Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Symmetry is decidable — page 7

Unusually for a physical science, the central questions here have exact answers computable in integer arithmetic. There is no tolerance to choose and no residual to interpret.
p4: a domain of 38 cells with 7 walls. The fundamental domain of p4 on a grid of 12ths, with the walls it shares with its neighbouring copies marked. Each wall names the element that carries this copy onto the copy across it, and there are 7 distinct such elements. Those elements generate the whole group — checked by closing them up and requiring every coset and the whole translation lattice to be reached, not assumed — which is Poincaré's theorem, and it means the generators of a wallpaper group can be read off a picture. The domain is pixelated rather than a polygon, so the wall count is a property of this domain and not of the group. Operations

Every wall names a generator

The copies of a fundamental domain tile the plane and stand in one-to-one correspondence with the elements of the group. So the elements that carry the home copy across a wall generate everything — and the generators of a wallpaper group can be read off a picture rather than looked up.

 75 rod groups over 27 axial classes. Each axial crystal class, with the number of rod groups it carries: every consistent choice of translation along the axis, in every way the class can sit on the rod, with two groups counted as one when a shift of the origin along the rod or a turn about it carries one onto the other. The total is 75, and every row as well as the total agrees with the International Tables, which are compared with this enumeration rather than used to produce it. The classification

Seventy-five ways to be a thread

Eighty layer groups and seventy-five rod groups are usually quoted, both as numbers from the literature. The second is derived here, class by class — and the total alone turned out to be no check at all, because two errors of six groups each give seventy-five as well.

 (17, 5) and (23, 7) reduced in 3 steps. Lagrange's reduction, run on the basis (17, 5), (23, 7). Each step subtracts a whole multiple of the shorter vector from the longer and swaps them; after 3 steps neither can be shortened by the other and the pair is reduced. The faint arrows are the intermediate bases and the solid pair is the answer, of length 1.41. The procedure always terminates and always finds the shortest vector, and in the plane that is a theorem rather than a hope. Lattices

The shortest vector, and where it stops being easy

Two moves find the shortest vector of a plane lattice, and they always terminate. Nothing on this site has ever needed more, because every lattice here has two or three dimensions. In general the same question is NP-hard, the best polynomial procedure returns an answer that may be exponentially too long, and an entire branch of cryptography is built on the gap.

 625 tiles, 32 directions. The subdivision applied 4 times to one right triangle with legs 1 and 2, giving 625 tiles of one shape and size. They point in 32 distinct directions — the tint follows the direction — and the count grows every time the rule is applied, without bound. Order without repetition

The tiling that points every way

A Penrose tiling never repeats and its tiles still point in only ten directions, which is why its diffraction pattern has ten-fold symmetry. One triangle, cut into five copies of itself, breaks that — and the difference between it and a tiling with eight directions is which diagonal of one small rectangle gets drawn.

 p4: the map comes back. p4 written on two bases related by an integer matrix of determinant one, and about two origins. The two descriptions share no coordinate; they are the same group. The matrix and the origin shift were then recovered from the two operation sets alone — which is what Bieberbach's theorem promises, carried out as a search over the integer matrices and the origins the lattice permits, and checked by applying what was found. What a lattice forbids

The same group means the same pattern

Seventeen patterns is not the same statement as seventeen groups. Two patterns that look nothing alike could in principle have symmetry groups that are abstractly the same, and then the classification would be a classification of drawings. Bieberbach's theorem says they cannot — and the affine map that proves it can be recovered from the two operation sets alone.

 5 units of 70.53°: 7.36° left. 5 tetrahedral units of face-centred cubic metal, each the mirror image of its neighbour in a {111} plane, arranged about a common ⟨110⟩ edge. The angle between two such planes is arccos(1/3) = 70.53°, computed from the plane normals rather than quoted, and 5 of them come to 352.64°. The shaded sector is what is left over: 7.36°, or 2.04 per cent of a full turn, which must be taken up by strain, by a gap, or by a defect along the axis. What a lattice forbids

Five copies, and the gap they leave

Gold, silver and silicon grow particles with a five-fold axis down the middle, out of a lattice that forbids one. Nothing is violated: five tetrahedral pieces of ordinary face-centred metal, each the mirror image of its neighbour, come to three hundred and fifty-two and a half degrees rather than three hundred and sixty — and the seven degrees left over have to go somewhere.

 a lattice triangle: 1 inside, 6 on the edge, area 3. a lattice triangle on its lattice, with the 1 points strictly inside it in the first colour and the 6 points on its boundary in the measured colour. Pick's theorem says the area is the interior count plus half the boundary count less one, which is 1 + 6/2 − 1 = 3; the shoelace formula on the same integer coordinates gives twice the area as 6. The two agree, and both sides are integers, so the check has no tolerance in it. The theorem holds for a non-convex polygon and a polygon with no interior point alike, neither of which the usual triangle-and-square picture makes obvious. Lattices

How many points a shape holds

Draw a polygon on a lattice, count the points inside, then double the polygon and count again. The counts are not approximately a polynomial in the scale — they are one, exactly, with the area as its leading coefficient and a constant term of one for every polygon there is.

 18 extension classes, 17 groups. Each of the thirteen arithmetic classes with the number of ways translations may be attached to it — its cohomology — the shape of that group, and how many distinct plane groups the classes come to once the changes of basis that are mere relabellings are quotiented out. The two columns differ in exactly one row, 2mmp, where four extension classes are three groups because two of them are the same group with the axes swapped. No lattice is drawn anywhere in this computation. The classification

Seventeen, without a picture

Every other count of the plane groups has a plane in it — a pattern generated, a domain folded, an orbifold's curvature spent. The same seventeen come out of pure algebra: attach translations to a point group, keep the assignments that close, throw away the ones that differ only by where the origin was put, and add up over the thirteen arithmetic classes.

 p4m: 10 of 36 wavevectors have to be visited. The Brillouin zone of the square lattice with a grid of 36 wavevectors on it, of which 10 are drawn solid: one per star, which is everything a calculation over a p4m-symmetric operator has to visit. The share is 27.8 per cent against the 12.5 per cent that the order of the point group would give, and it is larger for a reason worth naming — the wavevectors on the boundary of the wedge have short stars, so they are over-counted by any argument that only divides by the group order. The identity that is checked is that the star sizes add to the whole grid. Operations

The domain in reciprocal space

A fundamental domain is the piece of a pattern the group repeats, and this collection has drawn several. The same idea in reciprocal space is what makes a calculation over a crystal affordable — and its share of the zone is larger than one part in the group's order, for a reason worth measuring.

 P2₁/c from 27 marks. The marks of P2₁/c's plan, counted by kind, and what they rebuild to. Each mark is reduced to what a reader can see and handed to a closure with the matrices withheld: an axis gives its direction, its position and how far one turn advances along it; a plane gives its normal, its position and its slide. The lattice supplies the candidate matrices, the closure supplies the rest, and what comes back is the group — 4 operations against 4, with nothing missing and nothing extra. Into space

The plan contains the group

A space-group diagram has always been treated here as a picture of the group. It is more than that: hand back the marks alone — no matrices, no operations, not even the centring — and the group comes out exactly, forty-five times out of forty-five.

 Two structures on 8 sites with the same vectors. Two arrangements of 4 atoms on a ring of 8 positions. They are not the same arrangement — no rotation of the ring and no reflection carries one onto the other — and every interatomic vector occurs the same number of times in both. The bars below are the shared vector counts, which is the Patterson function of each: the tall one at the origin is the atom count and carries no information, and everything else is what a diffraction experiment measures. Their diffraction patterns are identical in every intensity, so no measurement of intensities, at any resolution, distinguishes them. How it is known

Two structures, one Patterson

Eight arrangements of four atoms on a ring of eight sites, and only seven distinct sets of interatomic vectors between them. Two of the arrangements are genuinely different and no measurement of intensities can tell them apart — at any resolution, for ever.

 Dropping one invariant of 3m makes two orbits agree. Every lattice point within four cells of the origin, coloured by the values a proper subset of 3m's invariants takes on it — the 2 generators with the first one removed, over a window of 4 cells. With the full set, the 25 orbits of the group take 25 distinct sets of values, one each, so the invariants are a complete set of coordinates on the quotient. With one removed, the two circled points — in different orbits, so no operation of the group carries one to the other — take the same values and become indistinguishable. That is the whole content of the statement that a complete set of invariants separates orbits: the completeness is what is doing the work. Operations

An orbit is what the invariants cannot tell apart

Two points of the plane lie in the same orbit of a group exactly when every invariant polynomial takes the same value on both. One direction of that is a definition; the other is a theorem, and it is checked here by comparing every pair of points in a window both ways.

 15 classes may rotate light, 11 of them chiral. A crystal is chiral when its point group contains no improper operation, and there are 11 such classes. A crystal may rotate the plane of polarisation when its class permits a non-zero gyration tensor, and there are 15. The four in the difference — 4̅, m, 4̅2m, mm2 — are achiral and may still rotate light, which is why the two words are not synonyms. In each of the four, symmetry forces the tensor to be traceless, so the rotation changes sign with direction and cancels in any average over directions. What symmetry decides

Fifteen may rotate light, and eleven are chiral

Optical rotation and handedness are treated as the same thing and are not. Eleven crystal classes are chiral; fifteen permit a crystal to rotate the plane of polarisation; and the four in between are the reason quartz and sodium chlorate are the examples everybody uses.

 8 tiles over 5 colours. Wang tiles: unit squares with a colour on each edge, which may be laid side by side only where the touching edges agree, and which may never be turned or reflected. That last restriction is what makes them a computational object rather than a jigsaw — an edge colour is a symbol passed from one tile to its neighbour, and turning a tile would let a symbol change direction. The set here was generated from a stated seed. The classification

Nothing decides whether a set of tiles tiles the plane

This collection rests on decidability — generate a pattern, forget the group, rediscover it, compare. One question in the same subject has no procedure at all: given a finite set of tiles, whether they cover the plane cannot be decided by any algorithm whatever. What can be done is two half-searches, and measuring what they leave behind.

 orders 5 and 7 reach a site of symmetry 1 and no more. A molecule whose only symmetry is one n-fold axis, and the highest site symmetry it may occupy in any of the 45 space groups this site builds. The site's symmetry has to be a subgroup of the molecule's, so the site's order must divide n and the site group must be cyclic. Orders 1, 2, 3, 4 and 6 reach a site of their own order. Orders 5 and 7 reach one, because no site symmetry in any space group contains an operation of order five or seven — the orders available are 1, 2, 3, 4, 6, computed by asking every operation of every group whether it moves a point. A five-fold molecule keeps its axis; the crystal simply has no use for it. What a lattice forbids

What a molecule gives up to sit in a crystal

A molecule brings its own symmetry. A crystal offers sites with symmetries of their own, and the two have to be compatible — the site's symmetry must be a subgroup of the molecule's. So a molecule may always keep more than its site offers, and a molecule with a five-fold axis may sit only where the crystal offers nothing at all.

 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. Into space

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.

 Sums of two squares, arriving as superstructures. Which indices admit a sublattice of the same shape as the square lattice, drawn as a bar per index whose height is how many there are. The pattern is not a pattern about lattices at all: an index works exactly when it is a sum of two squares, because a similar sublattice of the square lattice is multiplication by a Gaussian integer and its index is that integer's norm. The indices that work up to 30 are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, and the same list is produced here a second time by factorising rather than by searching, with the two required to agree. Lattices

The sublattices that are the same shape

Thinning a lattice usually changes its shape. Sometimes it does not: the sublattice is the parent rotated and scaled, and a drawing of it alone would be a drawing of the parent. Which indices allow it turns out to be a question Fermat answered in 1640.

 p4m: which distortions the atoms of each position can make. Every Wyckoff position of p4m, with the number of independent displacement patterns of each symmetry its atoms supply. A zero is a distortion those atoms cannot make however their amplitudes are chosen — an atom pinned at a rotation centre cannot move in a way that keeps less symmetry than the centre has. The row sums, weighted by the dimensions along the top, come to twice the number of atoms in the cell, which is the check that nothing has been lost. The last column is the number of free coordinates the position has, and it equals the multiplicity of the identity representation in that row: a displacement that keeps every symmetry is exactly a move of the position within its own Wyckoff set. Operations

Which modes a site can carry

An atom on a mirror cannot move in a way that breaks the mirror while its images move with it: the displacements of a Wyckoff orbit carry a representation, and some of its pieces have multiplicity zero. The count of those pieces is a character, and the one that breaks nothing is the position's own freedom.

 Y-pentomino: A B C D E F, with 6 arcs. The boundary of the Y-pentomino cut into six arcs. A runs from one corner to another and D is the same arc traversed backwards, so D is a translate of A and the translation is (3, 1) cells. Each of B, C, E and F is carried onto itself by the half turn about its own midpoint, and those midpoints are the four marked dots — That is Conway's criterion, and a shape meeting it tiles the plane by translations and half turns. The classification

A tiling of the whole plane, decided on one tile's edge

Whether a shape tiles the plane is a question about an infinite object, and there is no procedure that answers it. There is a procedure that answers it *sometimes*, and it reads nothing but the shape's own boundary — a closed path of a few dozen steps, cut into six arcs. When the cut exists the tiling exists, and the cut names the group that makes it.

 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. How it is known

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.

 Ten ways for space to be flat. The thirteen groups, with each mirror-image pair counted once, because a shape and its mirror image are the same shape. 3 of the ten arrive that way — the three-fold, four-fold and six-fold screws, which are the enantiomorphic pairs this collection already counts among the two hundred and thirty. Six of the ten are orientable and four are one-sided. Into space

Ten ways for space to be flat

Thirteen of the two hundred and thirty space groups hold no point still, and folding space along one of them gives a shape with no curvature anywhere. There are ten such shapes, not thirteen, and the difference is the same eleven pairs that separate 230 from 219.

 Arrangements per site, falling towards the exact value. The number of configurations of an L × L torus obeying the ice rule, taken to the power of one over the number of sites. The largest computed here is 4,484,823,396 configurations on a 7 × 7 torus. The values fall towards Lieb's exact 1.54 from above, and every one of them is above Pauling's estimate of 1.5 — which undercounts, because it treats the vertices as independent and they are not. Order without repetition

The arrangements a crystal keeps at absolute zero

Ice has a residual entropy, and the number a calorimeter measures is the logarithm of a count of arrangements. Pauling's one-line estimate of that count is out by two and a half per cent; the exact count in two dimensions is available, falls towards its limit from above, and the whole disorder is invisible to a diffraction experiment, which sees only the average.

 Equilibrium and growth are different shapes. Two predictions for the habit of the same cubic crystal, computed through the same intersection of half-spaces. The equilibrium shape puts each face at a distance proportional to its surface energy, which is Wulff's construction; the growth shape puts it at a distance proportional to its growth rate, taken here from this site's own spacing rule. They differ — the equilibrium shape carries {111}, {110}, {100} and the growth shape {100} — and the difference is between two rules rather than between two pieces of code. A crystal on a bench has grown; a crystal annealed long enough has relaxed; the two look different and neither picture is wrong. Symmetry at work

The fast faces are the ones that vanish

A crystal has two predicted shapes and they are not the same. One minimises surface energy and is what a crystal settles into; the other is what growth leaves behind, and in it a face that grows quickly grows itself out of existence.

 Rotation orders 1, 2, 3, 4, 6 and no others. Every net in this collection, with the orders of the rotations its own symmetry group has, and the degrees of its vertices beside them. The orders are 1, 2, 3, 4, 6 — the crystallographic restriction, arrived at with no length anywhere in the argument: the translations of a net are ℤ² by construction, an automorphism carries translations to translations, so it acts on ℤ² by an integer matrix, and an integer trace in the interval from minus two to two is one of five numbers. The degree column is there because the two are constantly confused: a net may perfectly well have vertices of degree five, and one here does. What a lattice forbids

The restriction, with no lattice assumed

The proof that only two-, three-, four- and six-fold rotations are possible is usually stated about a lattice, and every step of it turns out to need no lengths at all. A periodic graph has the same theorem, proved the same way — and a graph may have a five-fold symmetry the plane cannot receive.

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