The four motions of the plane

Every rigid motion of a flat surface — every way of picking the plane up and putting it down again without stretching it — is one of exactly four kinds. Not four kinds that have been found so far. Four, provably, with no fifth.

The result is old — Michel Chasles stated it in 1830 — and the proof is a page of case analysis on how many points a motion leaves fixed. What matters here is not the proof but the consequence: because the list is finite and short, every question about the symmetry of a flat pattern is a question about combinations of four things, and combinations of four things can be enumerated.

Slide #

A translation moves every point by the same vector. Nothing stays where it was.

It is the least interesting motion and the most important one, because it is the motion that makes a pattern a pattern. A design with no translational symmetry at all is a rosette — a badge, a snowflake, a ceiling boss — and rosettes are classified in about four lines. A design with translations in one direction only is a frieze, and there are seven of those. A design with translations in two independent directions is wallpaper, and there are seventeen.

The translations of a pattern form a structure of their own: the set of all vectors that map the pattern onto itself. That set is the lattice, and it is the single most constraining feature of the whole subject. Every other operation has to be compatible with it, and most candidate operations are not.

Compatibility is a stronger demand than it sounds. A rotation that is a symmetry of the pattern must also be a symmetry of the pattern’s set of translations, because rotating a translation gives another translation and the rotated one has to be in the set. So a rotation cannot merely look plausible on the motif; it has to map the entire infinite grid of repeat vectors onto itself. That is the demand that kills five-fold rotation, and it is worth noticing that the demand is about the translations, not about the drawing. A five-pointed star tiles nothing not because stars are awkward but because no lattice survives being turned by a fifth of a turn.

There is one more thing to say about translations before moving on, because it is easy to mis-state. A pattern’s shortest translation is not the same as its unit cell, and neither is unique. Two different pairs of repeat vectors can generate exactly the same lattice, and choosing between them is a matter of convention rather than of fact — a point taken up in the essay on unit cells.

Turn #

A rotation fixes one point and turns everything about it. The fixed point is the rotation centre, and it is the only point the operation does not move.

The interesting quantity is the order: how many times the rotation must be repeated before everything is back where it started. A half turn has order two, a quarter turn order four, a sixth of a turn order six. Order one is the identity, which is a rotation by nothing at all.

For a pattern with a lattice, the available orders are startlingly few. Only one, two, three, four and six are possible; five is not, seven is not, and neither is anything above six. That is the crystallographic restriction, it has a one-line proof, and it is the constraint from which the finiteness of the whole classification ultimately comes.

A rotation preserves handedness. A left-handed motif stays left-handed however far it is turned, which is worth holding on to, because the next two motions do not have that property and telling them apart from rotations is otherwise surprisingly hard by eye.

Rotation orders also nest. If a pattern has a sixfold centre then repeating that rotation twice gives a threefold rotation about the same point, and repeating it three times gives a half turn — so a sixfold centre is automatically a threefold centre and a twofold centre as well. This is why the group symbols are not simply a count of centres: p6 has, per cell, one sixfold centre, one further threefold centre that is not on it, and further twofold centres again, and the numbers only make sense once the nesting is understood. The notation records the highest order present rather than every order implied.

Flip #

A reflection fixes a whole line — the mirror — and swaps the two sides of it.

Reflections reverse handedness. This is the one visual cue that reliably distinguishes them from rotations, and it is the reason every pattern figure on this site draws reflected copies of the motif in a second colour. A pattern drawn with a symmetric motif hides the distinction completely, which is a hazard with its own essay.

The composition rule for reflections is the most useful fact in elementary symmetry: two reflections in lines crossing at angle $\theta$ compose to a rotation through $2\theta$ about the crossing point. Two reflections in parallel lines a distance $d$ apart compose to a translation by $2d$. Every rotation and every translation can therefore be built from reflections, which makes reflections in a sense the primitive motion — though nothing in the classification depends on choosing them as such.

Flip and slide #

The fourth motion has no fixed point and no fixed line. A glide reflection reflects across a line and then translates along that same line, and it is a symmetry of patterns in which neither the reflection nor the translation is one.

The standing example is a trail of footprints. Reflect the trail across its centre line and every left print becomes a right print in the wrong place; slide it half a stride and the prints land correctly. Neither operation alone does anything useful. Together they map the trail onto itself.

Glides are the reason the classification is awkward and the reason it is interesting. Five of the seventeen wallpaper groups contain glides that are not compositions of a mirror already present with a translation already present — the glide is essential, not incidental — and those five are exactly the ones that ornamentalists working by eye tended to miss. A pattern in pg looks, to an untrained eye, like a pattern with mirrors in it, because the local impression of a flip is there. The mirror is not.

Distinguishing a glide from a mirror is not a matter of looking harder. It is a matter of asking whether the pure reflection, with no slide at all, maps the pattern onto itself. Every generator on this site answers that question by computing it: the operation’s own matrix and translation determine whether the translation runs along the axis, and a translation along the axis is precisely what a glide is.

The friezes as a miniature of the whole argument #

The four motions are easier to see acting together on a strip than on a plane, and the strip case is a complete classification in its own right — the shortest one in the subject.

A frieze has translations in one direction only. The remaining motions have to be compatible with that single direction, which cuts them down brutally: the only rotation available is a half turn, since anything else would tilt the repeat direction into a direction it does not have; the mirrors available are one across the strip and one along it; and the glide runs along the strip. That is four possible extras, and the combinations that are consistent come to seven.

Seven, not sixteen, because the combinations are not free. A horizontal mirror and a vertical mirror force a half turn, by the composition rule already noted. A half turn and a vertical mirror force either a horizontal mirror or a glide, depending on where the twofold centre sits relative to the mirror. Each forced consequence collapses two apparently distinct candidates into one, and by the time the forcing is exhausted only seven remain.

That is the entire logic of the wallpaper classification, rehearsed at a scale small enough to check by hand. The seventeen is the same argument with a second translation direction, five lattice types instead of one, and considerably more bookkeeping.

How each is identified, without looking #

Every symmetry element drawn on this site is located by a computation rather than by an author’s judgement, and the computation is short enough to describe.

An operation is a matrix $M$ together with a translation $t$, acting on a point $x$ as $Mx + t$. The determinant of $M$ decides handedness: $+1$ means the motion preserves it, so the operation is a translation or a rotation; $-1$ means it reverses it, so the operation is a reflection or a glide.

For the handedness-preserving case, the trace of $M$ gives the rotation order directly — trace $2$ is the identity or a translation, and the other allowed traces $-2, -1, 0, 1$ give orders $2, 3, 4, 6$ respectively. That correspondence is not a coincidence; it is the crystallographic restriction in the form it is actually proved.

For the handedness-reversing case, the axis direction is the eigenvector of $M$ with eigenvalue $+1$, and the translation splits into a part along that axis and a part across it. The across-part merely shifts where the axis lies. The along-part is the glide. If it is zero the operation is a mirror; if it is not, it is a glide. There is no judgement anywhere in that, and no threshold.

Who gets fooled by a glide #

It is worth being concrete about how often the glide is missed, because the answer is: constantly, including by people who are careful.

The failure has a specific shape. A pattern in pg presents alternating mirror-image motifs, and the eye reads mirror images as evidence of a mirror. The reader locates, correctly, that handedness is reversing; infers, incorrectly, that a reflection is present; and labels the pattern pm. Nothing in the visual impression distinguishes the two cases, because both do contain reversed copies — the question is only whether any pure reflection, with no accompanying slide, works.

The same failure runs the other way in three dimensions, where a screw axis is routinely read as a plain rotation. In both cases the fix is identical and unglamorous: apply the pure operation and check, rather than judging from the impression the pattern makes. This is also the reason the diffraction route matters so much in practice — a glide announces itself in a diffraction pattern by removing alternate reflections along a row, and those absences are visible in a way the glide itself never is.

Why exactly four, and not five #

The argument that the list is complete is a case analysis on fixed points, and it is short enough to sketch.

An isometry of the plane is determined by where it sends three non-collinear points. If it fixes all three, it is the identity. If it fixes exactly two, it fixes the whole line through them, and the only such motion other than the identity is the reflection in that line. If it fixes exactly one, it is a rotation about that point. And if it fixes none, it is either a translation or — the case that is easy to overlook — a glide reflection.

The fourth case is the one Camille Jordan got wrong in his 1869 attempt at the classification of motions, and it is the reason his enumeration of the plane groups came out at a different number from Fedorov’s. Missing a motion propagates: a group whose distinguishing feature is an essential glide simply does not appear if glides are not on the list of motions to check for.

Handedness is the thing to draw #

There is a practical rule that follows from all of this, and every figure on this site obeys it.

The four motions divide into two that preserve handedness and two that reverse it. That division is the most informative thing a pattern figure can convey, and it is invisible if the motif has a mirror of its own. A dot, a square, a circle, a five-pointed star — all of them are their own mirror images, so a reflected copy is indistinguishable from a rotated one, and a reader has no way to tell which operations are present.

So the motif here is a small scalene flag with a head, drawn in one colour where handedness is preserved and another where it is reversed. It is the algebraic version of the comma on a nineteenth-century ornament plate, and the plates were right for a reason their draughtsmen could not have articulated: a dot is too symmetric to illustrate most groups, and the resulting picture is quietly of a different group from the one in its caption.

What changes in three dimensions #

The plane has four motions. Space has more, and the extra ones are the reason crystallography is harder than wallpaper.

Space keeps translations, rotations and reflections, and it keeps the glide — now a reflection in a plane combined with a slide within that plane. It adds the screw axis, which is a rotation combined with a slide along the rotation axis, and the rotoinversion, which is a rotation combined with an inversion through a point. A screw axis is to a rotation what a glide is to a mirror: the composite is a symmetry although neither part is.

That is where the seventeen becomes two hundred and thirty. The extra motions multiply the combinations, but the logic does not change at all, and neither does the crystallographic restriction: space permits rotations of order one, two, three, four and six, and nothing else, for the same reason the plane does.

The count itself has a slightly comic history. Fedorov derived the two hundred and thirty space groups in 1891; Arthur Schoenflies derived them independently at almost the same moment; each had errors, and the two men corrected one another by correspondence until the lists agreed. William Barlow, an English amateur with no university position, had arrived at the same enumeration by a different route. Three people, three methods, one number — which is about as good a demonstration as a piece of mathematics ever gets that the number is a fact rather than a taxonomy.

Where the ladder goes next #

The natural next step is to stop treating these four as a list and start treating them as an algebra. They compose, the compositions are forced, and the forcing is what makes the classification finite — which is the argument for calling the collection a group.

After that, the constraint: patterns that repeat sit on a lattice, lattices come in exactly five kinds, and the compatibility of the four motions with those five kinds is the entire content of the classification.

What the pictures here cannot show. A single figure shows an operation applied once or a few times; it cannot show that no other operation is present, which is a statement about an infinite search. That statement is made by the detector rather than by the drawing, and where a figure asserts a group, the assertion is the computation’s, not the eye’s.

The same theorem one dimension up #

The four are a fact about the plane, and the corresponding statement in space is worth having beside them, because it is the same argument and it produces the objects a crystal is actually made of.

Split by handedness first, as here. A rigid motion of space either preserves handedness or reverses it, and the two halves behave quite differently.

Every handedness-preserving motion of space is a screw. Turn about an axis and slide along that same axis, in either order — they commute, so the order does not matter. The translation is the case where the turn is nothing; the pure rotation is the case where the slide is nothing. That is Chasles’ theorem proper, and the surprise in it is that the two ingredients can always be made parallel: a rotation about one axis followed by a translation in some unrelated direction is a single screw about a shifted axis, exactly as a turn followed by a slide is a single turn about a shifted centre here.

Every handedness-reversing motion is a glide reflection or a rotary reflection. The first is this essay’s fourth motion with a plane in place of the line — reflect in a plane and slide inside it. The second has no analogue here at all: turn about an axis and reflect in the plane perpendicular to it. Its special case, a half turn composed with the perpendicular reflection, is the inversion through a point, which sends every point to its opposite through a centre.

So the plane’s four become space’s six, and the extra two are exactly the operations that make the two hundred and thirty possible where seventeen were: the screw, which is a glide with a rotation in it, and the rotary reflection, which is a reflection with a rotation in it. Both have the same character as the glide — a composite with no fixed point of its own, invisible to anyone looking for its ingredients separately.