A corridor of mirrors counts their angle, not their leans

A mirror’s top edge waits for an eye above it ended in a barber’s shop. Two facing mirrors give a person standing between them a long row of their own images, and the row curves away and leaves the glass. Between upright mirrors it never would: a path from the eye back to itself starts and ends at one height, and an upright mirror leaves height alone. What curves it is a mirror that leans, which tips every line of sight it reflects; along the corridor the tips add, and the path sinks quadratically with the number of bounces until its middle falls off the bottom of the glass. With one mirror leaning back by τ, the corridor’s count came out close to $\sqrt{8\,\Delta h/(g\,\tau)}$ — the eye’s height above the glass’s lower edge, the gap and the lean — and counting images turned out to measure a lean of a tenth of a degree that no spirit level would notice.

Real shops hang their mirrors independently, and both may lean, by different amounts and either way. The essay ended by asking what the corridor says then: whether its count follows the same law with the two leans combined, and whether a person can recover both leans separately by counting twice, once from near each mirror, on the expectation that the corridor’s deepest point sits nearer the mirror that tips each path more.

The first part holds, and in a stronger form than asked. The second does not, and the reason it does not points to what does work.

Two reflections are a turn by the angle between the mirrors #

The shop is the earlier essay’s: mirrors 2.5 metres apart, glass from 0.9 to 1.9 metres up, a person standing 1 metre from the front mirror with their eye at 1.6 metres. Each mirror now leans by its own angle about its bottom edge, back (top away from the person between them) or forward. Every self-image is traced in three dimensions, as before: the eye reflected in each mirror of the path, last first, a straight line from the eye to that image, and every crossing required to land on its glass.

The drawing shows the row of self-images laid out where each one lies: the image after two reflections a few metres beyond the front glass, after four twice as far, and so on. With the front mirror leaning back a quarter of a degree and the back mirror upright, the row bends down steadily and the twenty-two images nearest the eye are seen before one leaves the glass. With both mirrors leaning back an eighth of a degree it bends exactly as much. With the front mirror back 0.4 degrees and the back mirror forward 0.15 it bends exactly as much again.

What they share is the sum of their leans back, and the reason is in two mirrors make one turn. A reflection in one mirror followed by a reflection in another is a rotation about the line where the two mirrors’ planes meet, by twice the angle between them. Two facing mirrors leaning back by $\tau_0$ and $\tau_1$ have planes that meet far below the floor at an angle of $\tau_0 + \tau_1$. So every round trip across the gap turns a line of sight by twice that sum, and what was a quadratic sink for one leaning mirror is a quadratic sink set by the angle between two. How that angle is shared between the mirrors does not enter.

The count depends on the sum alone #

The claim can be tested against every arrangement rather than three.

Forty-two pairs of leans, each mirror leaning anywhere from half a degree forward to a degree back in quarter-degree steps, give counts that fall on one curve when plotted against the sum of the two leans. Pairs that share a sum differ by at most one image: six pairs summing to a quarter of a degree back all give 22, seven summing to half a degree all give 15, four summing to a quarter forward give 14 or 15. A sum forward sends the corridor up and out of the top of the glass instead of down and out of the bottom, and follows the same curve, measured from the 0.3 metres between the eye and the top edge instead of the 0.7 between the eye and the bottom.

So the first half of the question has its answer: the square-root law holds with the sum of the leans back in place of one lean. In the terms the question used — leans in either direction — leaning both back adds their tips and leaning one back and one forward subtracts them. A shop whose two mirrors lean back a tenth of a degree each has the corridor of a single mirror leaning back a fifth.

Counting twice gives the same number twice #

The second half asked whether counting from near each mirror would tell the two leans apart, on the expectation that the deepest point of each path sits nearer the mirror that tips it more.

It does not. From anywhere along the gap — 20 centimetres from the front mirror to 20 from the back — a front mirror leaning back a quarter of a degree with the back one upright gives 22 or 23 images. The reverse arrangement, front upright and back leaning a quarter, gives 22 or 23 from the same places. Two mirrors leaning a tenth each give 25 from every one of eight places.

The expectation was reasonable and wrong in an instructive way. Moving along the gap does move where along each path its deepest point falls, but not how deep it is. A path’s sink is set by how much its line of sight has turned by the middle of its journey, and the turn per round trip is the angle between the mirrors wherever the round trip starts. What moves the count by one image now and then is only where that deepest point lands between two bounces. A count, from anywhere, is a reading of the angle between the mirrors and of nothing else.

Two mirrors tilted together keep every image #

The question also predicted that a pair leaning in opposite directions by the same angle — one back, one forward — would send the corridor straight on, as if both were upright. Such a pair is two parallel mirrors tilted together: their planes never meet, the angle between them is nought, and a round trip turns a line of sight by nothing at all.

The prediction holds. Two mirrors tilted together by a quarter of a degree, or by four-tenths, keep every one of the first sixty self-images on the glass, as upright mirrors do. But the corridor is not as if both were upright, and the profile shows the difference: the row runs straight, as for upright mirrors, but tilted, along the two mirrors’ common normal. Each image after a round trip lies lower than the one before by twice the gap times the tilt, and the whole row points a quarter of a degree below level. The path to every image strikes each mirror square-on to its tilted face and lands on the glass at the eye’s own place, which is why none is ever lost.

So there is a component of a pair’s leans that no count can see at all: the tilt they share. A shop whose mirrors are both hung a degree out of plumb, the same way, has a corridor as long as a perfect shop’s.

A photograph reads what a count cannot #

The count is a single number, and it can only report one thing. The corridor holds more than a count: every image is somewhere, and a photograph taken from the eye records where.

The image after each round trip appears lower than the one before by exactly the sum of the two leans: a quarter of a degree a round trip for every pair here that sums to a quarter. That is the angle between the mirrors read as a slope, from the images’ directions rather than from where the row ends. The rows of different pairs with one sum are parallel and differ only in where they start. To first order the start is $-\tau_0 + (x/g)(\tau_0 + \tau_1)$ for an eye $x$ from the front mirror in a gap $g$: the front mirror’s lean, less the sum weighted by how far across the gap the eye stands. With the front mirror back a quarter of a degree and the back upright, the first image appears 0.40 degrees below level; with the leans swapped, 0.15; with an eighth each, 0.275. The two tilted together give a flat row at 0.25 below level — the shared tilt itself.

So a straight line through the images’ elevations, read against the number of round trips, gives both leans: the sum from its slope, and the share from where it starts, once the person’s place in the gap is known — which a tape measure gives.

Reading the first six even images’ elevations to 0.03 degrees — a pixel or two on a phone camera’s ordinary lens — the sum comes back to 0.0074 degrees and each lean to 0.026 to 0.033. At 0.01 degrees, the sum to 0.0025 and each lean to about a hundredth. The count, by comparison, fixes the sum only to about two parts in the count it gives, 0.023 degrees for a corridor of 22, and fixes neither lean. A photograph is three times the instrument a count is for the angle between the mirrors, and the only instrument of the two for how that angle is shared.

The camera has to know where level is #

The share has a catch, and it is the same one the shared tilt revealed. Every elevation is measured from level, and a camera that is itself tilted by a tenth of a degree adds a tenth of a degree to every image alike. The slope through the images does not change, so the sum comes back as well as ever, 0.0074 degrees. Where the row starts moves by the camera’s tilt, and both leans move with it, in opposite senses: with the camera tilted a tenth of a degree, each lean comes back a tenth of a degree out, 0.104 and 0.106 against 0.026 and 0.033 from a level camera.

That is not a defect of the method. It is the shared tilt again, seen from the camera: a pair of mirrors tilted together and a camera tilted the opposite way produce identical photographs, because the photograph only records directions relative to the camera. The angle between the mirrors is relative to nothing and is read from the picture alone; how the mirrors stand relative to gravity needs gravity — a spirit level on the camera, a plumb line in the picture, or the shop’s door frame trusted to be vertical. A mirror is a second camera placed a virtual eye behind each reflection; a corridor is a row of such eyes, and like any row of cameras it can be read for their arrangement among themselves without a single external reference, and for their arrangement in the world only with one.

How a fitter would use the two readings #

The two readings divide a fitter’s job in the order a fitter would do it. Hanging a pair of mirrors to face each other properly is two separate tasks: making them parallel, so the corridor runs on, and making them plumb, so the room’s reflection is not tipped. The corridor separates the tasks as cleanly as the geometry does.

Parallel comes first and needs nothing but the corridor. Stand anywhere between the mirrors, count, adjust either mirror’s lean, and count again: the count grows as the angle between the mirrors closes and is unlimited when they are parallel, whichever mirror was adjusted. A count of 22 says a quarter of a degree between them, 35 a tenth, and a corridor that runs to the side edges before it reaches the top or bottom says the mirrors are parallel to better than the count can tell. No level, no tape, no knowledge of which mirror was wrong.

Plumb comes second and needs a level, because nothing inside a pair of parallel mirrors knows which way is down. A spirit level against either mirror finishes the job, since the mirrors are now parallel and one plumb mirror makes both plumb. Or a photograph of the corridor taken with a levelled camera reads the shared tilt from the row’s direction alone.

The order matters because the other order wastes work. Plumbing each mirror with a spirit level first leaves each a little out by the level’s own error — a good builder’s level reads to a twentieth of a degree — and two mirrors each a twentieth out in opposite senses leave a tenth between them, a corridor of 35 images with a visible droop at its end. The corridor is the finer instrument for the angle between the mirrors, by the earlier essay’s arithmetic; the level is the only instrument for their attitude. Each should be used for what only it can read.

There is an older relative of this division. Measured down from the waterline used still water as a mirror that reverses height and keeps everything else, the opposite of an upright mirror; a pond is plumb by gravity and needs no fitter. And two mirrors show fewer images than they make found that glass cut short in its own plane removes images by bounce count. A corridor removes them by bounce count too, but for a different reason — each round trip adds to the turn — and the turn, unlike the glass’s length, is something the fitter can change.

What the corridor measures, and what it does not #

Put together: a corridor of self-images between two leaning mirrors ends where it leaves the glass, at a count set by the sum of the leans, the angle between the mirrors, by the earlier essay’s square-root law. The count is the same from anywhere along the gap, so counting twice cannot divide the angle between the mirrors. Two mirrors tilted together keep the whole corridor, tilted; a count is blind to the tilt they share. A photograph of the corridor reads the angle between them three times better than a count, from the slope of its images’ elevations, and reads how the angle is shared from where the row starts — as well as the camera knows its own level, and no better.

Two mirrors are three cameras used a pair of mirrors as extra viewpoints in a single photograph, and a mirror that is not parallel to the wall measured what a mirror turned about a vertical axis does to one reflection. The corridor combines both: it is many viewpoints, arranged by the mirrors’ relative attitude, and every one of them a little more turned than the last. Its length is a reading of the turn; its direction is a reading of the attitude; and its direction means something only against level.

What the tracing assumes #

The mirrors are flat and parallel along their width. Each leans only about its bottom edge, so the angle between them lies in the vertical plane through the eye. A mirror also turned about a vertical axis makes the corridor curve sideways as well, by the horizontal angle between the mirrors, and leave through a side edge; the turn per round trip is then the full angle between the two planes, and the count reads it in whatever direction it points.

The eye is a point and the person transparent. As in the earlier essays, a real head blocks the straight-back path, and the corridor is usually seen past the shoulder, slightly off the normal. Seen off the normal the round trips also step sideways, but the turn per round trip is unchanged, and so is everything a count or a slope reads.

The elevations are read independently. The photograph’s error is taken as independent from image to image. A lens’s distortion is not: it bends the row of images by an amount that grows towards the edge of the frame, and a corridor that runs towards a corner of the picture would read a lens’s barrel as a bend. A reader should keep the corridor near the centre of the frame or correct the lens first.

The share is first order. The start of the row is $-\tau_0 + (x/g)(\tau_0 + \tau_1)$ to first order in the leans; the mirrors pivot on their bottom edges, so at eye height the gap itself changes by the eye’s height times each lean, which shifts the start by a few ten-thousandths of a degree at these leans — far below the photograph’s own error, but not nothing for a mirror leaning a degree or more.

Still open: a corridor seen past the shoulder in a narrow wedge #

The facing pair is the limit of two mirrors meeting at an angle as that angle closes, and two people between mirrors see each other equally counted images in wedges of fifty degrees. Between the two lies the case every barber’s customer actually sees: two mirrors that are not quite facing, a few degrees off parallel about the vertical, so that the corridor curves sideways as well as down and leaves through a side edge long before it leaves the bottom.

The measurement that settles what such a corridor says traces the self-images between two mirrors turned a stated angle from facing about the vertical and leaning by stated angles back, and asks whether the count still depends only on the full angle between the two planes, whichever way it points; which edge — side, top or bottom — the corridor leaves through, and at what count, as the turn and the lean are traded; and whether a photograph, which sees the row curve in two directions at once, reads the turn and the lean separately from the two components of the row’s bend, as it reads the sum of the leans here, without needing to know where level is for the angle between the mirrors, and needing it only for how that angle is shared.