Two mirrors make one turn

A single mirror is a map from the room to itself, and a mirror is a second camera measures what kind: an isometry that reverses handedness, with the mirror plane fixed pointwise and nothing else fixed at all. Two mirrors are two such maps, and the interesting object is not either of them but everything they generate between them.

That object turns out to be a rotation group, and the pictures a kaleidoscope makes are its orbits.

Two reflections compose into a rotation #

Reflect a point in one plane and then in another. Each reflection reverses handedness, so the composition preserves it: two negatives make a positive, and a handedness-preserving isometry with a fixed point is a rotation.

Which rotation is the useful part. The two mirrors meet along a line, and every point of that line is fixed by both reflections and therefore by their composition — so the composition is a rotation about that line. And the angle is not the angle between the mirrors; it is twice it.

Measured on three probe points in a pair of mirrors thirty-six degrees apart, the composition turns by 72.000000 degrees, with a spread of zero across the probes and a displacement along the axis of 4 × 10⁻¹⁶ metres. The factor of two is exact and it is worth pausing on, because the collection meets it three times in this field alone.

It is the same two that makes a mirror pair’s baseline twice the camera’s distance from the glass, as one shutter, two views records — reflect a point across a plane and it moves twice its distance to that plane. And it is the two that makes a corner reflector’s tolerance twice its mirror’s, which is the next rung of this ladder. Every one of them is the same fact: a reflection is an operation whose effect is measured from the mirror, and composing two of them measures from both.

Every image is on one circle #

The images of a point in a dihedral pair are generated by applying both reflections to everything found so far until nothing new appears. What comes out is a set, and the set has a shape that costs nothing to prove and is worth drawing anyway.

A reflection is an isometry and it fixes the mirrors’ common line pointwise. An isometry cannot change a distance, so it cannot change the distance from a point to a line it fixes. Every image is therefore exactly as far from the axis as the original, which is to say they all lie on one circle in the plane perpendicular to the axis.

The measurement is 4.4 × 10⁻¹⁶ metres of spread across all radii at thirty degrees, and zero at sixty — the arithmetic floor, because nothing here is fitted. This is the same kind of statement as what a flat map leaves alone makes about the fixed structure of a projective map: what an operation preserves is a stronger fact than what it does, and it is usually the fact worth carrying.

The circle is also why a kaleidoscope looks the way it does. The images sit at equal angular steps around it, so a pattern drawn on one wedge appears repeated at regular intervals — and the repetition is exact rather than approximate, because it is a group orbit rather than a construction.

Where the taught rule is right #

Every account of a kaleidoscope gives the same formula for how many images two mirrors at $\theta$ produce: $360/\theta - 1$. At sixty degrees, five images; at forty-five, seven; at ninety, three.

Generating the orbit rather than quoting the formula gives the same answers at those angles and different ones elsewhere.

The angles where they agree are the ones that divide a half turn. The group generated by two reflections at $\theta$ contains the rotation by $2\theta$, and it closes when some whole number of those rotations comes back to the start — so the group has order $2m$ where $m$ is the smallest whole number with $2m\theta$ a multiple of $360°$. The reflected copies of the wedge between the mirrors then cover $2m\theta$ degrees. That is exactly one full turn when $m = 180/\theta$, which needs a half turn to divide evenly by the angle.

At seventy-two degrees it does not. Three hundred and sixty divides by seventy-two, so the taught rule confidently answers four; a hundred and eighty does not, the rotation by a hundred and forty-four takes five steps to close, the group has ten elements, and the orbit has nine points. The copies of the wedge go round the axis twice and land on top of each other.

At fifty degrees the rule answers 6.2, which is not a count at all, and the orbit has thirty-five.

Why the coverage is the right quantity #

There is a temptation to fix the taught rule by patching it — to say “and if three hundred and sixty over the angle is odd, double it”. That works for the cases above and is a rule about arithmetic rather than about the arrangement, which is the kind of repair this collection is written to distrust.

The quantity that actually governs it is the coverage: how many turns the reflected copies of the wedge sweep out before the pattern closes. At every angle the group has $2m$ elements and each element carries the wedge to a copy of itself, so the copies span $2m\theta$ degrees. Divide by three hundred and sixty and the answer is one at thirty, thirty-six, forty-five, sixty and ninety; two at seventy-two and at a hundred and twenty; five at fifty.

A coverage of one is a tiling. A coverage of two is a double cover, which is a perfectly good group action and an impossible physical arrangement, because the plane a viewer is looking at only has one turn in it. A coverage of five is worse in the same way and not in a new way.

Stating it as a coverage rather than as an arithmetic condition also says what to do about it. Nothing can be done: it is a fact about which dihedral groups act on the plane without overlap, and the angles that work are the ones a hundred and eighty divides evenly by, which is a short list. Kaleidoscope makers have that list.

The rule that replaces it, in one fraction #

The coverage is described above as a quantity to compute rather than an arithmetic patch, and it has a closed form short enough to be the replacement rule outright.

Write $180°/\theta$ as a fraction $p/q$ in lowest terms. Then the group has $2p$ elements, the orbit has $2p - 1$ images, and the copies of the wedge cover the turn $q$ times.

Every number in this essay follows. At 60° the fraction is $3/1$: six elements, five images, one cover. At 45°, $4/1$ — eight, seven, one. At 36°, $5/1$ — ten, nine, one. At 72° the fraction is $5/2$: ten elements, nine images, and a double cover, which is the case the taught rule answers four for. At 120°, $3/2$ — six, five, two. And at 50° the fraction is $18/5$: thirty-six elements, thirty-five images, and five covers, against the taught rule’s 6.2.

So the arrangement tiles the plane exactly when $q = 1$, which is exactly when $\theta$ divides 180° — the condition the section above states in words, now as the denominator of a fraction. And the image count is $2p - 1$ rather than $360/\theta - 1$, the two agreeing precisely when $q = 1$ and $p = 180/\theta$.

One fraction, reduced, gives all three numbers. The taught rule is that fraction with its denominator assumed to be one, which is why it is right on a kaleidoscope maker’s short list of angles and wrong everywhere else.

What a kaleidoscope maker knows and the rule does not say #

The practical version of this is old and correct: kaleidoscopes are built at sixty, forty-five, thirty and twenty-two and a half degrees, and not at seventy-two. The reason usually given is that the pattern “does not line up”, and the reason is the one above.

What a real instrument shows at seventy-two degrees is a seam. Two copies of the wedge occupy the same region of the field and the eye is shown both, which reads as a discontinuity rather than as a doubling. The orbit is still a perfectly good group orbit — the geometry has not failed — but the physical instrument cannot show nine images in a turn that only has room for five, and what a viewer counts is the visible ones.

So there are two numbers here and the essay is careful to keep them apart. The orbit is the mathematical object, and it is nine. The visible count is what survives the mirrors’ own occlusion, and it is smaller. The taught rule is a statement about the second at the angles where the two coincide, and is quoted as though it were a statement about the first.

The group, and what it is not #

It is tempting to describe the images as “reflections of reflections” and leave it there. Naming the group is worth the sentence it costs, because it says which operations are in the set and which are not.

The set generated by two reflections in planes meeting at $\theta = 180°/m$ is the dihedral group of order $2m$: $m$ rotations about the axis, at multiples of $2\theta$, and $m$ reflections. Half of the images are the object turned and half are the object turned and mirrored, which a viewer can check on any asymmetric object — alternate images in a kaleidoscope read the wrong way round.

That alternation is the handedness reversal a mirror is a second camera measures at 315 pixels, arriving as a parity rather than as a displacement. An even number of reflections preserves handedness and an odd number reverses it, and there is no third case.

What this is worth to a photograph #

A camera between two mirrors sees several virtual cameras, and every one of them is a genuine extra view of the scene taken in the same exposure. That sounds like a large gain over the one extra view of one shutter, two views, and it is a smaller gain than the count suggests.

The virtual centres are the real centre reflected and re-reflected, so by the argument above they all lie on one circle about the mirrors’ common line, at the same distance from it. A set of viewpoints confined to a circle is a narrow set, and it constrains nothing along that circle’s axis — a scene displaced along the axis looks the same from every one of them.

That is the same complaint square to the camera is the worst mirror makes one level down: what an arrangement supplies is a spread of viewpoints, and the count is a poor proxy for it. Two mirrors buy a great deal of angular spread about one axis and none at all along it.

The reader’s own instrument #

The whole of this can be checked with two hand mirrors and a coin, and the checking is worth describing because it distinguishes the two counts the essay has been careful to separate.

Set two mirrors on a table with their edges touching, put a coin between them, and open them until the images are evenly spread. At sixty degrees there are five and the coin makes six with itself. Alternate images show the coin’s reverse reading correctly and its obverse reading backwards, which is the parity above: an even number of reflections preserves handedness and an odd number does not.

Now open the mirrors to seventy-two degrees. The group says there are nine images. What appears is fewer, unevenly spaced, with a visible seam where two copies of the wedge fall on the same part of the field — and the seam is at a definite place, which is where the coverage runs past a full turn. That is the difference between the orbit and the visible count, made by hand in about a minute.

The reason to insist on the distinction is that it is the difference between a claim about geometry and a claim about an instrument, and this collection keeps those apart everywhere. The sixty-degree cone of vision is the same shape of confusion from the other end: a rule about what a viewer will tolerate, quoted as though it were a rule about projection.

The boundary, stated #

Two flat mirrors, exactly. Every claim above uses the fact that a reflection is an isometry fixing a plane, and a curved mirror is not one — the composition of two curved reflections is not a rotation, there is no axis, and the images do not lie on a circle. A curved mirror has no eye is the single-mirror version of that failure and everything in it applies.

Real mirrors are also not perfectly flat and not perfectly aligned, and the orbit is correspondingly approximate. How approximate is a question with a sharp answer in the three-mirror case, where an error in one face has a known and rather large effect on the whole arrangement, and that is the next rung.

The axis is the only thing left fixed #

One more property of the composition is worth stating, because it is the property that makes the whole family manageable.

A single reflection fixes a whole plane — every point of the mirror stays where it is. The composition of two reflections fixes much less: only the line where the two mirrors meet, and nothing else in the room. Everything off that line moves, and moves along a circle about it.

That drop from a plane of fixed points to a line is the same kind of accounting the foundations field does for a projective map, where the census of what is left fixed tells a reader which maps can be composed into which. Here it settles a practical question: two mirrors have one axis, so a scene arranged along that axis is unaffected by the whole arrangement, and a scene arranged across it is fully exercised by it.

It also says what happens when the mirrors are moved apart so that they no longer meet. Two parallel mirrors have no intersection line, the composition has no fixed point at all, and it is a translation — by twice the separation, along the normal, which is the same factor of two once more. The images then run away in both directions without end, which is the barber-shop corridor, and the group is infinite rather than dihedral.

What is measured here #

Four numbers.

The composition of two reflections in mirrors thirty-six degrees apart turns by 72.000000 degrees, against a predicted seventy-two, with a spread of zero across three probe points and a displacement along the axis of 4.4 × 10⁻¹⁶ metres. The orbit’s images all lie at one distance from the axis, spread 4.4 × 10⁻¹⁶ metres over twelve radii at thirty degrees. The taught count agrees with the generated orbit at six of nine angles tried and disagrees at fifty, seventy-two and a hundred and twenty. And the deliberate wrong prediction — that the composition turns by the dihedral angle rather than twice it — is rejected by thirty-six degrees, which is what says the first number is a measurement.

The short version #

Two reflections compose into a rotation about the line where the mirrors meet, by twice the angle between them — the same factor of two that puts a mirror pair’s second eye twice the distance to the glass away.

Every image of a point lies on one circle about that line, because an isometry fixing the line cannot change the distance to it. The number of images is the order of the group less one, and the taught rule for it is correct exactly at the angles that divide a half turn; at seventy-two degrees the orbit has nine members and the rule says four.