Three reflections, and never four

A mirror is the simplest motion there is. It has an equation with two numbers in it, it is its own inverse, and anybody can see what it does. Everything else the plane can do — turning, sliding, the awkward fourth thing that slides while it flips — is built out of mirrors, and the surprising part is how few are needed.

Three. Never four, whatever the motion.

That count is worth more than it first appears, because it is not a fact about mirrors. It is the classification of the four motions written as a single integer:

• nought mirrors — the identity;
• one — a reflection;
• two — a rotation, or a translation;
• three — a glide.

Nothing occupies a fifth line, and no motion of the plane needs a fourth mirror. The list is exhaustive because that is what the four motions being four means, and it is ordered in a way the usual list is not: a rotation and a translation cost the same, which says something about them that no picture does.

Two mirrors, and the only two things they can be #

Take two mirrors and do one after the other. There are exactly two cases and no third, because two lines in the plane either meet or they do not.

If they meet, at a point p and at an angle θ, the product fixes p — both mirrors fix it, so their product does — and it turns everything else through twice θ. The doubling is the reason mirrors at 30° generate a six-fold rotation and the reason the operations of a group come in twos so often.

If they are parallel, at a distance d, the product fixes nothing at all and is a translation through 2d, perpendicular to both. Again the doubling, and again the freedom: the pair may be slid together anywhere across the direction of travel.

So a product of two mirrors is a rotation or a translation, and nothing else. That single sentence does most of the work below, and it is worth noticing that it is a statement about the plane rather than about groups: it holds for any two lines whatever, symmetries of nothing in particular.

The translation, which is the same story with the lines apart #

A translation is the case that makes the ordering of the list look strange. It moves everything and fixes nothing, it looks nothing like a turn, and it costs exactly what a turn costs.

The reason the two cost the same is visible in the product of two half-turns: two half-turns about different centres compose to a translation through twice the distance between them, and each half-turn is itself two mirrors. Four mirrors, of which the middle two are the same line and cancel — leaving two. A translation is what a rotation becomes when the mirrors stop meeting, and the cancellation is how the count survives the transition.

Why a rotation cannot be done in one, or in three #

Two arguments, and they are different in kind.

The first is parity. A mirror has determinant −1 — it reverses handedness, taking a left-handed comma to a right-handed one — so a product of k mirrors has determinant (−1)ᵏ. An orientation-preserving motion is therefore an even number of mirrors and an orientation-reversing one an odd number. That refuses, with no search anywhere, every claim that a rotation is three mirrors or a glide is two. It is the cheapest kind of impossibility proof: an invariant, checked once.

The second is fixed points. A reflection fixes a whole line; a rotation fixes one point; a translation and a glide fix nothing. Those are properties of the motion, so they are the same whatever decomposition is used, and they separate the cases parity leaves together. A translation and a rotation are both two mirrors and are told apart by whether the mirrors meet — which is the geometric half of the same statement.

The glide, which is where the third mirror is spent #

A glide reverses handedness and fixes nothing. Parity says it needs an odd number of mirrors; one mirror fixes a line, so one is out; therefore three, and three is enough.

The construction is direct. A glide is a mirror along some line ℓ together with a slide along ℓ, and the slide is the part of the translation no origin removes. Reflect in ℓ, then perform the slide as two mirrors perpendicular to it and half the slide apart. Three mirrors, and their product is the glide.

The decomposition is not unique, and neither is any of the others: the perpendicular pair may sit anywhere along ℓ, and the rotation’s pair may be turned together about its centre. What is unique is the count. A motion has one number of mirrors and many sets of them, which is exactly the situation in which counting is the useful thing to do.

What was checked, and how #

Every decomposition drawn here was built from the motion’s own invariants — the mirror line and the slide for a glide, the centre and the angle for a rotation — and then multiplied back out and compared with the motion it came from, to one part in a thousand million. A decomposition that fails that comparison is never reported: the figure refuses to be drawn rather than drawing three lines whose product is something else.

That check is worth stating plainly because a figure of three mirror lines is easy to draw and impossible to inspect. The lines could be at the wrong angle, in the wrong order, or half a unit off, and the picture would look exactly as convincing. Multiplying them back is the only thing that separates a diagram of a decomposition from a decomposition.

The same routine was then run over every operation of every one of the seventeen, which is seventy-eight operations counting each group’s identity once: each one classified, decomposed, rebuilt, and compared.

Finding the mirrors, rather than recognising them #

The decompositions above are constructions and not searches, and that is worth spelling out, because it is what makes the count usable rather than merely true.

An isometry arrives as a matrix and a vector: x ↦ Ax + b, with A orthogonal. Everything needed follows from two questions asked of that pair.

What is the determinant of A? Positive gives an even number of mirrors, negative an odd one. This is one multiplication.

What does the motion fix? Solving (A − I)x = −b either gives a point, a line, or nothing at all, and which of the three it gives is read off the same elimination that solves it. A motion with a fixed point and a positive determinant is a rotation about that point; with a negative determinant and a line of fixed points, a reflection in that line; with nothing fixed, a translation or a glide according to parity.

From there the mirrors are written down. For a rotation about p through θ: any line through p, and a second through p at θ/2 to it. For a translation by v: any line across v, and a second parallel to it at |v|/2. For a glide: the mirror line, then the slide as a translation, by the recipe just given.

No case analysis is left over. Two questions, four answers, four constructions, and each construction is checked by multiplying it out. The classification of the four motions is not a list to be memorised but a decision procedure with two branches, and the number of mirrors is what it returns.

The count is a rank, and that is where the bound comes from #

The four constructions above are four cases, and cases invite the suspicion that a fifth is hiding. It is not, and the reason is a single formula that returns the count for every motion in every dimension at once.

Ask two things of x ↦ Ax + b. What is the rank of A − I? And does the motion fix anything?

If it fixes a point, the number of mirrors needed is exactly the rank of A − I. If it fixes nothing, it is that rank plus two.

Every line of the list is that formula evaluated. The identity has rank nought and a fixed point: nought mirrors. A reflection has rank one — $A - I$ kills the mirror line and doubles everything across it — and a fixed point: one. A rotation has rank two and a fixed point: two. A translation has rank nought and no fixed point: two, which is where the ordering that looked strange comes from, since the rank says a translation is doing less than a rotation and the missing fixed point costs it the difference. A glide has rank one and no fixed point: three.

The bound follows from the formula rather than being imposed on it. A motion with no fixed point must have $A - I$ singular, because $(A - I)x = -b$ is solvable whenever $A - I$ is invertible. So the rank of a fixed-point-free motion is at most n − 1, and its count is at most n + 1. That is Cartan and Dieudonné’s bound, and the reason the awkward motion in each dimension is the expensive one is now visible: it is the motion that spends the most rank and still has nowhere to stand.

Space is the same formula with one more row. A rotation is rank two with a fixed point, so two mirrors. A screw is rank two with none, so four. A rotoinversion is rank three — $A - I$ is invertible, which is exactly the statement that a rotoinversion has a fixed point — so three, and never four. The plane’s list and space’s are not two lists.

The count is a distance, and the surprise is that it stops #

There is a standard way to turn a generating set into a distance, and doing it here is what makes the bound strange rather than merely tidy. Take any group and any set of generators; the length of an element is the fewest generators whose product it is, and the distance between two elements is the length of the one that carries the first to the second. The count above is exactly that length, for the isometry group of the plane generated by all of its reflections.

Most word metrics are unbounded. The integers generated by one is the obvious case — the length of n is |n|, and there is no largest. The isometry group of the plane generated by its reflections is not like that at all: every element is at distance three or less from the identity, and the group is infinite. The ball of radius three is the whole group, and the ball of radius two is the subgroup of orientation-preserving motions together with nothing else.

That is what Cartan and Dieudonné’s theorem says, read as a statement about a metric rather than about a construction, and it is the reason the count is a classification. A bounded word metric has only finitely many spheres, so listing the spheres lists everything — and the four motions are the four spheres. Nothing in the geometry of the plane guarantees a bound of this kind; the orthogonal group’s does, and it comes from the rank formula above rather than from any picture of lines.

The mirrors need not belong to the pattern #

There is a confusion available here that costs nothing to avoid and a great deal to make, and it is worth separating before the next rung.

The decompositions above are statements about the group of all isometries of the plane, which is generated by reflections. They are not statements about the symmetry group of any particular pattern, and the mirrors a decomposition produces are usually not symmetries of anything.

The pattern with a glide and no mirror makes the point sharply. Its glide is a symmetry; the three mirrors the construction writes it as are not. Reflecting the pattern in the glide’s own line does not give the pattern back — that is precisely what makes the group pg rather than cm — and the same holds for the perpendicular pair. The decomposition happens in the ambient group and lands, at the end, on an operation the pattern happens to have.

So a count of mirrors is not a count of symmetries, and a group containing a glide is not thereby a group containing reflections. Four of the seventeen contain no reflection at all and every one of their glides still costs three.

The distinction has a name worth carrying, because it decides which theorems apply. A reflection group is one generated by the reflections it contains; the full isometry group of the plane is one, and so are p4m and p6m and pm. A group like pg is not, and nothing about being a subgroup of a reflection group makes it one. The count above is a fact about the ambient group; which walls a pattern’s own mirrors make is a fact about the pattern, and the two agree only when the pattern is a reflection group to begin with.

One consequence is a warning about the count’s arithmetic. Reflection length is subadditive and not additive: composing two glides gives three mirrors and three mirrors, and the product is a translation costing two, because four of the six cancel in pairs. The only part of the count that survives composition intact is its parity, which is a homomorphism onto two elements and is exactly the determinant. Everything else about the count has to be recomputed from the product’s own matrix, and a figure showing six lines and asserting a translation would be showing six lines that are not the decomposition of anything.

What is left over: the two-mirror theorem, and where it bites #

The statement that two mirrors give a rotation or a translation and nothing else looks like an observation about the plane. It is the load-bearing part of the whole subject, and its consequence appears one dimension up.

In space a mirror is a plane; two planes either meet in a line, giving a rotation about that line, or are parallel, giving a translation. The same two cases, the same doubling. But space has a motion the plane has no room for — a screw, a rotation about an axis together with a slide along the same axis — and a screw is neither a rotation nor a translation. So a screw is not two mirrors. It preserves handedness, so it is an even number, so it is four, and space’s answer to the same question is one larger than the plane’s.

The general statement is Cartan and Dieudonné’s: every isometry of n-dimensional space is a product of at most n + 1 reflections. The plane’s three and space’s four are the two smallest cases, and the bound is attained in each — by the glide and by the screw respectively, which is a satisfying way of saying that the awkward motion in each dimension is the one that costs the most.

The counts either side of that step are worth having in the same sentence, because they say how much room the extra dimension buys. The seventeen plane groups contain seventy-eight operations between them, of which seven are glides and nothing needs a fourth mirror. The forty-five space groups this collection builds contain six hundred and seventeen, of which a hundred and twenty-six are screws — so the motion that attains the bound is not a rare specimen introduced to attain it. It is the second commonest orientation-preserving operation in the catalogue, behind the ordinary rotation and ahead of everything else.

The point inversion, whose count depends on the dimension #

One motion is worth following across the step, because it is the case where the plane’s intuition is not merely incomplete but reversed.

Send every point to its opposite through a fixed centre. In the plane that is the half turn: it is two mirrors at right angles, it preserves handedness, and it is an ordinary rotation with nothing special about it. In space the same instruction is the inversion, and it is three mirrors — the three coordinate planes, applied in any order — so it reverses handedness and is not a rotation at all.

The reason is one line of parity. Sending every point to its opposite in n dimensions negates n coordinates, so its determinant is (−1)ⁿ, and the count of mirrors has to have the same parity as n. Two in the plane, three in space, four in four dimensions. Nothing about the operation changes; the ambient dimension decides whether it is even or odd, and therefore whether it is a motion a hand can perform.

That parity is why a centre of symmetry is a real constraint on a crystal and a half turn is not. Twenty-seven inversions occur among the space groups this collection builds and each of them refuses a structure the chance of being one-handed, while every one of the thirty rotations in the plane groups leaves handedness alone. The count says which is which before anything is drawn: an odd number of mirrors is an operation no rigid object can carry out on itself without being its own mirror image.

What the picture cannot show #

Three things, and each is a place where the count is more honest than the drawing.

The freedom. Every figure here draws one decomposition out of infinitely many, and nothing in the picture says which parts of it are forced. The angle between a rotation’s two mirrors is forced; their absolute orientation is not. A reader looking only at the drawing would have no way to tell those apart, and the caption has to say it.

The order. Reflecting in ℓ₁ and then ℓ₂ gives a rotation one way; the other order gives the rotation the other way. The two are different motions and the same two lines. A static figure showing both lines and the final position is compatible with either, which is why the intermediate image is drawn in every figure above.

The identity. Nought mirrors is a real entry in the list and cannot be drawn at all. It is not a gap in the figures: it is the empty product, and it is what makes the list a statement about a group rather than about four unrelated kinds of motion.

Where it came from #

Reflections as the building blocks of everything else is an old idea made precise slowly. Euler knew in 1776 that a rotation of space about a point is a rotation about an axis; Chasles gave the general reduction of a rigid motion in 1830. Élie Cartan and Jean Dieudonné proved the general theorem in the twentieth century, in the setting where it belongs — an orthogonal group over an arbitrary field, where the bound is the dimension and one more allows the affine case.

For the plane the result is older than the theorem and belongs to the ornamentalists as much as to the geometers. The rule that a pattern’s mirrors at 30° force a six-fold centre is the doubling rule applied without ever being stated, and it appears in Islamic ornament centuries before anybody wrote down a group.

Where the ladder goes next #

Two directions, and the collection takes both.

Upwards in dimension: every motion of space is one screw, which is Chasles’ theorem and the fourth mirror’s home.

Sideways into the classification: the doubling rule is what makes the classification proof finite, because it turns “which mirrors can a pattern have” into “which angles between them are permitted”, and the permitted angles are those whose doubles are rotations a lattice allows. Everything about the seventeen that looks like geometry is this arithmetic in disguise, and the arithmetic is the count of mirrors.