Two sheers make a moiré that walks with the viewer

Every moiré computed so far has been between two grids pressed together. Watered silk is two plies of one ribbed cloth folded and flattened at a small angle; the reed’s mark is a grouping laid over a weave repeat in the same warp; a colour order beats against the weave it is threaded on. In each the two grids touch, and a beat needs something to differ between them — an angle, a pitch, a period.

Two sheer curtains hung a few centimetres apart need neither. They can be cut from the same bolt, hung square to each other and to the window, and they still make a moiré, because one of them is further from the eye than the other. A grid further away looks finer, and that is all the difference a beat requires.

A grid further away looks finer #

Seen from a distance V, a grid of pitch p lying a further D behind a near grid of the same pitch subtends the angle a grid of pitch p·V/(V + D) would subtend at the near one’s distance. Project the far grid onto the near layer along the rays to the eye and that is its pitch there: a fraction D/(V + D) finer.

Two grids of pitches p and p·V/(V + D) beat at the product of the two over their difference, which works out to exactly p·V/D. Nothing else enters. The layers need not be at an angle, their threads need not differ, and the result holds at any distance at which the layers can be told apart from a single sheet.

The numbers are ordinary. A voile of about thirty-three ends to the centimetre has a pitch of 0.3 millimetres. Two layers of it five centimetres apart, seen from across a room three metres away, are in register every eighteen millimetres — sixty thread pitches, which is the gain the geometry supplies free.

A pair of insect screens, of the ordinary eighteen threads to the inch, two and a half centimetres apart and seen from two metres, beat at a spacing of about eleven centimetres. That is the broad, slow band anybody who has looked through a double screen will recognise, and it has the same arithmetic.

The fringes are the same size from everywhere #

The spacing p·V/D grows with the distance, so stepping back spreads the fringes across the curtain. The angle they cover at the eye does not change at all: it is the spacing divided by the distance, which is p/D, and V has cancelled.

At a five-centimetre gap that angle is a third of a degree, about two thirds of the width of the moon. From one metre the fringes are six millimetres apart on the curtain; from six metres they are thirty-six. A person walking towards the window sees the pattern stay exactly the same size while it shrinks on the cloth.

That is the first thing that separates this moiré from every other one on a textile. A watered figure’s size is fixed on the cloth — it was made by a fold at an angle and it stays where the pressing put it — so it looks larger close to and smaller from across a room, the way the weave does. The double curtain’s pattern does the opposite of the cloth it is made from.

The gap is the only lever. Halving it doubles the angular size of the fringe, and a pair of curtains that touch in places and hang apart in others shows its largest fringes where they are closest — up to the point where they touch, where there is no fringe at all.

Why the pattern walks #

A person walking past a double sheer sees the bands slide across it, keeping pace. That is not an impression and it is not approximate.

Move the eye sideways by h. The rays to the far layer swing, and its projection onto the near layer slides by h·D/(V + D) — a small fraction of the step, because the far layer is only a little further away. But the moiré magnifies any displacement between its two grids by its own gain, which here is (V + D)/D. The two factors cancel exactly.

A fringe on two identical layers moves across the near layer by precisely the distance the eye moves, at every distance and for every gap.

A feature that moves one for one with the eye keeps the same direction from the eye as the eye moves. That is what the horizon does, and the moon from a moving car: the fringes of two identical layers have the parallax of an object infinitely far away. A person standing still and swaying sees the curtain’s threads shift against the window frame and sees the fringes hold their place in the view, as though they lay beyond the glass.

Whether a pair of eyes places them there is a question about vision, and nothing here models it. The geometry is simpler than the perception. The two eyes are six and a half centimetres apart, so the pattern each sees on the curtain is the other’s displaced by exactly that — which is the disparity of something at infinity. But the fringes are eighteen millimetres apart, and a pattern that repeats more finely than the distance between the eyes can be paired fringe to neighbouring fringe as readily as fringe to fringe, which is the old wallpaper illusion; which pairing a viewer’s eyes settle on is not something arithmetic can decide.

A fringe fixed to the viewer and a fringe fixed to the cloth #

That gives a test anybody can make, and it separates two things that look identical.

The vernier essay found that a moiré is an instrument: it reads a pitch difference at the magnification of its own gain, and it reads the cloth’s irregularity at that magnification too. A banded appearance in a sheer could be either — an irregularity in one layer, magnified by a beat between the two, or a fault in a single layer that no beat is involved in at all.

Step sideways. A fault in the cloth stays on the cloth. A beat between two identical layers stays in the view and slides across the cloth as the eye moves. A beat between two layers of different pitch does something in between, at a rate the next section computes — and the rate is itself a measurement.

A stretch, and the one distance it cancels #

Real curtains are not identical, even when they come off one bolt. A sheer hung from a rod carries its own weight, the width it is gathered to changes its spacing, and a woven cloth pulled in one direction narrows in the other, so a layer hung under a little more tension than its partner is a layer of slightly different pitch.

Suppose the far layer is coarser by a fraction ε. Perspective makes it look finer by D/(V + D), and the stretch makes it coarser by ε. At one distance the two cancel exactly, the layers look identical, and the beat has an infinite period. That distance is D/ε.

For a five-centimetre gap and a one per cent stretch the null is at five metres. Walking back from the window, the bands widen, swell until one fills the whole curtain, and then begin to narrow again as the walker continues past five metres.

If it is the near layer that is coarser, there is no null: perspective and the stretch both make the far layer look finer than the near one, the two effects add, and the fringes never vanish at any distance.

The null says which layer is coarser, and by how much #

That asymmetry makes the double curtain a measuring instrument of an unusual kind: one read by walking.

A pair whose fringes swell to nothing at some distance has its coarser layer at the back, and the distance at which they vanish gives the stretch directly: ε = D/V, with no pitch in it and no count of threads. Five centimetres of gap and a null at five metres is a one per cent difference in pitch; a null at two and a half metres is two per cent.

The same distance is where the motion reverses. Short of the null the fringes walk with a viewer and faster than one; beyond it they walk against. At three metres, with the far layer one per cent coarser, they travel two and a half times as far as the eye does. At eight metres they travel one and two thirds times as far, the other way.

Which is a second reading of the same quantity, and it needs no null to be reached. A step sideways and a glance at how far the bands went gives D/(D − Vε), and with the gap and the distance known that is ε again. A thread counter reads a sett to about one part in twenty-five; this reads a one per cent difference between two layers from across a room.

A mesh beats in both directions at once #

Everything above has been about one family of threads, and a woven sheer has two. Its ends are a grid running up the cloth and its picks a grid running across it, and each beats with its counterpart in the other layer by exactly the same perspective. So a double sheer shows two families of fringes at right angles, and the moiré is itself a grid, with the proportions of the cloth that made it.

A square mesh makes a square grid of fringes. A cloth set more closely one way than the other does not. An insect screen of eighteen threads to the inch one way and sixteen the other has pitches of 1.41 and 1.59 millimetres, and two panes of it two and a half centimetres apart, seen from two metres, beat at 113 millimetres in one direction and 127 in the other — a moiré grid an eighth longer than it is wide, which is the mesh’s own ratio magnified eighty times.

The stretch argument then splits in two. A sheer hung from a rod is pulled down by its own weight and gathered across its width, so its two pitches change by different amounts. Walking back from such a pair, one family of fringes can swell and vanish at its own null while the other goes on narrowing, and the distance at which the first disappears measures the stretch in that direction alone. The bands running up the curtain and the bands running across it are two separate instruments sharing a pair of layers, and they read two separate numbers.

How far away the fringes survive #

The fringes cover a fixed angle from every distance, but they do not keep their quality. Their magnification is the gain V/D, and the gain grows as the viewer steps back.

The vernier essay’s finding applies without change. A moiré stays countable while the spacing irregularity of its grids times its gain is under 0.84 — inversely, because a single misplaced thread against the pitch difference is what splits a fringe. For two identical layers that puts a ceiling on the distance: V = 0.84·D/cv.

For a five-centimetre gap and a one per cent irregularity that is 4.2 metres. Close to, the pattern is ruled and regular; from further off it keeps its apparent size and starts to wander, fork and close up, which is what watered silk does at a much higher gain. A wider gap pushes the ceiling out in proportion, and a pair of layers nearly touching keeps a legible pattern only a short step away.

This runs against expectation in a small and useful way. Most textile effects become easier to see from further off, because a random error hides and a periodic one shows once the eye stops resolving threads. The double curtain’s fringes become harder to read, because the thing that makes them visible is the same magnification that shows the irregularity.

What a legible double curtain says about its cloth #

The vernier essay ended on a measurement nobody had made. No figure for the spacing irregularity of a real cloth exists anywhere in its arithmetic, and the inference it could draw was a bound: watered silk exists, so a loom lays its picks far more evenly than a yarn’s diameter varies.

The double curtain gives the same kind of bound, from an object in a great many windows. A pair of sheers five centimetres apart that shows clean, countable bands from four metres has a thread-spacing irregularity below about one per cent. From eight metres, below half of one.

That is an inference about the cloth from an appearance, and it carries the vernier essay’s caveats with it: the simulation behind 0.84 is one-dimensional, its spacings are independent, and “countable” is a sharper criterion than “looks tidy”. What it adds is a second route to the same missing number, and one that needs no pressing, no fold and no silk — only two layers and a room to walk back across.

A fold beats harder than a gap #

Curtains are rarely hung flat, and a fold changes the arithmetic more than the gap does.

A thread spaced along a layer that slopes away from the eye is foreshortened: seen square-on, a slope of β brings its threads closer by the cosine of β. At twenty-five degrees that is nine per cent. The gap at three metres makes a layer five centimetres further back look finer by under two per cent. The slope of a gentle fold beats more than five times harder than the gap it hangs across.

So the pattern on a folded pair is a map of the folds rather than of the gap. The fringes crowd into fine bands wherever the far layer is steepest and open out across its crests and troughs, and they do so symmetrically either side of a crest, because the foreshortening does not care which way a slope faces.

Where the near layer is folded too, its own slopes foreshorten it in the same way — which works the opposite way round, making the near layer look finer and the far one comparatively coarser. Two layers whose steepest places lie one in front of the other cancel their foreshortening there and leave only the gap; two whose steep places fall in front of each other’s crests add a beat of their own at every slope. A real pair of gathered curtains is a mixture of both, which is why its pattern is a patchwork of fine and coarse bands that changes with every draught.

How the fringes were found #

The closed forms are four lines of geometry, and every one of them was checked by doing what an eye does rather than by rearranging the algebra.

A point on the near layer is joined to the eye by a ray, the ray is continued until it meets the far layer, and the positions of the two grids at those two points are counted in their own pitches. Where the difference passes a whole number the gaps line up. Scanning across the near layer at an eighth of a pitch and interpolating those crossings gives the fringes, and their mean spacing agrees with p·V/D to a fifth of a per cent at one, three and six metres.

The motion of a fringe was found the same way: the fringe nearest the line of sight located, the eye moved a quarter of a millimetre, the scan repeated, and the fringe found again. For a folded far layer the ray has to be solved rather than written down, because where it meets the layer depends on how deep the layer is where it meets it; a few iterations settle it, and the far grid’s threads are counted along the curve rather than across the room.

Where the model stops #

The layers are one-dimensional grids. A sheer has warp and weft, and each system beats with its counterpart in the other layer. The perspective argument applies to both equally, so a square mesh behind a square mesh makes a square grid of fringes — but a cloth set more closely one way than the other makes a moiré grid with the cloth’s own proportions, and a stretch in one direction moves one family of fringes and not the other.

The eye is a point and looks square-on. Viewed obliquely the projected pitches change by a cosine as well, and a cloth has a thickness that closes its own holes at an angle, both of which alter what is seen at a steep angle and neither of which is computed here.

Nothing is said about contrast. How dark a fringe is depends on how much of each layer is thread, which is what two layers pass on average and nowhere in particular; a sheer of low cover makes faint fringes, and one whose threads let light through makes fainter ones still, for the reason opacity is not cover.

And the irregularity is still unmeasured. Every distance quoted for the ceiling is conditional on a coefficient of variation that no measurement here supplies. The ceilings are drawn at three values for that reason, and the bound above runs from the appearance to the cloth rather than the other way.

Still open: what a fringe does between two cloths of different weave #

Every result here takes the two layers to be the same grid. The commonest double window is not: a net in front of a voile, or a lace panel over a lining, is a coarse grid and a fine one, and those beat not at the difference of their pitches but at the difference between one pitch and a multiple of the other. The perspective shift then moves which multiple is nearest as the viewer walks back, so the pattern should jump between families of fringes at particular distances rather than growing smoothly — a prediction with the same geometry and a harmonic in it, and not worked out here.