Two mirrors show fewer images than they make

Two mirrors make one turn counted the images two mirrors make by generating them: reflect the object in each mirror, reflect the reflections, keep going until nothing new appears. At thirty-six degrees that gives nine images, every one of them the same distance from the line where the mirrors meet, to four parts in ten thousand million million. The count is exact and it is arithmetic rather than a fit.

It also found that the rule everybody is taught — three hundred and sixty over the angle, less one — is wrong at most of the angles it was tried at. At seventy-two degrees the rule says four and the orbit has nine. At fifty degrees the rule says six point two, which is not a count of anything, and the orbit has thirty-five.

That essay’s conclusion was that the rule is a rule about the angles which divide a half turn, quoted for the angles which divide a whole one. The conclusion was too kind to the orbit and not kind enough to the rule, and this essay is the correction.

An image is somewhere light comes from #

The orbit is a set of points computed from two planes and a point. Nothing in that computation mentions an observer, and there is no reason it should: the group two reflections generate is a property of the two mirrors, and it has the same members whether anybody is in the room or not.

A viewer counting images is answering a different question. An image at a given place is seen only if light leaves the object, strikes the glass, and arrives at the eye along the direction that place lies in. That is a condition on the path, and a path has requirements the point does not — each bounce has to land on a mirror that is actually there.

The test is the standard one and it is worth stating because everything below is a consequence of it. Straightening a folded path is the same operation as reflecting the object, once for each bounce. So to ask whether the image made by a particular sequence of reflections is visible, run the sequence backwards from the eye: draw the straight line from the eye to where the image appears to be, find where that line crosses the first mirror, reflect the target in that mirror, and repeat. The image is seen exactly when every crossing lands on glass — at a positive distance from the apex, and not past the end of the mirror.

Applied to every sequence, that turns the orbit into two sets: the images light delivers, and the images the group makes and nothing shows.

The count is not a property of the mirrors #

At forty-five degrees the orbit has seven members and the glass is long. An eye standing on the bisector reaches all seven. An eye standing three degrees from one mirror reaches six.

Nothing about the mirrors changed. The object did not move. One image simply stopped having a path, because the straight line from the new eye position to that image no longer crosses the first mirror on the near side of the apex, and no other sequence of reflections produces it either.

Sweeping the eye from one mirror to the other gives the whole shape of it: six images for the first two fifths of the way across, seven for the rest, with the step where one path’s first bounce passes through the apex. The step is not gradual and cannot be — a path either lands on the glass or it does not — so the number a viewer reports is a step function of where the viewer stands.

That is already enough to say the taught rule cannot be a rule about the mirrors alone, because the quantity it is supposed to predict is not a function of the mirrors alone. It is also enough to explain why the rule has survived being wrong: a person who checks it stands in the middle, sees seven, and is satisfied.

Half the glass is not half the images #

The second thing a path needs is glass to land on, and a real pair of mirrors is finite.

Hold the angle at forty-five degrees, keep the object and the eye where they are, and cut the mirrors back toward the apex. The images go out in order of how many reflections make them, because each extra bounce reaches further along the glass: at three tenths of the object’s own distance from the apex, one image of the seven survives; at half, three; and all seven are delivered once the mirrors reach three quarters of that distance. Longer mirrors buy nothing more.

So a pair of mirrors has a length past which it shows everything it can, and the length is set by the object’s distance rather than by the mirrors. It is a modest number — the glass does not have to be long, it has to reach past the object — and it is the reason a shop’s two-mirror corner shows the whole ring while a pair of hand mirrors held close together does not.

A reader who wants the general statement can have it in one line. A path with $k$ bounces alternates between the two mirrors and its bounce points march outward, so the furthest bounce of the deepest visible path sets the requirement; and since the images all lie on one circle about the axis, that requirement scales with the object’s distance and with nothing else.

At seventy-two degrees, four images nobody can see #

The angle at which the taught rule looked worst is the angle at which this reading is sharpest.

Seventy-two degrees does not divide a half turn. The reflected copies of the wedge go round twice before they close, so the orbit has nine members rather than the four the rule predicts, and two of the copies land on top of each other — the seam that essay named.

Trace the paths and the nine split cleanly. Five of them are reachable: the object’s direct reflections in each mirror, their reflections in the other, and one more. The remaining four — at forty-seven and a half degrees and ninety-six and a half degrees on both sides — are reachable from nowhere in the wedge at all. Every sequence of reflections that produces them requires a bounce behind the apex, at every eye position across the whole opening.

Those four are exactly the second-wrap copies. The orbit’s extra members, the ones that made the taught rule look wrong, are the members no eye was ever going to count.

The rule is right about a quantity nobody named #

With the two counts separated, the rule can be measured against each of them.

Seventeen angles from twenty-five degrees to a hundred and twenty, the object fixed, the eye on the bisector. Against the orbit the rule is out by as much as sixty-six images: at sixty-five degrees the orbit has seventy-one members and the rule says four and a half. Against the count an eye on the bisector reports, the worst miss over the same seventeen angles is six tenths of one image, and at the five angles which divide a half turn — thirty, thirty-six, forty-five, sixty, ninety — it is exact.

That is not a rule that is nearly right. It is a correct rule about a different quantity. Three hundred and sixty over the angle, less one, counts how many copies of the wedge fit into a full turn around the apex, and a straight sight-line from an eye inside the wedge can reach a copy only if it is less than a half turn away in one direction or the other. The two conditions are the same condition, which is why the arithmetic lands where it does.

The six tenths of an image is worth a sentence rather than an apology. It is a residual that cannot be removed by a better rule, because the quantity being predicted is an integer that depends on the viewer’s position and the object’s, and no function of the angle alone can track it. A rule that was exact everywhere would be a rule about something simpler than what is happening.

The half turn, and why it is a half turn #

The agreement between the taught rule and the visible count is close enough to want an argument rather than a table, and the argument is short.

Unfolding the wedge lays the copies out side by side around the apex: the object’s own copy occupies the wedge from zero to $\theta$, the copy across the first mirror occupies $-\theta$ to zero, the next occupies $\theta$ to $2\theta$, and so on in both directions. An image made by $k$ bounces is the object’s position inside the copy $k$ steps away, and the path that produces it is the straight segment from the eye to that copy — which is the whole content of the unfolding.

A straight segment between two points can subtend at most a half turn about a point it does not pass through. So a copy more than a half turn away from the eye’s own copy, measured around the apex, cannot be reached along a straight segment, in either direction. That is the condition, and it is a condition about angles rather than about lengths, which is why more glass does not repair it.

Counting the copies within a half turn on each side gives $180/\theta$ each way, so the number of copies other than the eye’s own is close to $360/\theta - 1$, with the remainder deciding whether the last copy on each side is reached. At an angle which divides a half turn exactly, the two half-turn limits land on the same copy from opposite directions and the count is exact; otherwise the last copy on each side is reached from some eye positions and not others, and the count wobbles by one. That is the six tenths of an image measured above, and it is the whole of the discrepancy.

The derivation also says which images are lost first, and it matches the traces: the images lost are the ones furthest around the fan, which are the ones made by the most bounces. A viewer standing off the bisector has moved the whole half-turn window round with them, so they gain nothing at one end and lose at the other — because the copies are not uniformly spaced about their own position, only about the apex.

Where the viewer enters, and where the viewer does not #

The distinction this essay turns on is a familiar one here, and it has never before had to be made against a group.

The point to stand at computes where a perspective picture is correct from, and standing in the wrong place measures what moving away costs. In both, the picture exists and the viewer’s position decides how wrong it looks. An anamorph has one eye is the same question at its limit, where the answer is a point rather than a region.

Here the viewer’s position decides whether part of the picture exists at all, and the quantity it decides is an integer. That is a harder dependence than a tolerance, and it has none of a tolerance’s gradations: there is no eye position from which the fifth image at seventy-two degrees is faint, or partly there, or nearly right. It is delivered or it is not.

It also has the opposite sign from the corner that answers every eye, which is the arrangement in this field where the viewer’s position stops mattering entirely. Three mirrors at right angles return every ray to its source, so every position is correct and the count of what is seen is one — the viewer’s own eye. Two mirrors at an angle sit between that and an ordinary picture: a count that is neither fixed nor continuous, stepping as the viewer crosses the opening.

Why the orbit looked like the thing to count #

Two mirrors make one turn computed the orbit because the orbit is what the geometry offers. Reflection is an isometry, the composition of two of them is a rotation, the set of images is the orbit of a point under the group they generate, and every part of that sentence is exact and checkable. A count with that pedigree is hard to doubt.

What it is not is a count of anything a person has ever done. Nobody has counted the images in a pair of mirrors by applying a group to a point; they have counted them by looking, and looking is subject to two conditions the group knows nothing about — where the looker is, and how much glass there is.

The same shape has turned up here before under other names. The rule that draws another room found a construction taught for centuries that is exactly right for one ratio of depths and silently wrong for the rest, and the sixty-degree cone of vision found a taught limit whose number turns out to be about the page rather than about the eye. Dividing depth by eye is the third, where the taught division is exact for one spacing and drifts for every other. All three survived because the case people check is the case they are right in. This one comes off best of the three: it is right about the thing it is used for, and wrong only in what it is said to be about.

Where the mirror’s other job is unaffected #

None of this disturbs what a mirror is worth as a second camera.

One shutter, two views reads a photograph containing a mirror as a stereo pair whose fundamental matrix is skew-symmetric, and two matches are enough reduces the whole fit to two correspondences and a straightedge. Both of those use the first reflection, which is the one image that is always visible from anywhere in front of the glass, and neither depends on how many further images the arrangement generates. Square to the camera is the worst mirror adds the one condition on that first image which is about angle rather than existence, and it is a condition on the mirror’s orientation rather than on the viewer’s place.

Two mirrors give more than one second camera, and that is the interesting consequence. An image reached by two bounces is a view from a camera twice reflected, with its own centre and its own baseline against the real one; and the visibility test says which of those cameras a photograph actually contains. A single photograph of a two-mirror corner is therefore a multi-view rig whose size a count of the orbit would overstate — by four at seventy-two degrees, and by sixty-five at fifty-five.

The two conditions are independent, and only one is about the angle #

Separating the two failures matters for anybody predicting what an arrangement will show.

An image can fail to be visible because the sight-line to it would cross the mirror’s backward extension rather than the mirror — a failure of angle, which is what the seventy-two degree case is, and which more glass does not repair. Or it can fail because the bounce lands past the end of the mirror — a failure of size, which longer mirrors do repair and which has nothing to do with where the copies fall.

The first failure is what the taught rule is unknowingly about. The second the rule cannot see at all, and it is the one that decides what a real pair of hand mirrors shows: the count rises as the mirrors are moved closer to the object or made larger, and stops rising at the value the rule predicts.

A reader who wants to check the whole of this without computing anything can do it with two mirrors and a coin. Stand the mirrors at sixty degrees and the coin between them, and count from the middle: five. Move to one side, keeping the coin still, and one image goes. Slide the mirrors apart until the coin sits near their ends and images go in order, the deepest first.

What this does not settle #

Three things, and the first is the reason the numbers above are stated for a particular object.

The count depends on where the object stands as well as on where the eye does, and only the eye has been swept here. Moving the object toward one mirror moves the whole orbit round the circle, so the same argument applies with the roles exchanged, and the joint dependence is a surface rather than the curve drawn above.

Nothing here treats the object as having size. A real object is not a point, so different parts of it are reached by different numbers of bounces and an image can be partly delivered — a fact anyone who has looked into a two-mirror corner has seen as images cut off along a vertical edge. The point model gives the count and says nothing about the cropping.

And the mirrors are perfect and unbacked. A real mirror has a glass thickness in front of its silvering and a reflectance below one, so each bounce both displaces the image slightly and dims it; the deep images in a real corner fade out before they run out. A curved mirror has no eye covers the other departure from the ideal, where the plane assumption fails outright and there is no second camera to have.

Still open: which images survive being the object #

The measurement above sweeps the eye and holds the object, and the natural completion sweeps both. That is a surface rather than a curve, and its shape is worth having: the count is an integer, so the surface is a set of flat plateaux separated by curves, and those curves are the configurations at which a path’s first or last bounce passes exactly through the apex.

The question with more in it asks what happens when the object is a second viewer. Two people standing between two mirrors each see a number of images of the other, and the visibility test is not symmetric in an obvious way: the path from the first to the second reversed is a path from the second to the first, so the count ought to match, while the depths at which each sees the other need not. Measuring it would place two points in the wedge, count the images each reaches of the other, and test whether the two counts are equal at every position — and if they are, ask what the shared count is a function of, since it cannot be a function of either position alone.

The short version #

The orbit of a point under two mirrors is a property of the mirrors. The number of images an eye sees is not: it depends on where the eye stands, and on how far the glass reaches, and at forty-five degrees it is six from near one mirror and seven from the middle of the same pair.

Separating the two makes the taught count correct. Three hundred and sixty over the angle, less one, is out by up to sixty-six images against the orbit and by at most six tenths of one image against what an eye on the bisector reports, exactly right at every angle which divides a half turn. At seventy-two degrees the four images the rule appeared to miss are four images no position in the wedge can reach.