A mirror's top edge waits for an eye above it

Two people between mirrors see each other equally often traced every path of light that joins two people standing between a pair of mirrors, and found the counts always equal: a path walked backwards is the same path, so if one person can catch the other’s eye along it, the other can catch theirs. With glass long enough, the shared count depended on only two numbers made of both people’s positions — the difference and the sum of their angles about the line where the mirrors meet — and not at all on how far either stood from it. All of that was in the plane square to the meeting line, which is exact for upright mirrors tall enough that nothing runs off their top.

The essay ended on the edge it had set aside. Real mirrors have a top, and the familiar corridor of images in a barber’s shop does not go on for ever: it curves away and leaves the glass. The question was at what difference in height between two people the top edge starts to remove images the plane count delivers, and whether the images lost first are those with the most bounces or those with the longest paths.

The top edge removes nothing at any difference in height, until one of the eyes is above it. When images do go, they go in an order neither guess named. And the barber’s corridor is not an edge at all.

Every bounce lies on a straight climb #

The arrangement is the earlier essay’s: two mirrors meeting at fifty degrees, cut 1.4 metres along the floor, one person 0.8 metres from the meeting line and eighteen degrees round from the first mirror, the other 0.55 metres out at thirty-four degrees. The glass now has a bottom and a top, 0.9 and 1.8 metres above the floor. The first person’s eye is 1.6 metres up; the second person’s eye moves.

The tracing is done in three dimensions, by the same unfolding the plane measurements used: reflect the second eye in each mirror of a path, last mirror first, draw the straight line from the first eye towards that reflection, and require that it cross each mirror on the glass — along the floor between the meeting line and the mirror’s end, and up the face between its bottom and top edges. With the height limits removed it returns exactly the plane count, six paths for these positions.

The drawing explains the result before any count is made. An upright mirror leaves height alone: it reverses the horizontal part of a line of sight and keeps the vertical part — the reverse of the still water measured down from the waterline used, which reverses height and keeps everything else. So once a path is unfolded into the straight line from one eye to the other’s reflection, its height changes evenly along it, from the first eye’s height at one end to the second eye’s at the other. A bounce a third of the way along the unfolded path is a third of the way from one height to the other.

So every bounce lies between the two eye heights. If both eyes are inside the glass’s height, every bounce is too, and no path is touched by the top or the bottom edge however different the two heights are. The slider shows it: with the second eye at 1.6 metres the paths are flat and all six images survive; at 1.75 they climb gently and all six still survive, the highest bounce at 1.75. Only past 1.8 metres, with the second eye above the glass, does a bounce rise past the top edge — first the two-bounce path’s second bounce, at 1.808 metres when the eye is at 1.85.

The earlier essay’s expectation was that the count would become a function of both heights and start to fall as they diverged. It is a function of both heights, but in a strong way: it is exactly the plane count for every pair of heights inside the glass, and depends on them only outside it.

Nothing is lost until an eye is above the glass #

The count is a step function, and its first step is never below the top of the glass. With the first eye at 1.6 metres, the second can be anywhere from a metre up to 1.8 metres and both see six images of each other. The first image goes when the second eye passes 1.841 metres, and by 2.06 metres there are none: every path’s last bounce is then above the glass.

With the first eye lower, at 1.2 metres, the losses begin later, at 1.923 metres, and the count falls more slowly. The reason is the same straight climb. A bounce’s height is the two eye heights mixed in the proportion of its place along the path, so a lower first eye pulls every bounce down, and the second eye has to rise further before the bounce nearest it reaches the top. A tall person standing between a pair of mirrors with a child loses fewer images of the child than of another adult whose eyes are just below the glass’s top, because the child’s low eye drags every path down.

The exchange the earlier essay found survives untouched. Every count in the figure is the same in both directions, at every height, because a path that runs off the top of the glass is one path and runs off it for both people. A person who has stretched up until their eyes are above the mirrors loses images of their companion at exactly the moment the companion loses images of them.

The bottom edge behaves the same way by the same arithmetic, upside down. A person whose eye is below the glass’s bottom — a child at a pair of mirrors hung for adults — loses images in the same order, because the paths now descend towards the low eye and the bounce nearest it is the first to drop below the glass.

The order is set by the last bounce #

When images do go, which goes first? The earlier essay offered two candidates. Cutting the glass short in its own plane had removed images in order of bounce count, since each extra bounce lands further out along the glass; and in a narrow wedge the longest unfolded paths are not always those with the most bounces.

The straight climb settles it without either. A path loses its image when its highest bounce passes the top edge. With the second eye above the glass and the first below, the highest bounce is the one nearest the second eye along the unfolded path — the last bounce, as the light travels from the first eye. If that bounce sits a fraction $t$ of the way along, its height is $h_A + (h_B - h_A)\,t$, and it passes a top edge at $T$ when the second eye reaches $h_A + (T - h_A)/t$. The image is lost at a height set by $t$ alone.

The measurement places forty random pairs of positions between the mirrors, the first eye always at 1.6 metres, and for each of the 241 images the plane count gives, finds the height of the second eye that loses it. Every one lies on the curve 1.6 + 0.2/t. In every one of the forty arrangements, the first image lost is the one whose last bounce lies furthest along its path. The image with the most bounces was never first; nor was the image with the longest path.

The colours in the figure show why the guesses fail. Images with one, two, three and four or more bounces are spread along the whole curve. A one-bounce path whose single bounce happens to fall near the far eye — a mirror close to the second person — is lost before a three-bounce path whose last bounce is halfway along. For the positions in the profile figure, the two-bounce path with its last bounce at 0.83 of the way is lost first, at 1.841 metres; the single bounce off the second mirror, at 0.74 of the way, at 1.872; and the three-bounce paths at 1.851 and 1.887, according to where their last bounces fall.

So the order is a statement about where along each path the light last touches glass, which is a statement about how close the second person stands to the mirror that delivers their image. A person above the glass who steps towards one mirror loses first the images whose final reflection is in that mirror. Unlike the gradual tolerance standing in the wrong place measured for a picture, each of these losses is a step, and it is taken by both people at once.

The corridor that leaves the glass is a lean #

The barber’s corridor is two facing mirrors — the earlier essay’s arrangement in the limit as the angle between the mirrors closes — and a person between them sees a long row of their own images that curves away and leaves the glass. The straight climb says that cannot be the top edge’s doing. A person looking at their own eye has a path that starts and ends at one height; between upright mirrors it is flat, every bounce at eye height, and no top or bottom edge can touch it however many bounces it has.

What curves the corridor is a mirror that is not quite upright. A mirror leaning back by an angle tips every line of sight it reflects by twice that angle, downward. On a single reflection that is a small change in where an image appears. Along a corridor of many reflections the tips add. A mirror is a second camera put a virtual eye behind each reflection; in a corridor each virtual eye is placed by reflecting the one before, so a lean that misplaces the first by a little misplaces the tenth by much more.

The bars are each self-image’s path, from its lowest bounce to its highest. With the mirror upright every bar is a point at 1.6 metres and the corridor runs on until it reaches the mirrors’ side edges — forty images and more. Leaning half a degree, each path sinks from the eye, deepest in its middle, and comes back up to the eye at its end, and the sink grows with the number of bounces until the sixteen-bounce image’s path dips below the bottom of the glass. Fifteen images are kept. Leaning two degrees, seven.

A mirror leaning back sends the corridor out of the bottom of the glass, and one leaning forward sends it out of the top. Either way the edge it leaves through is an edge, but the reason it reaches the edge is the lean. A mirror that is not parallel to the wall measured what a mirror turned about a vertical axis does to a single reflection; a lean about a horizontal axis is the same mistake turned through a right angle, and a corridor multiplies it by every bounce.

The corridor’s length goes as one over the square root of the lean #

The sink grows as the square of the bounces, because each reflection off the leaning mirror adds twice the lean to the line of sight’s slope, and a slope that grows steadily makes a path that falls quadratically. The path returns to the eye at its end, so its deepest point is at its middle, a quarter of the way into the quadratic. The corridor ends when that middle falls past the bottom edge, which makes the count of images about $\sqrt{8\,\Delta h/(g\tau)}$, where $\Delta h$ is how far the eye is above the glass’s bottom edge, $g$ the gap between the mirrors and $\tau$ the lean in radians.

The traced counts follow the law closely. From an eye 0.7 metres above the glass’s bottom edge, a lean of a tenth of a degree ends the corridor at 35 images, a fifth of a degree at 25, half a degree at 15 and a degree at 11 — each fourfold increase in lean halving the count. Raise the eye to 1.8 metres and the counts grow by the square root of the extra height to the edge; lower it to 1.3 and they shrink the same way. Moving the mirrors further apart shortens the corridor too: for an eye 1.6 metres up, standing two-fifths of the way across, the same quarter-degree lean leaves twenty-nine images between mirrors a metre and a half apart, twenty-two at two and a half, seventeen at four.

That makes the barber’s corridor an instrument. Two mirrors are three cameras read a pair of mirrors as extra viewpoints in one photograph; a corridor is a row of them, and the row’s end is a reading of how the mirrors stand. A lean of a tenth of a degree is invisible to anyone standing in the shop: the mirror is upright to the eye and to a spirit level held against it. But it ends the corridor at a few dozen images, and counting them gives the lean — thirty-five images from an eye 0.7 metres above the glass’s lower edge in mirrors 2.5 metres apart says a tenth of a degree, to the precision a count allows. The corridor’s end measures the mirror’s lean about as well as any tool a fitter carries, and with no tool at all.

What a mirror’s height takes, and what it does not #

Put together, the measurements answer the question the two-viewer essay left, and reverse half of its expectation. Between upright mirrors a path’s unfolding climbs straight from one eye’s height to the other’s, so the top edge removes nothing while both eyes are inside the glass’s height, at any difference between them. Once an eye is above the glass, images are lost in the order of where along each path the last bounce falls, and that order is neither the bounce count nor the path length. The exchange of glances survives every loss.

Two mirrors show fewer images than they make found that a mirror cut short in its own plane removes images in order of bounce count, because each bounce lands further out along the floor. The height of the glass does something different because height is the coordinate an upright mirror does not touch: a bounce’s height is fixed by where it falls between two eyes, not by how many bounces came before it. Only a lean puts the bounce count back into the vertical, and then — as in the corridor — the images are lost in order of bounces again, now because each bounce adds to the slope.

Two mirrors make one turn found the images two mirrors make arranged on a circle about the line where they meet. For upright mirrors every image on it sits at the height of the thing it is an image of, which is why every bounce sits at the height the straight climb gives it. A lean tilts the line the reflections turn about, and the images no longer keep their height: in the corridor they march down, one bounce at a time.

The idealisations the counts rest on #

Eyes are points. A count of images of an eye is a count of paths ending on that point. Seeing someone’s face needs a neighbourhood of paths to land on glass, and near the top edge part of a face can survive while the eye’s path is lost. The order of loss for the eye is the order for any point on the face, with that point’s height in place of the eye’s.

The mirrors are flat and their edges sharp. A real mirror has a frame, and a bevelled edge reflects a little past where its flat face ends. Both move the loss heights by a few millimetres and change nothing about the order.

The wedge’s mirrors are upright. Leaning both mirrors of a fifty-degree corner moves the cross-section of the corner at eye height, because each mirror’s face moves outward as it rises, and that changes the plane count itself before any edge does. The facing pair is the clean case of a lean, and the one whose corridor is long enough for the lean to matter.

Nothing occludes. As in the earlier essay, each person is transparent to the other’s light. A person between facing mirrors blocks their own corridor’s straight-back images with their head, which is why the corridor is usually seen past one’s shoulder — off the straight path, where the facing pair becomes a narrow wedge again.

Still open: what the corridor says when both mirrors lean #

Here one mirror leans and the other is upright. A shop’s mirrors are hung independently, and both may lean, by different amounts and in either direction. Leaning both back adds their tips; leaning one back and one forward subtracts them, and a pair that lean in opposite directions by the same angle should send the corridor straight on, as if both were upright.

The measurement that settles what a corridor can say about a pair of leans traces the self-images between facing mirrors leaning by two independent angles, from a tenth of a degree to a degree in either direction, and asks whether the corridor’s count still follows the square-root law with the difference of the leans in place of one lean — and whether a person can recover both leans separately by counting twice, once from near each mirror, since the corridor’s deepest point sits nearer the mirror that tips each path more.