Crystallography & Symmetry
patterns derived from their groups
Concept

Plane group — where it appears

Another name for a wallpaper group: a group of plane isometries whose translations form a two-dimensional lattice. The two names are used interchangeably, and the second is preferred where the contrast with a layer group or a space group matters.

Named by 18 essays across 7 fields — each of them below, with the objects they name alongside it.

Two half-turns make a translation. The half-turn about (0.25, 0.25) followed by the half-turn about (0.75, 0.5) is the translation by (1, 0.5) — twice the vector between the two centres, and not the vector itself. The open lens is a third centre, and it is not the midpoint of the two drawn: it is where the half-turn about the first lands when it is composed with one repeat vector of the lattice, which is half a repeat along. That is the step that puts two-fold centres on the half lattice and gives a p2 cell four inequivalent ones. Both the translation and the forced centre are computed from the operations and compared with the construction in exact rational arithmetic.

Where the product is

Composing two symmetries lands on a third — and the third one is somewhere. Two half-turns make a translation by twice the distance between their centres, and that single fact puts the lattice into a pattern before anybody chooses one.

operations · Composition p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

The two that fold into a surface

Fold a wallpaper pattern along its own symmetries and what is left is usually a shape with corners and edges nobody drew. For two of the seventeen it is a plain surface with no marks on it at all — a torus and a Klein bottle — and which two is decided by a single question asked of every operation.

classification · Flat space a general quadrilateral tiles. A general quadrilateral — convex, with no equal sides and no parallel edges — with copies placed by half-turns about edge midpoints. The patch was checked by sampling 2000 points inside a disc: every one of them lies in exactly one tile, so there is no gap and no overlap anywhere in the region tested.

Which shapes tile by themselves

Every triangle tiles the plane. So does every quadrilateral, convex or not. Six sides admits three families, seven sides admits nothing at all — and the five-sided case took a hundred years and finished with a computer search. The bound at seven needs no search: it is Euler's relation with the curvature set to zero.

classification · Tilings At which indices a group contains a copy of itself. A filled circle where the group has a subgroup of that index which is the same plane group again. The groups with no rotation past a half-turn take every index — the lattice can be stretched along one direction by any factor. The four-fold groups take the sums of two squares and the three- and six-fold groups take the Loeschian numbers, because a sublattice invariant under a quarter or a third of a turn is an ideal in the Gaussian or Eisenstein integers and its index is a norm. The groups with mirrors take fewer still, and p4g takes only the squares.

The same group in a bigger cell

A subgroup usually gives something up. An isomorphic subgroup gives up nothing but scale — the same plane group again, on a coarser lattice — and the indices at which that is possible turn out to be the values of a quadratic form.

space-groups · Isomorphic subgroups pg scatters as pmm. The orbit of a motif under pg, and the set of all ordered differences between its points brought to a common origin. The vector set was handed to the detector with no indication of where it came from, and came back as pmm: 31 peaks from 6 atoms, on the same lattice.

Seventeen groups, seven vector sets

A map of interatomic vectors is more symmetric than the structure it came from, twice over: it always acquires a centre, and it loses every translation part. So a glide becomes a mirror, seventeen plane groups collapse onto seven — and the collapse is verified by handing the vectors to a detector that has never heard of Patterson.

diffraction · The Patterson function Four ways to lay a second row on the first. The same shape four times, with the upper row related to the lower one by a translation, a half turn, a glide and a mirror. In each case the upper row is pushed down until it touches, and the number is the density that results. The three that keep the shape the same way round come out within two per cent of one another; the mirror packs at 78 per cent of the best of them — 22 per cent less dense — because it presents a protrusion to a protrusion. Stated the other way round, the best of the four is 28 per cent denser than the mirror; the two percentages are the same measurement against two different bases, and neither is the other.

The four plane groups a molecule packs in

A molecule is not a disc: it has bumps and hollows, and packing it tightly means getting one molecule's bump into another's hollow. A mirror puts a bump against a bump. Filter the seventeen by that one observation and four survive — and the space groups the structural literature is mostly made of are the three-dimensional version of the same four.

applied · Packing 21 superspace groups in (2+1) dimensions, from 31 names. The whole count, in the order it is built. Thirteen arithmetic classes of the plane; six of them admit an incommensurate wavevector; those six give ten sign assignments; each assignment contributes the plane cohomology times the internal cohomology, which is thirty-one names; and the names are merged by the changes of basis that are relabellings — a change of the plane basis, which moves the sign assignment and the internal cocycle with it, and the choice of q against −q. The last row is what the count would be if the two factors were quotiented separately, which over-counts because the merge is not independent of the internal part.

Superspace groups in the plane

A modulated crystal has no space group, and in a space of one more dimension it has one. Counting them in the plane is the seventeen's own extension arithmetic with a third coordinate on which the point group acts by a sign — and the sign has to be plus or minus exactly, which kills the three-fold, four-fold and six-fold classes before a single extension is counted.

aperiodic · Modulation The domain is a polygon, and its edges are elements. The Dirichlet domain of a point whose stabiliser is trivial: the set of points at least as close to it as to any other point of its orbit. It is a convex polygon, it is a fundamental domain, and each of its edges lies on the bisector of the base point and one image of it — so each edge already carries the element that produced it, with no search. Edges are drawn by kind: paired with another edge, fixed pointwise by a reflection, or folded in half by a half turn.

The relations a polygon dictates

Poincaré's theorem has two halves. The walls of a fundamental domain name the generators, which is the half this collection already computes; walking round its corners names the relations, which needs a domain with corners rather than a domain made of pixels. Building the Dirichlet polygon exactly gives a presentation of each of the seventeen — and coset enumeration says every one of them is right.

operations · Fundamental domain The parity argument loses 36 pairs it had won alone. The argument that refutes ten of the twenty-one species walks round a polygon of odd size: the ring of polygons about it is a closed walk of odd length in a graph the species decides, and a bipartite graph has no such walk. With two species at a vertex the flanking pairs come from the union of two graphs, and a union of bipartite graphs need not be bipartite — so the walk stops being constrained. The fourth row is the cost: pairs whose members the argument kills on their own and which it cannot kill together.

The argument that closes eleven

Twenty-one vertex species satisfy the angle equation; a parity argument kills ten before anything is drawn, and the eleven survivors are all built. Asking the same question of tilings with two kinds of vertex, the parity argument evaporates — it constrains a walk in a graph one species decides, and two species decide the union of two graphs, which need not be bipartite. What is left is a search, and a search cannot close a count.

classification · Decidability One group refuses two colours and three refuse three. The two counts side by side, with the rows that refuse a number of colours marked. p3 is the only group with no two-colouring; p4, p4m and p4g are the only ones with no three-colouring. Neither list is a subset of the other and both come from the same arithmetic — a rotation order that divides nothing the symmetric group has.

What a half-turn does to three colours

Ten of the seventeen plane groups have no three-colouring, because a half-turn cannot permute three colours cyclically — that is the first rung of this ladder and it is true. Drop the word cyclically and the answer changes completely: a half-turn permutes three colours perfectly well by swapping two and fixing one, and only the three four-fold groups refuse three colours at all.

classification · Colour One achiral motif in p4, chiral along one line and achiral along two. The same motif — three points with a mirror and no other symmetry — repeated by p4, the plane group of quarter-turns with no mirror, and placed with its mirror along three different lines of the square lattice. Along the first line the pattern has no operation that reverses orientation: it is chiral, although every piece of it is achiral. Along the second and third the motif's mirror is also a mirror of the whole pattern, and the detected groups are p4m and p4g; the mirror lines of each pattern are drawn. All three verdicts come from detecting the symmetry of the points and agree with whether the motif's mirror normalises p4.

A hand made of pieces that have none

Quartz is built from tetrahedra that have no handedness, and every quartz crystal is left-handed or right-handed anyway. Put a piece with a mirror into a pattern whose group has none, and the pattern keeps the piece's mirror only if that mirror lies on one of a few lines the group's normaliser draws. Anywhere else, the arrangement has a hand its parts do not.

space-groups · Chirality Seventeen plane groups, and one chiral sheet over each. The seventeen plane groups, whether each is chiral as a pattern in the plane, how many sheets can be built over it by giving each operation a sign on the sheet's normal — 63 in all — and which of those sheets is chiral in space. There is always exactly one. For the five groups chiral in the plane it is the sheet whose two faces differ and nothing turns it over. For the twelve achiral in the plane it is the sheet turned over by exactly the operations that reverse orientation in the plane, so that every mirror line becomes a half-turn axis lying in the sheet.

Chiral in the plane is not chiral in the room

A pattern with mirrors all over it can be a sheet with a hand, and a pattern with no mirror can be a sheet without one. Whether a layer is chiral depends on what each of its operations does to the side of the sheet, and over every one of the seventeen plane groups exactly one sheet is chiral in space.

space-groups · Chirality Whether a rolled sheet ever comes back round. Three plane lattices, each with the same rolling vector C = 3a₁ + a₂ drawn from the origin and the line through the origin perpendicular to it. A translation of the rolled pattern straight up the tube, with no turn, is a lattice vector on that line. The square lattice has one, marked T, and the tube repeats every 10 turns. The general rectangular lattice has none in this direction — only along its cell edges — and the general oblique lattice has none in any direction at all, so its rolled pattern climbs forever without returning to the same angle.

Most sheets roll into a tube that never repeats

Rolling the honeycomb along a lattice vector always gives a tube with a repeat, and that is a property of the honeycomb rather than of rolling. Over the seventeen plane groups, 567 of 1,008 rolling directions give a tube with no translation along its axis at all — and every direction of an oblique pattern is one of them.

restriction · Finite groups Free going down, two conditions going up. The edge between p2 and p4 in the diagram of maximal translationengleiche relations, read in both directions. Downwards it costs nothing: the quarter-turns are discarded and the lattice is exactly the lattice that was there, so every p4 pattern contains a p2 pattern. Upwards the added quarter-turns must carry the lattice onto itself, which forces the cell to have equal edges at a right angle — two conditions on a general oblique cell, which has only two parameters to give. So a p2 structure has a p4 supergroup exactly when its measured cell happens to be square, and the question is about the metric rather than about the group.

Going up costs the cell a parameter

The usual asymmetry — finitely many maximal subgroups below, infinitely many minimal supergroups above — is false in both halves for a plane group. Both directions are infinite and equinumerous index by index. The real asymmetry is that 17 of the 31 edges cost the lattice a parameter going up and nothing going down.

operations · Subgroups Two symmetric structures a diffraction pattern cannot separate. Two arrangements of 6 atoms on a 6 × 6 torus, each invariant under the plane group p6m, drawn beside the Patterson they share. No translation and no inversion carries one onto the other, so they are different structures; every one of the thirty-six interatomic vector counts is the same, so every diffracted intensity is the same and no measurement at any resolution separates them. Of the 4 structures with this symmetry and this many atoms, there are only 3 Pattersons — so imposing the most symmetric of the seventeen plane groups has not removed the ambiguity.

Symmetry does not rescue a Patterson

Every homometric pair found so far sits on a bare ring with no operations imposed, and a real crystal sits in a space group. Impose one and the ambiguity does not go away: 12 of the 13 groups searched still have pairs, and at six atoms the hexagonal groups are indistinguishable two to three times as often as the general position.

diffraction · Homometry Fifty-four of the seventy-five rod groups are a rolled plane pattern. Every rod group, one dot each, grouped by its crystal class. A dot is filled when some plane pattern rolled along some lattice vector has exactly that group, and the 106 crystallographic rollings of the seventeen plane groups fill 54 of them. The eighteen improper classes are full: every one of the 43 achiral rod groups is reached. The nine proper classes are not, and the twenty-one groups named on the right are what is missing — the sixteen whose screw is one of a left- and right-handed pair, and the five bare axes with no climb at all, p1, p112, p3, p4 and p6.

A rolled sheet is never one of a pair

Rolled up along every lattice vector that gives a crystallographic tube, the seventeen plane groups reach fifty-four of the seventy-five rod groups: every achiral one and eleven of the chiral. Not one of the sixteen screws that come in left- and right-handed pairs is among them, and the reason is a single fact about how far a rolled lattice can climb.

restriction · Finite groups The closure is a lattice exactly when the orders allow one. Twelve pairs of rotation orders, with a centre of each order placed one unit apart and the group they generate closed out to words of length 6. The linear parts reached are exactly the least common multiple of the two orders, every time — two rotations generate rotations, and the angles they generate are the multiples of the smaller of two fractions of a turn. A lattice admits rotations of order one, two, three, four and six and no others, so the closure can be a plane group exactly when that multiple is one of those five. The pairs where it is not are the pairs where the translations keep getting shorter.

Closing the plane from two centres

Put two rotation centres down and close under composition: the result is a plane group or is not discrete, and nothing in between. What decides it is the least common multiple of the two orders, because two rotations generate rotations and the angles add — so the crystallographic restriction arrives as a condition on a closure rather than as one on a lattice.

operations · Composition The turns that keep the join discrete. Two copies of p4 on one square lattice, one turned against the other, with the shortest translation their union generates. At a turn whose cosine and sine are both rational the translations are a lattice, and its shortest vector is one over the square root of Σ — where Σ is the odd part of p² + q² for the rational point (p, q) — which the measurement reproduces to six places at every one tried. At a whole number of degrees other than a multiple of ninety there is no such point, and the search finds shorter translations the further it runs.

Two patterns laid over one another

Compose two plane groups rather than two operations. The group generated by both is one of the seventeen or is not discrete at all, with nothing between — and it takes two conditions, one on the rotation orders and one on the turn between the lattices. The turns that work have rational cosines, which by a theorem of Niven's means none of them is a whole number of degrees.

operations · Composition

Named alongside it

The objects these essays reach for when they reach for this one.

EnumerationChiralityCrystallographic restrictionOrbitCompositionGlide reflectionHalf-turnIndexQuotientRotation centreSubgroupClosure

All concepts