Two people between mirrors see each other equally often

Two mirrors show fewer images than they make separated two counts that the usual account of a pair of mirrors runs together: the images the two reflections generate, which belong to the mirrors, and the images an eye standing between them can actually reach, which belong to the eye. The second was measured by tracing each image’s path backwards from the eye and asking whether every bounce lands on glass, and it turned out to be a step function of where the eye stands. The taught count, three hundred and sixty over the angle less one, is a good rule for that second quantity and a poor one for the first.

That measurement held the object still and moved the eye. The question it left is sharper when the object is also a viewer. Two people stand between the same two mirrors, and each counts the images of the other. The path of light from the first to the second, run backwards, is a path from the second to the first, so it would be surprising if the counts differed — but a count of paths is one thing and what each person sees is another, and it is not obvious in advance which parts of the picture the two share.

One set of paths, used both ways #

The hero figure puts the mirrors at fifty degrees and the two people at different distances from the line where the mirrors meet and at different angles from the first mirror. Every path of light that joins them by bouncing off the glass is drawn.

There are six paths: one straight to each mirror and back, two that bounce twice, two that bounce three times. The first viewer sees six images of the second and the second sees six of the first. The dots mark where each set of images appears, and they are not the same dots — the images one person sees are spread across a different part of the view from the images the other sees — but there are six of each, and every one of them lies on one of the six drawn paths.

The measurement does not assume this. Each path is traced twice, once from each end, by the test the earlier essay used: draw the straight line from the viewer to where the image would appear, find where it crosses the first mirror, reflect the target, repeat, and require that every crossing lands on glass. A sequence of mirrors that delivers an image of the second viewer to the first delivers, in reverse order, an image of the first to the second. Over forty pairs of positions chosen at random, at four different angles between the mirrors, the counts agreed every time, and so did the individual sequences: none was seen one way and not the other.

That is the principle a physicist would state without measuring: light paths are reversible, so the set of paths joining two points is the same set whichever end is called the source. What the tracing adds is that the finite glass, the apex, and the half-turn limit on how far round the fan a straight line can reach — every condition that removes an image — remove it for both people at once. A person who sees one fewer image of a friend on stepping sideways takes one image of themselves away from the friend’s view at the same moment. An exchange of glances in a pair of mirrors is always an exchange: if one person can catch the other’s eye along some path, the other can catch theirs along the same path.

The same distance, a different direction #

Each path has two ends, and a viewer at either end sees the other at the far end of the path, straightened. The table takes the six paths one at a time.

The apparent distance of an image is the length of its path once unfolded, and a path walked backwards has the same length, so each image the first viewer sees has a partner the second viewer sees at exactly the same distance: the six agree to $4\times10^{-16}$ m. The two-bounce image the first viewer sees at 0.922 m corresponds to a two-bounce image of the first viewer, 0.922 m away, in the second viewer’s view. The earlier essay’s guess that the depths “need not” match is refused. They must.

The directions are another matter. On a straight line of sight, two people looking at each other look in exactly opposite directions. Along a folded path they do not: each sees the path’s last leg from their own end, and after one or more bounces those legs point in unrelated directions. In the table the pairs of directions fall between ten and eighty degrees short of opposite. The first viewer might find the second’s three-bounce image nearly straight ahead, while the second has to turn well to one side to find the partner image. The two counts are one count; the two pictures are two pictures. The point to stand at found that a picture has one place it is correct from; a folded path has two, one at each end, and they see different pictures of one exchange.

That asymmetry is worth keeping in mind against a mirror is a second camera, which treats each reflection as a view from a virtual eye behind the glass. Here each path is two such virtual eyes, one for each person, and they are generally not placed alike. The reversibility of light fixes which pairs of virtual eyes exist and how far each is from the person it stands in for. It says nothing about where they stand. Two mirrors are three cameras counted the virtual eyes one photograph contains; two people in the same corner each carry their own set, the same size and differently placed.

What the shared count depends on #

If the count belongs to the pair of people rather than to either, it must be a function of both their positions. The obvious guess is that it depends on everything — two angles and two distances from the apex — and the measurement shows it depends on much less.

With glass long enough that no bounce ever runs off the end, the figure shades the shared count over every pair of angles the two people can stand at. At fifty degrees it takes the values five, six and seven, and the plateaux are cut by straight lines — every one of them at forty-five degrees on the map, running either parallel to the diagonal or across it. The map is symmetric about its diagonal, since exchanging the two people exchanges nothing about the count. And it does not change when either person moves toward or away from the apex. The distances do not enter.

The straight boundaries are the clue to why. Two mirrors make one turn found that the images two mirrors make lie on a circle about the line where the mirrors meet, arranged in two families: the images made by an even number of bounces are the object turned about that line by whole multiples of twice the wedge’s angle, and the images made by an odd number are the object reflected across the copies of a mirror. The earlier essay’s half-turn argument then decides which of those copies a viewer can reach: a straight line from the viewer can wind around the apex by at most half a turn, so a copy is seen exactly when it lies less than half a turn away, measured round the apex.

For a copy made by an even number of bounces, the second viewer’s image sits at their own angle plus some multiple of twice the wedge. Whether it lies within half a turn of the first viewer depends on the difference of the two angles and nothing else. For a copy made by an odd number of bounces, the image sits at some multiple of twice the wedge minus the second viewer’s angle, and whether that is within half a turn of the first viewer depends on the sum. Neither condition involves how far anyone stands from the apex, because a straight line between two points at any distances winds round the apex by exactly the angle between their directions.

The difference and the sum #

That argument makes a prediction precise enough to be tested on people standing anywhere at all.

A hundred and sixty pairs of people placed at random between the mirrors — random angles and random distances from the apex, from fifteen centimetres to a metre and a half — are sorted by what the argument says should decide each part of the count. The images made by an even number of bounces, plotted against the second person’s angle less the first’s, fall on one step function. The images made by an odd number, plotted against the sum of the two angles, fall on another. Every one of the hundred and sixty pairs is on its line, whatever its distances.

So the shared count is a function of two numbers, and they are not the two people’s positions. One is how far apart their directions are about the mirrors’ meeting line; the other is how far their directions together are from the first mirror, which is the same as how far each is from the other’s reflection. The count is the sum of a step function of the first and a step function of the second.

This is what makes the question in the earlier essay’s closing section answerable: the count “cannot be a function of either position alone”, and it is not, but it is not a function of four numbers either. It is two functions of one number each. The boundaries on the map are where either number crosses a whole number of turns of the wedge away from a half turn — which is where some path’s first or last bounce passes through the apex, and that path is lost, for both people at once.

Short glass bends the plateaux, and keeps the exchange #

Real mirrors end. Two mirrors show fewer images than they make found that cutting the glass back removes the images in order of how many bounces make them, since each extra bounce lands further along the glass. With two people, that condition cannot depend on angles alone: whether a bounce lands before the end of the mirror depends on how far out it falls, and so on both distances.

With the mirrors cut to seventy centimetres and the two people at eighty and fifty-five centimetres from the apex, the map changes shape. Counts now run from two to seven, the boundaries curve, and the map is no longer symmetric about its diagonal, because the two people stand at different distances and exchanging them now exchanges something. What does not change is the exchange itself. In every one of the sixteen hundred cells the two people see as many images of each other. A path that runs off the end of the glass runs off it in both directions, since it is the same path; a person near the edge of a short mirror loses images of their friend exactly as fast as the friend loses images of them.

The two regimes differ in one respect a reader can use. With glass long enough to show every path — the earlier essay found three quarters of the object’s distance from the apex was enough for its arrangement — only the directions matter, and two people can predict their shared count from where they stand relative to the meeting line without measuring any distance. Below that length the distances matter too, and the count has to be traced.

Why the half turn gives a sum and a difference #

It helps to see the whole arrangement unfolded. Lay the copies of the wedge side by side around the apex: the real wedge from zero to the mirrors’ angle, its reflection across the first mirror on one side, across the second on the other, and so on round. Each copy holds a copy of the second person. In the even-numbered copies that person is simply turned about the apex; in the odd-numbered ones they are reflected, so their angle within the copy is measured from the other side.

The first person looks out from the real wedge along straight lines, and a straight line can reach any copy whose person lies within half a turn round the apex. Turned copies of the second person are spaced at twice the wedge angle starting from their own angle; reflected copies are spaced the same way starting from minus their angle. How many of each lie within half a turn of the first person is a question about the offset between the first person’s angle and each family’s starting angle — the difference for one family, the sum for the other.

That is the same structure a symmetric object is its own stereo pair found in one mirror: a reflected copy is characterised by how its position sits relative to the mirror, which is a sum of angles, and a rotated one by how it sits relative to the original, which is a difference. Two mirrors generate both kinds of copy at once, and the count of what two people see of each other keeps the two kinds separate.

Where the count is fragile #

The map’s boundaries are where the count changes, and they are the places two people should avoid if they want a steady view of each other — or seek out if they want to watch images come and go. Standing in the wrong place measured a tolerance that degrades gradually as a viewer moves; this one has no gradations, and it is shared. Crossing a boundary adds or removes a pair of images, one in each view, simultaneously.

The boundaries are also where the images at the edge of each view become hard to see. An image on the point of being lost is one whose path grazes the apex, where the two mirrors meet and where real mirrors have a seam, a frame, or a gap. The corner that answers every eye found that a corner of three mirrors returns every ray to where it came from, so that a person looking into it sees only themselves; two mirrors do not have that property, and the images near a boundary are the ones whose paths come closest to it. A real pair of mirrors will lose them a little before the geometry does, and lose them for both people at once. A mirror that is not parallel to the wall is the reminder that a real mirror is also rarely where it is supposed to be, and a tilt moves every boundary on the map.

What was measured and what was not #

The mirrors are ideal lines seen along the meeting line. Everything here is drawn in the plane perpendicular to where the mirrors meet. For mirrors that meet in a vertical line and are tall enough, a person’s height makes no difference: reflection in a vertical mirror does not change height, and a path’s unfolding happens entirely in the horizontal plane. For mirrors of finite height it does.

People are points. Two real people are not, and “seeing” the other means seeing some part of them. A count of eyes that can meet is the natural reading; a count of whole figures would require each path’s neighbourhood, not just the path, to land on glass.

Nothing occludes. Each person is transparent to the other’s light. Two people standing close together between mirrors block some of each other’s paths, and whether blocking is also exchanged is a separate question: a path blocked by the first person’s own body is blocked in both directions, since it is one path, but the count of paths blocked depends on the bodies’ sizes, which the geometry here does not have.

Still open: what a mirror’s top edge takes away #

Everything above is in the plane square to the line where the mirrors meet, and that is exact for mirrors that are tall enough, because a vertical mirror leaves height unchanged and every path’s unfolding lies flat. Real mirrors have a top edge, and the familiar corridor of images in a barber’s shop — a pair of facing mirrors, the limit of this arrangement as the angle closes — ends not in a point but in images that climb out of the top of the glass one after another, because each extra bounce carries the line of sight a little higher or lower.

With a top edge, two people of different heights are no longer symmetric in the same way: the path joining them rises steadily from the shorter to the taller, and each bounce lands at a height set by how far along the unfolded path it falls. The exchange is still guaranteed — it is still one path — but the count is now a function of both heights as well as the two angles, and the question with a number in it is how quickly: at what difference in height between two people does the mirrors’ top edge, at a stated height, begin to remove images that the plane measurement delivers, and are the images lost first the ones with the most bounces, as they were when the glass was cut short in its own plane, or the ones whose unfolded paths are longest, which in a narrow wedge is not the same thing.