Two mirrors nearly facing bend the corridor two ways, and each bend runs out at its own edge

A corridor of mirrors counts their angle, not their leans stood a person between two facing mirrors that each lean back a little, and followed the corridor of self-images until it left the glass. The count depended on the two leans only through their sum, which is the angle between the mirrors’ planes; counting from near either mirror gave the same number; two mirrors tilted together kept every image. A photograph did better than a count: each round trip lowered the image by the sum of the leans, so the slope through the images read that sum, and where the row started shared it between the two mirrors — given a camera that knew which way was level.

That essay assumed both mirrors faced each other squarely across the room, so the angle between them lay wholly in the vertical plane. The case a barber’s customer actually sees is different. The mirrors are a few degrees from facing about the vertical as well, the corridor curves away sideways as well as down, and the customer sees it past their own shoulder, where it leaves through a side of the glass long before the bottom. The essay ended by asking whether the count then depends only on the full angle between the planes, whichever way it points; which edge the corridor leaves through; and whether a photograph can read the turn and the lean apart.

Two bends, one for each part of the angle #

The shop is the earlier essay’s — mirrors 2.5 metres apart, glass from 0.9 to 1.9 metres up and now 1.2 metres wide, a person standing 1 metre from the front mirror with their eye 1.6 metres up in the middle of the glass — and each mirror now has a turn about the vertical through its middle as well as a lean about its bottom edge. A turn is counted positive when it swings the mirror’s right-hand end towards the other mirror, so two positive turns close the pair into a narrow wedge, just as two leans back close it into one opening downwards.

Two mirrors make one turn is the reason the earlier essay’s rule carries over. Two reflections, one in each mirror, are a rotation about the line where the two mirrors’ planes meet, by twice the angle between them. When the mirrors lean, that line runs horizontally far below the floor; when they are turned, it runs vertically far to one side; when they do both, it runs at a slant, and the rotation has a part about the horizontal and a part about the vertical. The part about the horizontal is the sum of the leans, and it steps each round trip’s image down by that much; the part about the vertical is the sum of the turns, and it steps the image sideways by that much. The photograph shows the two as two independent slopes.

In the barber’s shop drawn here — the front mirror leaning back a tenth of a degree and turned a degree and a half, the back mirror upright and facing — the images step 0.10 degrees down and 1.50 degrees sideways each round trip. Eight are seen before one leaves through the side of the glass. A mirror leaning back a quarter of a degree and not turned gives 22 images, stepping straight down, and loses the twenty-third through the bottom. Two mirrors turned together by equal and opposite amounts — parallel mirrors standing at an angle to the room — step the images sideways by nothing; the row only stands aside, as the earlier essay’s mirrors tilted together kept their row straight but tilted.

Each bend runs out against its own edge #

The count is where the question expected a single law, and there is not one.

The lean sinks every path towards the bottom edge of the glass, 0.7 metres below the eye; the turn swings it towards a side edge, 0.6 metres to the side. Each grows as the square of the number of bounces, by the earlier essay’s arithmetic, and each runs out against its own edge. The corridor ends at whichever runs out first, so its count is the smaller of two square-root laws. With leans summing to a quarter of a degree and no turn, 22 images and out through the bottom. Add a turn of a tenth of a degree: still 22, still the bottom — the sideways swing has not reached its edge by then. A quarter of a degree of turn: 21, out through the side, and from there on the lean makes no difference at all. Half a degree of turn gives 15 whether the leans sum to nothing, to a quarter or to a half. With leans summing to half a degree the bottom holds out until the turn reaches half a degree too, when the side takes over.

So the corridor’s count reads one part of the angle between the mirrors, the part that reaches its edge first, and is blind to the other part entirely. A barber’s mirror turned a degree and a half has a corridor that says nothing whatever about whether the mirrors lean.

The count does not follow the angle’s size #

The question asked whether the count depends only on the full angle between the planes. The same arithmetic says it cannot, and the measurement shows by how much.

Two mirrors whose planes meet at a quarter of a degree give 22 self-images when the angle is all lean and 21 when it is all turn — the bottom edge a little further than the side. Pointed between, the angle splits into a smaller lean and a smaller turn, each of which lasts longer against its own edge than the whole angle would against either, and the corridor outlasts both pure cases: 23 at fifteen degrees, 24 at thirty, 25 at forty-five, where the two parts run out together. At a tenth of a degree the counts are 35 all lean, 33 all turn and 39 between; at half a degree, 15, 15 and 17. The angle’s size is the same along each row and the count is not.

A round mirror with the eye at its centre would make the size of the angle the whole story, because every direction would have the same edge. A rectangular glass — which every barber’s mirror is — measures the angle’s two parts, each against the distance to the edge it heads for. The earlier essay’s single law was the special case in which one part is zero.

Where to stand #

The edges are measured from the eye, and the eye can move.

Between mirrors turned half a degree, a person standing in the middle of 1.2-metre glass sees 15 self-images. Standing 40 centimetres towards the side the corridor bends away from, 19; towards the side it bends towards, 8. Turned a degree, the same moves give 13 and 5 against 10 in the middle. Wider glass helps as the square root of its half-width: turned half a degree, 12 images on glass 0.8 metres wide, 15 on 1.2, 19 on 2 and 23 on 3. Along the gap, as the earlier essay found for the leans, standing nearer either mirror changes nothing: 14 or 15 images from anywhere between 30 centimetres from the front glass and 30 from the back.

That is the customer’s habit explained. Seen straight back, a person’s own head blocks the corridor; seen past the shoulder, the corridor runs alongside it. The side to look past is the side the corridor bends away from, which a customer finds by trying both, and the count they then see is a little larger for having moved.

A photograph reads both bends, and needs nothing to read them #

The count reads at most one part of the angle and is affected by where the person stands. The images’ directions are affected by neither.

The mirrors here lean back 0.2 and 0.05 degrees and are turned 0.3 and 0.2 — a corridor of 14 self-images before one leaves through a side. A photograph from the eye that reads the first six even images’ directions to 0.03 degrees, a pixel or two on a phone, gives the sum of the leans to 0.0076 degrees and the sum of the turns to 0.0068: two straight lines through the images’ elevations and azimuths against the number of round trips, one slope each. At 0.01 degrees the two sums come back to 0.0025 and 0.0023; at 0.1, to 0.025 and 0.023. Neither slope needs anything outside the picture, because a slope is a difference between images, and anything the camera does to every image alike leaves it where it was.

Where each row starts shares its sum between the two mirrors, exactly as the earlier essay found for the leans alone: to first order the row starts at minus the front mirror’s own angle plus the sum weighted by how far across the gap the eye stands. From a level camera that knows where it points, the front mirror’s lean comes back to 0.026 degrees and its turn to 0.025.

Gravity for the lean, the room for the turn #

The two shares need different things, and that difference is the new part.

A camera pitched a tenth of a degree, believing itself level, adds a tenth of a degree to every image’s elevation. The slopes do not move; the lean’s share does, to 0.103 degrees out, and the turn’s share does not move at all. A camera turned a tenth of a degree from where it believes it points adds a tenth to every azimuth. The slopes do not move; the turn’s share does, to 0.104 out, and the lean’s share does not move. Each share needs its own reference, and the two references are different kinds of thing.

The lean of a mirror is its attitude to gravity, so its share needs gravity: a spirit level on the camera, a plumb line in the picture, or a door frame trusted to be vertical. The turn of a mirror is its attitude to the room, and there is no gravity sideways. Its share needs the room: the line of the mirror’s own frame, the floor’s tiles, the wall the mirror hangs on, any straight thing in the photograph whose direction relative to the walls is known. Both are usually in a photograph of a barber’s shop. Neither is in the corridor, which knows only the angle between the mirrors — the one thing that is relative to nothing. A mirror is a second camera made the general point, that a row of virtual eyes can be read for their arrangement among themselves without any reference, and for their arrangement in the world only with one; the turn adds a second direction in which “the world” means something other than down.

The count against the photograph #

Set side by side, the two instruments no longer do the same job worse and better; they do different jobs. The earlier essay found a count fixed the sum of the leans to about two parts in the count — 0.023 degrees for a corridor of 22 — and a photograph’s slope three times better. Here the photograph’s two slopes are each about as good as its one slope was, 0.0076 and 0.0068 degrees at a pixel or two, while the count has become something narrower than an instrument for the angle: it measures whichever part of the angle reaches its edge first, against an edge whose distance from the eye changes when the person moves, and it says nothing at all about the other part. Two pairs of mirrors with the same count can differ by any amount in the part that did not end the corridor.

That is not an argument against counting. A count needs no camera and no reading of directions, and it is exactly what a person in the chair does. It is an argument for knowing what a count is: a reading of one bend, the bend that wins, measured against a glass whose shape decides which bend that is. Two mirrors are three cameras used a pair of mirrors to put several viewpoints in one photograph and read a scene from all of them at once; the corridor is the extreme of that, a row of viewpoints each turned a little further than the last, and it is the row’s turning, read across many of them, that a photograph measures and a count cannot.

There is a pleasing contrast with still water. Measured down from the waterline used a pond as a mirror that gravity holds level, so that its reflection needs no fitter and reverses only height. A pair of wall mirrors is the opposite case: nothing holds either of them, every part of the angle between them is somebody’s workmanship, and the corridor is the record of that workmanship in both directions at once.

How a fitter would hang them #

The earlier essay drew a fitter’s order of work from the leans: make the mirrors parallel with the corridor, then plumb with a level. Turns add a step, and the corridor’s two edges say in which order to take the two parallels.

A pair hung a degree or more out about the vertical has a corridor that leaves through the side after ten images or fewer, and no adjustment of the leans changes its count, because the side edge is ending it. So the turn comes first: swing either mirror about the vertical until the corridor stops curving sideways — until the images run straight down the glass, or the count stops growing as the turn closes. Then the leans: the corridor now runs to the bottom or the top, and its count reads the leans’ sum as before. Then plumb and square, each against its own reference — a level for the lean, and the walls for the turn. A fitter who levelled first and squared nothing would have a corridor of a handful of images, and a fitter who squared first and levelled nothing would have one of twenty or so, drooping.

The corridor’s places, and what they are worth #

Two people between mirrors see each other equally counted images in a wedge of fifty degrees, where every bounce turns a line of sight by a hundred and the images form a ring round the wedge’s apex. A facing pair is the same wedge with its angle closed to nearly nothing, and its corridor is the ring opened out into a nearly straight line, the apex far below the floor, far to the side, or far away at a slant. Two mirrors show fewer images than they make found that glass cut short removes images by bounce count; the corridor’s two edges remove them by the same rule, one edge for each part of the turn. And a mirror that is not parallel to the wall measured what a turn about the vertical does to a single reflection, which is to swing it; the corridor multiplies the swing by the number of round trips and shows it as a curve.

What the tracing assumes #

The eye is a point and the head transparent. The count is traced for an eye with nothing round it. A real head hides the straight-back corridor, and a real shoulder hides part of the row seen past it; the images hidden are the first few, and the count of the rest, and every slope read from them, are unchanged.

The mirrors are flat. A barber’s mirror of float glass is flat to within the angles here; a mirror bowed by its fixings is a weak curved mirror, and a curved mirror has no eye begins another family, whose corridors focus or spread rather than bend.

The angles are small. The first-order share is exact to a few ten-thousandths of a degree at these angles. A turn of several degrees moves the corridor’s images out towards the edge of a photograph, where a lens’s distortion is largest, and the azimuths would need the lens corrected before their slope is read.

Every image is told from its neighbours. At a degree and a half of turn the images stand a degree and a half apart in the photograph — about twenty pixels in a picture 690 pixels across through a 50° lens — and are easy to tell apart; at a few hundredths of a degree of turn and lean together, the far images crowd within a pixel of each other and the row’s slope is read from fewer of them.

Still open: whether a corridor seen through its own reflection of the shop reads the shop’s walls #

A photograph of the corridor needs the room to share the turn between the mirrors. But the room is in the mirrors too. Every self-image comes with a reflection of the shop around it — the far wall, the door, the floor’s tiles — and in a turned pair those reflections are turned with each round trip as the person is. The walls’ lines in each round trip’s reflection are a reading of the turn that comes with its own reference, since the walls are the reference.

The measurement that settles what that is worth traces, beside each self-image, the reflection of a straight line on the shop’s floor running square to the mirrors, and asks whether the directions of that line in successive round trips read each mirror’s own turn without a camera that knows where it points; how many round trips of the floor line it takes to read the turn’s share as well as a camera turned a tenth of a degree would spoil it; and whether a line on the wall behind the person, reflected the same way, reads each mirror’s own lean without a level.