Twomirrors — the series
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Two mirrors make one turn
Reflect a point in two mirrors meeting at 36° and the images arrive nine at a time, every one of them on a single circle about the line where the mirrors meet, to 4 × 10⁻¹⁶ m. The rule taught for the count — three hundred and sixty over the angle, less one — is right at six of nine angles tried and wrong at the rest, because it is a rule about angles that divide a half turn and it is quoted for angles that divide a whole one.
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The corner that answers every eye
Three mirrors at right angles compose into the point inversion, so a ray entering leaves antiparallel to itself whatever direction it arrived from — a picture surface with no correct viewpoint because every viewpoint is correct. Tilt one face half a degree and the worst returning ray is out by exactly one degree, twice the error; the best is out by a twelfth of it, which is why a tolerance quoted from one measured ray is a statement about that ray.
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Two mirrors show fewer images than they make
Two mirrors at 55° generate seventy-one images of a point and an eye between them can reach six. The count the field teaches — three hundred and sixty over the angle, less one — is out by as much as sixty-six against the orbit and never by a whole image against what a viewer standing on the bisector actually sees. It is a correct rule about the eye, quoted as a rule about the mirrors.
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Two people between mirrors see each other equally often
Put two people between a pair of mirrors and each sees some number of images of the other. The two numbers are always equal, and so is the apparent distance of each image against its partner — a path of light walked backwards is the same path. What is not shared is where each image appears, and what the shared count depends on turns out to be neither person's position but two quantities made of both: the difference of their angles about the mirrors' meeting line, and the sum.
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A mirror's top edge waits for an eye above it
Between two upright mirrors a path of light from one eye to another, unfolded, climbs in a straight line from one eye's height to the other's, so every bounce lies between the two. While both eyes are inside the glass's height its top edge takes nothing at all. Above it, the images go in a fixed order — not the one with the most bounces, not the longest, but the one whose last bounce falls furthest along its path, in forty arrangements out of forty. The barber's corridor that sinks out of the glass is a mirror leaning a fraction of a degree, and its length goes as one over the square root of the lean.
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A corridor of mirrors counts their angle, not their leans
Between two facing mirrors that both lean, the barber's corridor of self-images ends where it leaves the glass, and the count depends on the two leans only through their sum — the angle between the mirrors. Forty-two pairs of leans fall on one curve, within an image. Counted from near either mirror the number is the same, so counting twice cannot say which mirror leans, and two mirrors tilted together run the corridor on for ever. A photograph can: each round trip lowers the image by the sum of the leans, and where the row of images starts shares the sum between the two — provided the camera knows which way is level.
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Two mirrors nearly facing bend the corridor two ways, and each bend runs out at its own edge
Between two mirrors that lean and are also turned a little about the vertical — the barber's shop, seen past the shoulder — the corridor of self-images steps down by the sum of the leans and sideways by the sum of the turns on every round trip. Its count is not a reading of the angle between the mirrors: each bend runs out against its own edge of the glass, and the corridor ends at whichever does first. A quarter of a degree between the planes gives 22 images when it is all lean, 21 when it is all turn and 25 when it points between. A photograph reads both bends from how the images step, with no reference at all, and needs gravity only to share the lean between the mirrors and the room only to share the turn.
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A mirror corridor shows the room once, and the room's lines are a reference by where they are
Between two facing barber's mirrors the corridor of self-images goes on for a dozen round trips, but the shop around it does not: the floor never appears in a level photograph, and the walls appear in the first reflection only, because every later path has to start and end on glass and only what stands between the two panes can be reached. That one sight of the wall behind the sitter is enough. A rail and a door jamb on it, fitted with the self-images, read each mirror's own lean and turn to about a twentieth of a degree with the camera's heading, pitch and roll unknown — better than a camera believed level and a tenth of a degree out. The reference is not the lines' angles, which can be 0.2° off at almost no cost, but their places: a rail 5 mm higher than believed reads as a lean of 0.14°.