A net of three directions beats a voile one way at a time

A net over a voile beats through a harmonic. A net whose mesh is five times a voile’s pitch has no beat between the two fundamentals worth the name, but its fifth harmonic is a grid only a few per cent coarser than the voile, and the two make fringes as two sheers do — at a period set by how far that harmonic is from the voile as the eye sees it, with nulls wherever perspective carries the ratio through a whole number. That account treated both layers as grids of one direction and said, at its end, that the commonest curtain net is not.

A tulle’s mesh is hexagonal. Its threads run three ways at sixty degrees, and a voile’s run two ways at ninety. The question left over was what that pairing shows: one family of fringes, three — one for each of the net’s directions — or a lattice of them, and at what distances each vanishes.

The difference between the two layers is not a detail of manufacture. Of the seventeen plane groups a pattern can have, a draft can have only twelve: the five it cannot have are exactly the ones with threefold or sixfold rotation, because every symmetry of a woven cloth has to carry warp to warp and weft to weft on a square grid. A hexagonal net is a structure no loom can weave, which is why tulle is made on a bobbinet machine, and a net over a voile is a sixfold lattice laid over a fourfold one — two symmetries that cannot share a grid.

The answer is two families at right angles, and the two could hardly be less alike. One is exactly the one-directional beat. The other comes from somewhere a one-directional account has no place for, and it is wide, faint, and has a null where the first has none.

Two layers as sets of bars #

The model keeps the one-directional account’s simplifications and adds a direction. The net is three sets of thin straight bars at a pitch of 1.55 millimetres, each set a sixtieth of a turn from the next, covering a tenth of the pitch. The voile is two sets at 0.3 millimetres, at right angles, covering four tenths. The voile hangs fifty millimetres behind the net and the eye stands square in front, three metres away, so the voile is seen at a pitch of 0.295 millimetres.

Light passes where no bar of either layer is, so the pair’s transmittance is the product of all five sets’. A product of periodic patterns has a spectrum of two kinds of point, and both matter here.

Single-set points lie along each set’s normal at whole multiples of one over its pitch, with the one-dimensional harmonic amplitudes a bar grating has. Cross points are sums of two sets’ points, and their amplitude is the product of the two — for thin bars an order of magnitude weaker. A fringe is a beat between a net point and a voile point, at their difference: the closer the two points, the wider the fringes.

One net set beats exactly as a one-directional net does #

Hang the net with one of its thread directions lying along one of the voile’s. That set’s harmonics then lie along the same line in the spectrum as the voile’s first harmonic, and the pair is the one-directional problem.

The net’s fifth harmonic sits at 5/1.55 = 3.226 per millimetre and the voile at 1/0.295 = 3.388, and the difference makes fringes 6.13 millimetres apart. That is the one-directional account’s family to the last figure, confirmed against it at the same pitches and distances, and it is the strongest family the pair has.

The other two sets of the net lie at sixty degrees to that voile direction, and their harmonics run off along lines that never approach the voile’s point. They beat with it only at spacings finer than a thread, which an eye averages away.

The voile’s other direction meets no set at all #

The voile’s second direction is the interesting one. It is at right angles to the aligned net set, which means it is thirty degrees from each of the other two — and thirty degrees is as far as any direction can be from the nearest of three sets at sixty.

So no single set of the net has a harmonic anywhere near the voile’s second first harmonic. The nearest single-set point is close enough in length and thirty degrees off in direction, and the beat it makes is 0.58 millimetres — finer than two voile threads, a texture rather than a fringe.

If single sets were all there were, the answer would be one family. They are not. The net’s two unaligned sets together make points that are not on either set’s line: the sum of a harmonic of one and a harmonic of the other. Along the voile’s second direction those sums fall on a line of their own, at multiples of $\sqrt{3}$ over the net’s pitch — every one of them a cross point, none of them present in either set alone.

Two families, one strong and narrow and one faint and wide #

The cross line’s third point, at $3\sqrt{3}/1.55$ = 3.352 per millimetre, is within 0.0365 of the voile’s 3.388. That is a closer match than the aligned set’s fifth harmonic manages, so the faint family is the wide one.

The fringes are 27.4 millimetres apart, about four and a half times the strong family’s, at seven per cent of its strength. They run at right angles to the strong fringes: the strong family lies in bands across the voile threads parallel to the aligned set, and the faint one in bands across the others.

So the pair shows neither one family nor three nor a lattice. It shows one family in each of the voile’s two directions, produced by two different mechanisms — a single set’s harmonic one way and a line two sets make together the other — and the two differ fifteenfold in strength.

Sampling the superposition itself and reading its spectrum finds both families where the arithmetic puts them: at each predicted beat the sampled pattern has more than three times the power it has at the same frequency turned forty-five degrees, where no set of either layer lies. The profiles show the difference in character. Down the voile the fringes are regular and strong; across it they are a faint modulation that a 160-millimetre sample barely resolves.

That difference matters for the one use a moiré has beyond looking at it. A moiré is a vernier, and it magnifies the error too: a fringe reads a pitch difference at the ratio of its period to the pitch, and the ceiling on that gain is set by the thread-spacing irregularity of the layers. The faint family magnifies a voile pitch about ninety times where the strong one magnifies it about twenty, so it is the far more sensitive gauge of how a net and a voile sit against each other, and the far easier one to lose in a real net’s irregular mesh.

The two families vanish at different distances, and never together #

The one-directional account’s nulls came from perspective. With the net in front, the voile seen further back looks finer the closer the eye is, so the apparent ratio of net pitch to voile pitch falls as the eye recedes, toward the true ratio of 5.17. A family vanishes wherever that ratio passes the multiple it beats through.

The strong family beats through whole numbers: its nulls are at ratios six, seven and above, and the last of them is at 0.31 metres. Beyond a third of a metre it never vanishes. The faint family beats through multiples of $\sqrt{3}$, and $3\sqrt{3}$ = 5.196 lies just above the true ratio — so the apparent ratio passes it far out, at 8.76 metres, and the faint fringes swell without bound as a viewer walks back toward that distance and then shrink again beyond it.

The two families can never vanish at the same distance, because one needs the ratio at a whole number and the other at a whole number times $\sqrt{3}$, and $\sqrt{3}$ is irrational. A viewer who finds a distance at which the strong fringes disappear has found one at which the faint fringes are still there, and the reverse.

This is the same irrationality that makes a hexagonal net and a square voile what crystallographers call incommensurate: no finite repeat contains both lattices, so no amount of scaling one against the other brings every direction into register at once. Watered silk made its enormous fringe from two identical ribs at a tiny angle; here two lattices with no common repeat can make an enormous fringe in only one direction at a time.

A walker moves one family and leaves the other #

Two sheers make a moiré that walks with the viewer: fringes made by perspective move as the eye moves, because they are the parallax of the gap between the layers. A step of the eye shifts where the far layer appears only along the step, so it moves only the fringes whose bands stand across it.

The two families here have their bands at right angles, so a step moves one of them and leaves the other where it was. Hang the net with its aligned set horizontal, and the strong family’s bands are horizontal too: a person walking past sees the faint wide family slide across the window and the strong narrow one stand still, and only crouching or standing up moves the strong one. Hang the net a quarter turn round and the roles exchange.

Walking toward the window moves both, on their own schedules — the faint family swelling through its null at 8.76 metres while the strong one merely widens. Two layers are the product on average and nowhere, and the two families are the two directions in which this product is not its average, each answering a different motion of the eye.

Turning the net brings wide fringes back at other angles #

The net need not hang with a set aligned. Turned in its own plane, its sets and cross lines swing past the voile’s directions, and the families change with them.

Most angles give nothing wide: at 5, 15 and 25 degrees no family is wider than four millimetres, which reads as texture. But wide fringes return near 10.9 degrees and 19.1 degrees, and at neither angle is anything aligned in the ordinary sense. What has happened is that another line of the net’s lattice — made by two sets together, at an angle whose tangent is $\sqrt{3}/9$ — has swung onto one of the voile’s directions, and its length, $\sqrt{28}$ over the net’s pitch, happens to be within a per cent of the voile’s apparent frequency. The fringes it makes are some forty millimetres apart.

The angles belong to the distance. Move the eye and the voile’s apparent frequency moves, so different lattice lines come within reach and the wide-fringe angles shift. A net that shows no moiré from one chair can show a wide one from another, and the reason is which of its infinitely many lattice lines the voile happens to match.

A curtain complicates this further by not hanging flat. A curtain is gathered so that it is seen edge-on, and a gathered net presents its flanks at a spread of angles to the voile behind it, so every flank is a net turned by a different amount in depth — some near an aligned angle, some near a quiet one. The fringes a gathered net shows are a patchwork of the rotation chart, flank by flank, rather than one family across the window.

With the net behind, the roles move #

The same pair hung the other way round — voile in front, net fifty millimetres behind — puts the perspective on the net instead, and the families come out differently from the same spectra.

With the net behind, the strong family is 18.6 millimetres and the faint one 13.4: the strong family is now the wider. The ratio the eye sees rises toward the true ratio rather than falling to it, so different multiples sit near it. That is the one-directional account’s asymmetry between a net in front and a net behind, now carried into two directions with two different multiples, and it means which family dominates a window’s look depends on which layer is hung nearer the room — and, since a net in front gives the figure to the room, the hanging order was already a choice with consequences for privacy before it was one for fringes.

What was computed, and how #

Each layer is a set of straight bar gratings: the net three at 1.55 millimetres and a cover of a tenth, with normals at 90, 150 and 210 degrees to the first voile set’s normal; the voile two at 0.3 millimetres and a cover of four tenths, at right angles. The far layer is projected onto the near one by the distance over the distance plus the gap. The net’s spectrum is each set’s harmonics to the tenth order at amplitude |sin(πkc)|/(πk), and every sum of two sets’ harmonics at the product of their amplitudes, leaving out sums that land on a single set’s own line. Each fringe family is the net point nearest one of the voile’s first harmonics, among single-set points and among cross points separately; its spacing is one over the distance between the two, and it runs at right angles to their difference. A null is a distance at which the apparent pitch ratio equals a family’s multiple.

The aligned single-set family was confirmed equal to the one-directional harmonic beat at the same pitches and distance; the voile’s second direction was confirmed to have no single-set beat slower than three voile pitches and a cross beat slower than ten, on a line at multiples of $\sqrt{3}$ over the net pitch, at under a fifth of the strong family’s strength. The superposition was sampled at a fifth of a voile pitch over 160 millimetres with a raised-cosine window, and at each predicted family its power was confirmed more than three times its power at the same frequency turned forty-five degrees. No null of one family was found at a null of the other, and turned fifteen degrees the net’s widest family was confirmed under a fifth of the aligned pair’s.

Where the bars stop describing a tulle #

A bobbinet’s threads are not straight. A real tulle’s hexagonal holes are made by twisting and crossing threads, so each thread zigzags through the mesh rather than running straight across it. Its spectrum has the hexagonal lattice’s points, but their amplitudes are not those of three independent straight gratings, and in particular the cross points need not be an order weaker than the single ones. The two-family structure depends on the lattice and should survive; the fifteenfold strength ratio depends on the straight bars and may not.

The voile’s own harmonics are left out. A voile covering four tenths of its pitch has a second harmonic a third as strong as its first, and it makes fainter families of its own — one of them visible in the sampled profiles as residue between the predicted peaks.

The eye is square on and the layers are flat. A curtain seen obliquely foreshortens one direction and not the other, which changes the two families’ spacings by different amounts and could bring a new lattice line into register at a glance.

Visibility is not computed. A family’s amplitude is a Fourier coefficient, not a contrast a person can see, and whether a fringe seven per cent as strong as its neighbour is visible at all is a question about the eye.

Still open: what a bobbinet’s twisted mesh does to the faint family #

The whole distinction between a strong and a faint family rests on the net being three independent sets of straight bars, so that a cross point is a product of two small amplitudes. A bobbinet tulle is not that: its hexagonal mesh is made of threads that twist round one another at the corners of every hole, and its transmittance is not a product of three gratings at all. Computing its spectrum needs the mesh’s real geometry — the hole shape and the thread path round it — and the question it would answer is whether the $\sqrt{3}$ family a straight-bar net makes faintly is, in a real tulle, as strong as the aligned one. If it is, a tulle over a voile shows a check of two equally visible fringe families, and the account above has the right lattice and the wrong picture.

Who found it, and when #

Moiré between superposed gratings, including hexagonal ones, is old optics, and incommensurate lattices are a standard object in crystallography and surface physics. Applying the pair of layered sheers’ perspective arithmetic to a three-direction net over a two-direction voile, and finding one single-set family and one $\sqrt{3}$ cross family at right angles, their strengths and nulls, and the angles at which turning the net brings wide fringes back, was done here.