ + β away. The panels are trapezoids of top a, bottom b and length H. The two-way lane places alternate panels turned end for end with centres (a + b)/2 apart; the one-way lane places them b apart; the stagger is the lattice with vectors (0, H) and ((3b + a)/4, H/2).

On a finite width F, two-way lanes hold the most m with (m − 1)(a + b)/2 + b ≤ F, one-way lanes ⌊F/b⌋, and staggered columns ⌊(F − b)/((3b + a)/4)⌋ + 1 with alternate columns starting half a length along; the one-way length for n panels is the shorter of lanes and stagger, in whole or half panel lengths.

The lean was confirmed to look the same from both sides for upright fibres, to show more from the side leaned away from below the lay angle’s complement, and to be end-on at the lay angle. The price was confirmed against its closed forms at seven tapers from a triangle to a rectangle; the stagger was confirmed to meet the difference-body bound computed from the polygon’s own hull, to have a hexagonal difference body for every taper short of a rectangle, to place no panel over another, and to overlap when shrunk by half a per cent; a one-way lane was confirmed unable to close up; a panel and its turn were confirmed to make a parallelogram of exactly twice the panel’s area; the stagger’s extra was confirmed to be exactly half the lane’s; and no finite marker from one to twenty-four panels was shorter one way than two.

Where the arithmetic stops #

Real pattern pieces are not trapezoids. A trouser leg has a curved crotch, a sleeve a curved head, a bodice darts and a neckline. Curved pieces are not convex, Rogers’s theorem needs convexity, and the densest one-way packing of a real set of pieces is a nesting problem computed by marker software rather than a formula. The trapezoid is the case where the answer is exact, and it bounds the effect of taper; it does not price a real marker.

A marker holds many different pieces, and different pieces nest against each other in ways a single shape cannot. A two-way marker of mixed pieces is not perfectly dense either, so the true with-nap extra is a difference of two imperfect packings and can be smaller than either formula here.

The lean says a difference exists, not how large it looks. Nothing here computes the brightness of a nap from either side, and nothing measures a lay angle.

Still open: what a real marker’s with-nap allowance is #

The arithmetic predicts that the difference between a pattern’s two yardage figures should track the taper of its pieces — near nothing for rectangular patterns, a third or more for gored ones — and should jump in whole panel lengths as sizes change. Pattern envelopes print both figures across a range of sizes, so the prediction is testable against published numbers without cutting any cloth: tabulate the two yardages for a straight skirt and a six-gore skirt from the same publisher across every size, and see whether the gored pattern’s extra steps where the widths say it should while the straight one’s stays flat. Nothing here has done that tabulation.

Who found it, and when #

Laying napped and one-way cloths with every piece pointing the same way is ordinary cutting-room practice, and printing separate with-nap yardages is long-standing pattern-envelope convention. The difference-body criterion for lattice packing is Minkowski’s, and the equality of translation and lattice packing densities for plane convex shapes is Rogers’s, published in 1951. Applying them to a tapered pattern piece, finding the lane price (1 − r)/(1 + r), that the densest one-way lay recovers exactly half of it, and that on a finite width it arrives in whole panel lengths, was done here.