Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Exactly this many — page 4

Seven friezes, seventeen wallpapers, fourteen lattices, two hundred and thirty space groups. Each number is a theorem with a finite proof, and the word that matters in every one of them is 'exactly'.
Every property count, in every class. For each of these 32 classes, how many independent components a property may have: elastic constants from 21 down to 3, piezoelectric moduli from 18 down to 0, dielectric tensor from 6 down to 1, pyroelectric vector from 3 down to 0, gyration tensor from 6 down to 0. Every number is a character averaged over the class's own operations, computed in the lattice basis with integer arithmetic. A zero means the symmetry forbids the property outright; a positive number means it does not forbid it, which is a weaker statement than it is usually read as. What symmetry decides

A filter of great precision and no predictive power

The whole table in one place — thirty-two classes, six properties, 192 exact integers. What it settles, what it merely permits, and why knowing which of the two is happening at any moment is the entire skill of using it.

 One hundred and twenty-two magnetic point groups. The three kinds, counted. Thirty-two ordinary groups, which contain no primed operation; thirty-two grey groups, which contain time reversal on its own and are the symmetry of anything magnetically disordered; and fifty-eight black-and-white groups, one for each way of splitting a class into a subgroup of index two and its complement. The last number is the one that has to be computed: the index-two subgroups are found by closure inside each class, reduced up to conjugacy, and reduced once more by an equivalence that needs a rotation no lattice may have. 32 + 32 + 58 = 122, and every term is a measurement. What symmetry decides

The operation that reverses time

A magnetic moment is a current loop, so running time backwards reverses it and moves nothing. Admitting that as a symmetry operation turns the thirty-two crystal classes into a hundred and twenty-two — and eight of the merges needed to reach that number require a rotation no lattice may have.

 One layer, and the two ways to sit on it. A close-packed layer — large pale discs, each touching six others, which is as tight as one layer of equal spheres can be. Its hollows come in two sets, marked in the two smaller colours, and a second layer must take one set or the other; the two choices are mirror images and equally good. The third layer then faces the same choice again, and this time the two answers are genuinely different: over the first layer, or over the hollows the second did not use. That single decision, repeated, is the whole of close packing — and nothing in the geometry prefers either answer, because both give the same density and the same number of touching neighbours. Symmetry at work

Two stackings, one density

Stack spheres as tightly as they will go and the third layer has a free choice. Both answers fill exactly the same fraction of space and give every sphere the same twelve neighbours — and their space groups are Fm3̅m and P6₃/mmc, which is the only thing that tells them apart.

 The classes that permit a spontaneous magnetisation. Every magnetic point group permitting a spontaneous magnetisation — 31 of the 122 — with the number of independent components each allows. an axial vector, reversed by time reversal — a ferromagnet has one and nothing else does. The count comes from averaging the character over the group, with a primed operation's contribution multiplied by −1 because the property reverses when time does. That is Neumann's principle with one extra sign in it, and it reproduces the numbers the literature records without being given them. What symmetry decides

Which magnetism a class permits

Neumann's principle with one extra sign in it decides which of the hundred and twenty-two magnetic classes may have a spontaneous magnetisation and which may show the magnetoelectric effect. The answers are thirty-one and fifty-eight, and they come out of the same average that counts elastic constants.

 How many close packings there are of each period. Every cyclic sequence over three letters with no two adjacent alike is a close packing, and two sequences describe the same structure when one becomes the other by rotating the cycle, reversing it, or relabelling the three positions. Counting the classes that remain gives 1 of period 2, 1 of period 3, 1 of period 4, 1 of period 5, and 38 altogether up to period 10. Period two is hexagonal close packing and period three is cubic; everything above them is a polytype, equally dense and equally close packed, and silicon carbide has been found in more than two hundred of them. Nothing in the geometry chooses. What chooses is an energy difference of a few thousandths of an electron volt per atom, and this site computes no energies. Symmetry at work

How many polytypes there are

One free choice per layer, repeated, gives a family of structures with the same composition, the same density and the same twelve neighbours — differing only in a sequence. Counting them up to rotation, reversal and relabelling turns "silicon carbide has hundreds of forms" into an enumeration.

 The thirty-two, in both notations. Each class with the symbol crystallography uses and the symbol spectroscopy uses, both derived from the class's own matrices. The Hermann–Mauguin symbol is a report on three families of directions, read in an order the lattice system fixes. The Schoenflies symbol is a report on a construction: a principal axis of order n, whether there are n twofold axes across it, and which mirrors were added. Neither can be computed from the other without going back to the group, which is why the two lists are set beside each other rather than one derived from the other. What symmetry decides

One class, two names

Hermann–Mauguin names directions and Schoenflies names a construction, and the two are derived here from the same integer matrices by computations that share no step. Neither can be obtained from the other without going back to the group — which is why a molecule has one kind of symbol and a crystal has both.

 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. Operations

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.

 What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order. Operations

Counting what a group cannot tell apart

Sixty-five thousand ways of putting two species on sixteen sites; eight hundred and five structures. The difference between those numbers is not a division, because the symmetric arrangements have short orbits — and the count that gets it right is an average of fixed points.

 The first 4 zones of the square lattice. Zones one to 4, each in its own shade. The n-th zone is the set of wavevectors with exactly n − 1 reciprocal lattice points nearer to them than the origin is, so the boundaries are the perpendicular bisectors and nothing else. The zones get further out and break into more pieces — 1, 4, 8, 12 fragments — and every one of them has the area of a single cell. Lattices

The zones above the first

The second Brillouin zone is a scattering of disconnected fragments in a different part of reciprocal space from the first, and it has exactly the same area. So does the third, and the seventh. The reason is that each of them is the first zone, cut up and moved.

 Six integers that do not depend on the description. The same monoclinic lattice written in 4 different bases, each obtained from the last by an integer matrix of determinant one, and each reduced by Niggli's algorithm. Every one of them gives the same six integers — the squared lengths and twice the dot products of the reduced basis. That is what makes the reduced form a fingerprint of the lattice: two cells with no number in common are the same lattice exactly when their reduced forms agree, and the comparison has no tolerance in it. Lattices

The cell that settles the argument

Two determinations of one compound can report cells that share no number and describe the same lattice. Reduction is the procedure that decides — six integers that depend on the lattice and not on anybody's choice of axes, and that agree exactly when the lattices do.

 How much a count of descriptions over-counts. For each plane group that has any two-colouring at all: how many colourings it has, how many designs those come to, and the ratio between them. Over the seventeen the ratio is 1.61, and group by group it runs from 1.00 — where nothing is identified — to 3.50 at p2, whose seven colourings fall into one class of six and one of one. The tick on each row is that row's largest single class, and it is at least the bar and usually more. The largest class anywhere is p2's 6, and that same group over-counts by only 3.50, because a factor is a mean over the group's classes and a mean reaches its largest term only when every term equals it. Reading the largest class as the over-count is therefore an over-statement, always. And the factor varies from group to group, which is why no single correction turns a count of descriptions into a count of designs after the fact. The classification

Seventy-four colourings, forty-six groups

This site counts the two-colourings of the seventeen and gets seventy-four. The literature says there are forty-six two-colour wallpaper groups. Both numbers are right, and the gap between them is a disagreement about when two coloured patterns are the same pattern.

 The (2, 3, 7) group, in the Poincaré disk. A triangle with angles π/2, π/3 and π/7, reflected in its own three sides until depth 12: 380 triangles, alternating in handedness because every generator is a reflection. The sum 1/2 + 1/3 + 1/7 is less than one, so the triangle does not fit in the flat plane and the drawing is of the hyperbolic one, with the whole plane squeezed inside a disk. Every triangle has the same hyperbolic area; the ones near the edge look small because the model shrinks distances there, and the tiling stops at the edge of the drawing rather than at the edge of anything. The classification

Past two, the list does not stop

Conway's accounting says a wallpaper group costs exactly two dollars, and there are seventeen ways to spend it. Spend less and the answer is a finite group. Spend more and the list is infinite — but the cheapest thing past two costs two and one eighty-fourth, and nothing at all lies in between.

 Thirteen ways to hold a lattice. Every finite group of integer matrices in two dimensions, up to a change of integer basis: 13 of them. Ten different abstract groups appear, and three of the ten hold a lattice in two inequivalent ways — a mirror along an axis or along a diagonal, and the same for 2mm and for 3m. The enumeration is a search: every subgroup of the two maximal holohedries, merged by conjugacy under integer matrices of determinant ±1, with the answer checked for not depending on how wide the search was. What a lattice forbids

Thirteen ways to hold a lattice

The crystallographic restriction is about one matrix. A crystal has a whole group of them acting on one lattice at once, and asking how many such groups there are gives thirteen — not the ten of the plane point groups, and not the seventeen of the plane groups.

 The most of an icosahedron a crystal can keep. Every subgroup of the sixty rotations of an icosahedron, found by closure, with the crystallographic ones marked — those whose rotation orders are all among the 1, 2, 3, 4 and 6 that a three-dimensional lattice admits. The largest is 23, of order 12, at index 5; everything containing a fivefold axis is refused. So a crystal containing an icosahedral molecule may fix a twelfth of the molecule's own symmetry and no more, and the remaining 5 orientations have to be related by something other than the site's symmetry. What a lattice forbids

The most of an icosahedron a crystal can keep

C₆₀ sits in crystals and virus capsids sit in crystals, and neither of them stops being icosahedral. What a lattice can fix is a subgroup — and the largest crystallographic subgroup of the sixty rotations has order twelve, at index five. The five are Kepler's five cubes.

 N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does. How it is known

Whether there is a centre is a statistic

Everything else on this site is decidable: a pattern has a symmetry or it does not, and the detector settles it in integers. Whether a structure has an inversion centre is not like that. No single reflection carries the answer — the distribution of all of them does.

 Three shapes, and nothing else. The dielectric tensor of a crystal is an ellipsoid, and averaging a generic one over a point group leaves exactly three possibilities: a sphere, where all three principal values agree and the crystal is optically isotropic; a spheroid, where two agree and there is one optic axis; and a general ellipsoid, with two. The counts are 5, 19 and 8 of the thirty-two classes, and they were found by computing the eigenvalues rather than by sorting the classes by system. What symmetry decides

Three optical characters, and the arithmetic that assigns them

A cubic crystal cannot be birefringent, whatever it is made of. Between crossed polars it stays dark at every rotation, and the reason is that averaging any ellipsoid over a cubic point group leaves a sphere — a permission computed before anybody measures anything.

 p4g in 4 letters and 8 relations. The presentation of p4g, derived from the group's own operations. The two translations commute; each conjugation relation is read off a column of a matrix; and the point group's relations are corrected by the translation they actually come back as, which is what makes this group an extension rather than a semidirect product. Every relator is evaluated where the group lives and must be the identity, and coset enumeration on the letters alone returns 8, which is the order of the point group. Operations

A group in four letters

Every other essay here describes a symmetry group by what it does to the plane. There is a second description — a handful of letters and the words in them that are required to equal nothing — and it can be counted with no plane anywhere in the computation.

 p3m1 and p31m, told apart without a picture. The two groups this site returns to most often: same point group, same lattice, same number of operations, and distinguished in every other essay here by where their mirrors sit relative to the lattice — which is a fact about the plane. Abelianised, they are ℤ2 and ℤ6, which are not isomorphic. That difference is a fact about the groups: no change of basis, no redrawing and no relabelling can carry one to the other, and the argument never mentions a mirror line. Operations

What is left when the order is forgotten

Abelianising a group throws away the order of the letters in every word and leaves a small abelian group behind. It is computed by a Smith normal form, it never mentions the plane, and it separates p3m1 from p31m — which a picture can only illustrate.

 Every plane group from at most 4 operations. For each group, the fewest operations that generate the whole of it — the point operations and both lattice translations, since a group that does not reach its own translations is a different group. The floor is the abelianisation's number of invariant factors, which no group can beat, and the search is exhaustive over the operations within one cell of the origin. 14 of the seventeen meet their floor, which settles those exactly; the other 3 need more than the abelian argument can see, and p3m1 needs three where its abelianisation is cyclic. Operations

How few operations make a pattern

A plane group is infinite, and a handful of its operations is enough to rebuild all of it. How small a handful is a question with a floor from the abelianisation and a ceiling from an exhaustive search, and for fourteen of the seventeen the two numbers meet.

 Five shapes, and a lattice in space has no other. The five combinatorial types a Wigner–Seitz cell can have in three dimensions — cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, truncated octahedron — each drawn from a lattice that produces it. Fedorov proved in 1885 that there are no others, and that fourteen faces is the most any of them has, which is Minkowski's bound of 2(2ⁿ − 1) in three dimensions. Each solid here is cut out by the perpendicular bisectors of nearby lattice vectors and its volume checked against the primitive cell's, which is what catches a face that failed to appear. Lattices

Five parallelohedra, and no others

The cell that needs no basis and no convention has, in three dimensions, exactly five shapes. The fourteen Bravais lattices produce all five between them — and which one a lattice gives is not decided by which of the fourteen it is.

 Subgroups of index two, three and four. Every plane group with the number of subgroups it has at each small index, counted by enumerating the transitive actions on that many points. The zeros are the interesting entries: p3 has no subgroup of index two and the four-fold groups have none of index three, because a subgroup of index n gives an action on n points and the group has to have a quotient that can act. A rotation of order three has nowhere to go in a set of two, and one of order four has nowhere to go in a set of three that is not the identity — so the index is constrained by the point group before any geometry is done. Operations

How many subgroups of index three

Taking operations away and closing what is left finds the maximal subgroups and stops there. Counting instead the ways a group can act on three points finds all of them — and finds that a four-fold group has none of index three at all.

 Every group decided by a window of radius 1. For each of the seventeen, the radius at which a round window on the pattern admits exactly the group's own operations and no others — with the numbers it admits at each smaller radius beside it. Two opposite failures are visible. Most groups under-report at small radii, because an operation carrying points out of the window cannot be tested at all; cm over-reports, admitting operations the pattern does not have. The groups that take longest to settle are the ones distinguished by a glide, which moves a point half a cell before anything can be compared. The classification

How much pattern is enough

Every claim here about a pattern's group is a claim about an infinite pattern. A reader sees a patch. Measuring what a finite window can decide gives a number — about one cell's radius — and two opposite ways of being wrong on the way there.

 The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry. Lattices

How many vectors of each length

Counting the lattice points at each distance from the origin turns out to be a question about divisors, and the answer explains something a crystallographer meets every day: why a cubic powder pattern has no line at seven.

 The ball of radius 5 in p6. Every element of p6 reachable in at most 5 multiplications by a generator or its inverse, plotted at its translation part — so each dot is a lattice position and its size says how few steps reach it. The picture is the word metric's unit ball scaled up, and its shape is what fixes the growth: a diamond where the group supplies two short translations, and a hexagon where it supplies three. Every dot here required the word problem to be solved, because the search has to know when two products are the same element. Operations

Telling two words apart

There are finitely presented groups in which no algorithm can decide whether two products of the generators are the same element. The seventeen are not among them, and the procedure that settles it is short enough to state in a sentence — which then makes it possible to measure how fast each group grows.

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