What a symmetry actually is

Ask what makes a snowflake symmetrical and the usual answer describes the snowflake: it is balanced, it repeats, it looks the same all the way round. Every one of those statements is about the object. None of them can be checked, counted, or argued with, and a subject built on them goes precisely nowhere.

The move that turns ornament into mathematics is to stop describing the object and describe a motion instead.

A symmetry is an operation that leaves a pattern indistinguishable from itself. Turn the plane a quarter of a turn about a certain point; if what results is identical to what was there before, that quarter turn is a symmetry of the pattern. Nothing has been said about the pattern’s appearance, and everything has been said about what may be done to it.

Why the shift is worth making #

The gain is not philosophical. It is that operations can be listed, composed, counted and refuted, and adjectives cannot.

Consider the difference in practice. “This tiling is fourfold symmetric” is a claim that can only be assessed by looking, and looking is exactly what fails at this task — a pattern with the wrong description looks precisely like a pattern with the right one. “Rotation by 90° about the point at the centre of this cell maps the pattern onto itself” is a claim that can be tested by applying the rotation and comparing. It is either true or false, and finding out is arithmetic.

That is the whole reason this subject has theorems rather than a catalogue. Ornament had been made for six thousand years before anybody proved anything about it, and what unlocked the proofs was the change from noun to verb.

The motions available #

A symmetry of a flat pattern must not distort it. Distances have to be preserved, or the copy would not be indistinguishable from the original, so the candidate motions are the isometries of the plane: rigid motions, possibly with a flip.

There are exactly four kinds, and the essay on the four motions takes each in turn. In brief: a translation slides everything by a fixed vector; a rotation turns everything about a fixed point; a reflection flips everything across a fixed line; and a glide reflection does a reflection and a slide along the same line, in a combination that is a symmetry even though neither half is.

That last one is the surprise, and it is the reason the classification comes out at seventeen rather than something tidier. A pattern of footprints down a beach has no mirror line — the left foot is not a mirror image of the left foot beside it, because there is no left foot beside it — and it has no translation shorter than a full stride. What it has is a glide: reflect across the line of travel and slide half a stride, and the trail maps onto itself.

What does not count #

The restriction to isometries is a choice, and seeing what it excludes makes clear what it is for.

A scaling is not a symmetry. Doubling the size of a pattern about a point produces something recognisably related to the original and utterly different from it: distances have changed, so a scaled copy laid over the original does not match. There are patterns invariant under a scaling combined with a rotation — logarithmic spirals, and the inflation that generates Penrose tilings — and those invariances are genuinely interesting, but they belong to a different classification with a different arithmetic.

A shear is not a symmetry either, for the same reason. Nor is any smooth distortion, however gentle. The topologist’s rubber sheet is exactly what this subject refuses: symmetry here is rigid, and rigidity is what makes the count finite.

What survives the restriction is a set of motions small enough to enumerate. Every isometry of the plane is one of the four kinds already named — a result usually attributed to Michel Chasles in 1830 and provable in about a page — and that finiteness is the first domino. Four kinds of motion, constrained to be compatible with a repeat, give five lattices; five lattices give seventeen groups. Loosen the rigidity anywhere along that chain and the numbers become infinite.

The identity, and why it is not a technicality #

Doing nothing is a symmetry. That reads like a joke and it is load-bearing.

The reason is that symmetries are going to be composed — one done after another — and a system of things that can be combined needs a neutral element for the arithmetic to behave. More concretely: every rotation has an inverse rotation that undoes it, and doing both in succession must land somewhere. Where it lands is the identity, so the identity has to be in the collection.

This is the first hint that the collection of symmetries has algebraic structure rather than being a list, and that structure is what makes classification possible.

What “the same pattern” has to mean #

Two decisions have to be made before any of this is precise, and they are conventions rather than discoveries. Both are worth stating plainly, because a great deal of confusion in the subject comes from authors who made different choices silently.

The first is whether colour counts. A chessboard has a fourfold rotation about the centre of a square only if the squares’ colours are ignored; with colours, a quarter turn about a square’s centre maps black to black and white to white and works fine, but a quarter turn about a corner does not. Everything here treats the pattern as a set of points, uncoloured. Colour symmetry is a real and rich extension — it multiplies seventeen into forty-six two-colour groups — and it is a different subject.

The second is whether the pattern is infinite. Every classification statement in this subject is about a pattern that repeats for ever. A real tiled floor stops at the wall, and strictly it has almost no symmetries at all, because every candidate translation runs off the edge. The mathematics is about the idealisation, and the idealisation is the useful object: it captures what a bounded piece of floor is a piece of.

Both conventions have the same consequence for how the figures on this site are to be read. Every pattern drawn here goes on for ever and is drawn as far as the paper allows, so every edge in every figure is an artefact of the page rather than a feature of the pattern, and every statement made about a figure’s symmetry is a statement about the infinite object it is a window onto. That is why a figure showing four cells and a figure showing sixteen are pictures of the same thing, and why nothing in this subject is ever settled by counting what is inside a frame.

The operation is not the element #

Here is a distinction that trips up almost everybody at first, and getting it straight early saves a great deal of pain later.

A rotation by 90° about a particular point is an operation: a motion. The point itself is a symmetry element: a geometric feature of the pattern where an operation has its fixed point. Similarly a reflection is an operation and the mirror line is an element.

The distinction matters because the counting is different. A pattern in the group p4 has, per unit cell, one operation of order four in each direction of turn, one of order two, and the identity — four operations. It has considerably more than four elements, because rotation centres of order four occur at two inequivalent positions and order-two centres at another. Textbook diagrams mark elements; the algebra counts operations; and a reader who conflates them will find every number in the subject slightly wrong.

The International Tables marks are a notation for elements — a lens for a twofold centre, a triangle for threefold, a square for fourfold, a hexagon for sixfold, a solid line for a mirror and a dashed one for a glide. Those marks appear on every pattern figure on this site, for the good reason that a reader who learns them here meets the same symbols in every other source.

The gap between the two numbers is not an accident of p4 and it does not close in the other groups; it widens. The reason is that a lattice translation carries an operation to a new place without producing a new operation: the half turn about the origin and the half turn about the centre of the cell differ by a translation, so modulo the translations they are the same element of the group, while as marks on a plate they are two centres a reader can point at. Every element beyond the first is a translate of one already counted, which is precisely why the census can be derived rather than transcribed — and why a plate with a centre missing from it is a plate that can be caught.

Doing one after another #

One more property of operations, mentioned here and taken seriously two essays later, because it is what elevates the collection from a list to a structure.

Symmetries compose. Do one, then do another, and the combined motion is itself a symmetry — it must be, since the first left the pattern indistinguishable from itself and so did the second. So the collection of symmetries is closed under composition, and that closure has content: it means the operations constrain each other.

The clearest instance is that two reflections make a rotation. Reflect across one mirror line and then across a second that crosses it at angle $\theta$, and the result is a rotation about the crossing point through $2\theta$. Nothing was assumed about the pattern; it is a fact about the plane. Its consequence for patterns is sharp: a pattern with two mirrors at 45° must have a fourfold rotation, whether or not its designer intended one.

The forcing is what makes the collection worth a name. Take two of a pattern’s mirrors, do one and then the other, and the motion that results is not a new object that has to be added to the list — it is a rotation that was already in the list, sitting at the point where the two axes cross, whether or not anybody had noticed it. That is the difference between a set and a structure: in a set the members are simply present, and in this structure each member constrains the others, so a pattern cannot have two of anything without having a third thing it never asked for. The essay that takes this seriously does the composition explicitly and shows the third motion arriving.

That is why the seventeen cannot be extended by mixing. Adding an operation to a group forces every composition it makes with what was already there, and the forced consequences either land inside one of the seventeen or contradict the lattice. There is no eighteenth case waiting to be discovered by a sufficiently inventive ornamentalist, and the argument that closes off every branch is finite.

Where the exactness comes from #

There is a reason this site can make stronger claims than most, and it is worth saying explicitly at the start.

Symmetry, for periodic patterns, is decidable. Choose coordinates along the pattern’s own repeat vectors rather than along the page, and something remarkable happens: every symmetry operation becomes a matrix of whole numbers, and every translation becomes a fraction with a small denominator. A rotation by 120°, which in ordinary coordinates involves $\sqrt{3}/2$ and immediately drags a tolerance into every comparison, becomes the perfectly exact integer matrix that sends the first repeat vector to the second and the second to minus their sum.

In that basis, asking whether a pattern has a symmetry is asking whether two lists of rational numbers are equal. There is no tolerance to choose and no residual to interpret. The answer is yes or no.

It is worth seeing how sharp that is. A threefold rotation, written in the pattern’s own repeat vectors, is the matrix that sends the first vector to the second and the second to the negative of their sum — all four entries whole numbers, all four exact. Multiply it by itself three times and the identity comes back exactly, not to fifteen decimal places. Every question the subject asks reduces to manipulations of matrices like that, and manipulations of whole numbers do not accumulate error. This is unusual. Most of the sciences that draw pictures of computed things are comparing a residual against a threshold somebody chose, and choosing it badly is a standing hazard. Here there is nothing to choose.

That last column is the whole reason for the change of basis, and it is worth being clear that it is not a complaint about decimals. A residual of a few parts in a hundred thousand million million is floating-point arithmetic doing exactly what it promises. The difficulty is that somebody now has to decide how large a residual still counts as zero, and the decision has no good answer: set the bar low and a genuine sixfold rotation is refused because a square root was rounded twice on the way in; set it high and a rotation by fifty-nine and nine-tenths of a degree is accepted as a rotation by sixty. In the lattice basis the decision does not arise. There is no threshold to set well, because there is no threshold.

Every pattern on this site exploits that. Each is generated by applying a group’s operations to a motif; then the group is thrown away and a detector examines the bare point set, enumerating every operation the lattice permits and keeping the ones that map the set to itself; and the detected set must match the generating set exactly. A figure that fails that comparison is never drawn at all.

The direction that matters #

Both directions of that comparison earn their place, and the second is the interesting one.

Detecting fewer symmetries than were generated means the motif was not really invariant. That kind of error tends to look wrong, so a careful reader would catch it.

Detecting more means the motif was accidentally too symmetric, and the picture illustrates a different group from the one named underneath it. Nothing about the picture betrays that. It is the characteristic error of hand-drawn pattern figures, it is invisible to the eye, and it is the entire argument for computing the answer instead of judging it. An essay is devoted to it, because the fix — draw a comma, never a dot — turns out to be the rule every nineteenth-century ornament plate already followed for reasons its draughtsmen could not have stated.

Who thought of it this way first #

The verb-not-noun move has a history, and it is shorter than the subject’s age suggests.

Ornamentalists classified patterns by eye for centuries. Owen Jones’s Grammar of Ornament of 1856 organised hundreds of plates by culture and motif, with no notion that the arrangements might be finite in number. Camille Jordan attempted a classification of the motions themselves in 1869 and got it slightly wrong. Evgraf Fedorov, a Russian crystallographer, derived the seventeen plane groups in 1891, and George Pólya rederived and popularised them in 1924 — after which M. C. Escher, who read Pólya’s paper without following its mathematics, worked through the whole list by hand and made most of his reputation on it.

The pattern in that history is that the enumeration had to wait for the operations. As long as symmetry was an adjective, the question “how many kinds are there?” could not even be posed, because there was nothing to count. Once it became a list of motions with an arithmetic, the count fell out — and it fell out finite, which nobody had any right to expect.

Where the ladder goes next #

Three directions lead out of this essay, and they are genuinely independent.

The first is downward into the motions themselves: what the four kinds are, and why a glide reflection is neither of the two things it is built from.

The second is upward into the algebra: why a set of symmetries is a group rather than a list, and what closure buys that a list cannot.

The third is sideways into the constraint that makes everything finite: patterns that repeat must sit on a lattice, and a lattice is remarkably fussy about which rotations it will tolerate. It permits rotations of order two, three, four and six, and nothing else at all — a fact whose proof is one line of arithmetic and whose apparent violation, ninety years later, won a Nobel Prize.

What the pictures here cannot show. Every figure on this page is a finite scrap of an infinite object, and every claim made about it is a claim about the infinite version. The drawn edges are the page’s, not the pattern’s. And the marks denoting symmetry elements are notation rather than content: they record where the operations act, and a reader who has not yet learnt them is looking at a pattern with some triangles on it.

A bounded figure always has a centre #

The definition has an immediate consequence for finite objects, and it is worth taking because it explains why molecules and crystals are described with different vocabularies.

A symmetry of a figure permutes its points, so it takes the figure to itself as a whole while moving individual points around. Nothing in the definition asks it to hold any point still.

And yet a bounded figure’s symmetries all hold one point still. Take the figure’s centre of mass — the average position of its points. A symmetry is a rigid motion that carries the figure onto itself, so it carries the set of points onto the same set, so it carries their average onto that same average. The centre of mass is fixed by every symmetry the figure has.

So a bounded figure’s symmetry group is a group of motions about one point, which is what a point group means. In the plane the only motions fixing a point are rotations about it and reflections in lines through it, so the symmetries of any bounded figure are a cyclic family of rotations, possibly together with mirrors — the two families a snowflake, a flower and a molecule are described by.

The argument fails for an infinite pattern, and it fails at the one step where it must: an infinite set of points has no centre of mass to be preserved. That is exactly the room a translation needs. A pattern that repeats forever can be carried onto itself by a motion with no fixed point at all, which is why the classification of patterns is a longer story than the classification of figures, and why translations are the first thing this collection introduces.

The same definition, elsewhere #

The move from adjective to verb is not peculiar to ornament, and seeing it in one other place makes clear which part of the definition is doing the work.

A symmetry is an invertible transformation that preserves the structure in question. The plane’s structure here is distance, so its symmetries are the distance-preserving maps. Change the structure and the same sentence gives a different subject.

The symmetries of a graph are the relabellings of its points that preserve which points are joined. Nothing is being moved and there is no distance anywhere, and every argument in this collection about composing, forcing and orbits applies unchanged.

The symmetries of an equation are the changes of variable that leave it looking the same — which is how the subject of solving polynomials became a subject about groups, and how the question of whether the general quintic can be solved by radicals turned into a question about a group of order sixty.

The symmetries of a physical law are the transformations under which it takes the same form, and the standing result about them is that each one comes with a conserved quantity: invariance under a shift in time gives conservation of energy, invariance under a shift in position gives conservation of momentum.

What all four share is the shape of the definition, not the objects. Something is fixed; something is allowed to move; the moves that respect what is fixed are composed, and the collection of them is closed. Everything this collection proves about wallpaper is proved from that shape, which is why the arguments transfer and the pictures do not.