Crystallography & Symmetry
patterns derived from their groups
Operations

The four motions of the plane

Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.
15 min read 8 figures Generated, not drawnSymmetry is decidable

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.

A glideThe motif in the first colour, its images under a single glide in the second, and the symmetry element marked where the operation itself says it lies.flip and slide — neither motion alone is a symmetryglide
Fig. 1 The fourth kind, and the only one most people have never been told about. A reflection across the line, followed by a slide along it. Neither half is a symmetry of this pattern; the two performed together are.

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.

A translationThe motif in the first colour, its images under a single translation in the second, and the symmetry element marked where the operation itself says it lies.slide, and the lattice comes back to itselftranslation
Fig. 2 A slide by one repeat vector and by another. The pattern is infinite, so nothing has fallen off an edge; what looks like a modest rearrangement of a finite drawing is, in the object being described, no change at all.

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 of order 6The motif in the first colour, its images under a single rotation in the second, and the symmetry element marked where the operation itself says it lies.a sixth of a turn — the largest a lattice permitsrotation-6
Fig. 3 A sixth of a turn, applied six times. Six is the largest order a lattice permits, and a pattern with a sixfold centre necessarily also has threefold and twofold centres, because repeating the sixth-turn twice and three times gives those.

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 θ\theta compose to a rotation through 2θ2\theta about the crossing point. Two reflections in parallel lines a distance dd apart compose to a translation by 2d2d. 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.

A mirrorThe motif in the first colour, its images under a single mirror in the second, and the symmetry element marked where the operation itself says it lies.flip across a line, and handedness reversesmirror
Fig. 4 A flip across the marked line. The images are drawn in the second colour because handedness has reversed: no amount of sliding or turning will bring a reflected motif back into agreement with the original.

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 wallpaper group pgA pattern with the symmetry of pg, generated by applying the group's 2 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.pgrectangular lattice · 2 operations per cellelements marked
Fig. 5 The group pg, whose only symmetry beyond translation is a glide. The dashed lines are the glide axes. There is no solid line anywhere in this figure, and a great many published patterns labelled pm are actually this.

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.

The seven frieze groupsEvery way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.p1hopp11gstepp1m1sidlep2spinning hopp2mgspinning sidlep11mjumpp2mmspinning jumpsolid line: a mirror · dashed: a glide · lens: a half-turn centreseven, and no eighth
Fig. 6 All seven friezes, each generated from its own operations. Reading down, the extras accumulate: nothing, a glide, a vertical mirror, a half turn, a half turn with a vertical mirror and a glide, a horizontal mirror, and finally everything at once.

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 MM together with a translation tt, acting on a point xx as Mx+tMx + t. The determinant of MM decides handedness: +1+1 means the motion preserves it, so the operation is a translation or a rotation; −1-1 means it reverses it, so the operation is a reflection or a glide.

For the handedness-preserving case, the trace of MM gives the rotation order directly — trace 22 is the identity or a translation, and the other allowed traces −2,−1,0,1-2, -1, 0, 1 give orders 2,3,4,62, 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 MM with eigenvalue +1+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 wallpaper group pmgA pattern with the symmetry of pmg, generated by applying the group's 4 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.pmgrectangular lattice · 4 operations per cellelements marked
Fig. 7 The group pmg, which contains mirrors and glides at once and is where the confusion is most acute. The solid lines are true mirrors; the dashed ones are glides. A reader who cannot tell which is which by looking is in good company, and the distinction is entirely computable.

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.

Growing the pgg orbitOne motif, then more of the group's operations applied to it, until applying another produces nothing new. The pattern is the orbit; the drawing is only its shadow.1 of 43 of 44 of 4the pattern is grown from the group, never drawnpgg
Fig. 8 The group pgg, built up operation by operation. It contains half turns and glides in two directions and no mirrors at all, which is a combination that is very hard to see and completely straightforward to compute.

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.

Where this goes next

turn about a point, and nothing moves at that pointrotation What a symmetry actually is Not a property of a shape but a motion that leaves it alone. Once symmetry is a verb rather than an adjective, everything else in the subject follows — including why there can only ever be seventeen wallpapers. 1 of 43 of 44 of 4the pattern is grown from the group, never drawnp4 The orbit is the pattern A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out. p1hopp11gstepp1m1sidlep2spinning hopp2mgspinning sidlep11mjumpp2mmspinning jumpsolid line: a mirror · dashed: a glide · lens: a half-turn centreseven, and no eighth Seven friezes The same classification argument on a strip instead of a plane, where it is short enough to check by hand. Seven ways to repeat a motif along a line, with names like hop, step and sidle.