Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Exactly this many — page 3

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, dielectric tensor from 6 down to 1. 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

Twenty-one, thirteen, nine, three

The number of independent elastic constants runs 21, 13, 9, 7, 6, 5, 3 down the crystal systems. Two of those systems carry two numbers rather than one, and which classes take which is not predicted by counting operations — a class with six of them can have more constants than a class with six of them.

 4 lattices. 4 lattices: cubic P, with 48 symmetries; cubic C, with 16 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything. Lattices

Forty-eight becomes sixteen

Centre one face of a cube and the four threefold axes along its body diagonals are gone. That sentence is usually offered as a fact to accept; it is a computation whose answer is a number, and the number says which lattice you got instead.

 The subgroups of p4m of index 2. p4m has 7 subgroup(s) of index 2 with cyclic quotient. 3 of them keep every translation and lose operations — the lattice is untouched and the pattern loses a symmetry at every point. 4 keep every operation and lose translations, and each is named beside the basis of the sublattice it keeps, written in the parent's own axes. Each subgroup is the kernel of a homomorphism onto a cyclic group, found by enumeration; each name is found by searching changes of basis and origin until the operation sets match exactly. Operations

Two ways down from a group

A pattern can lose a symmetry by giving up an operation or by giving up a translation, and the two are different in kind. Sorting the seventy-four subgroups of index two among the seventeen splits them twenty-nine to forty-five — and a containment test that compares operations modulo one shared lattice can only see the twenty-nine.

 The seventeen, arranged by what they can lose. Each group at the height of its own order, joined to every maximal subgroup that keeps all of its translations. Reading downwards is a crystal losing operations at a phase transition. The edges are the maximal ones only — every other containment is a path through these — and the whole graph is enumerated by closing every subset of each group's operations, so nothing is here because a table said so. Operations

The descent with no shortcut

A subgroup can give up operations, or it can give up translations. Hermann's theorem says that a *maximal* subgroup does one or the other and never both at once — which is why a crystal losing symmetry can be followed one clean step at a time, and why every route from p6m down to p1 has exactly three steps.

 Friedel's law, as an equality rather than a resemblance. Each pair of bars is a reflection and its opposite for a structure of three atoms in no particular arrangement. They are the same height, and not approximately: with real scattering factors, negating the indices conjugates the structure factor, and conjugation does not change a modulus. The test behind this figure requires the largest difference over 40 pairs to be below 10⁻⁹ and it comes back exactly zero. This is why a diffraction pattern is centrosymmetric whatever the crystal is, and why the thirty-two classes collapse to eleven before a structure is even proposed. How it is known

The law that hides handedness

With real scattering factors, negating the indices conjugates the structure factor and leaves the intensity exactly alone — so every diffraction pattern is centrosymmetric whatever the crystal is. The escape is an imaginary component that the negation does not touch, and it is how the handedness of a molecule is measured.

 3 lattices. 3 lattices: hexagonal P, with 24 symmetries; rhombohedral P, with 12 symmetries; hexagonal R, with 12 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything. Lattices

A lattice described on somebody else's axes

R-centring a hexagonal cell does lower its symmetry, from twenty-four to twelve — and the lattice that results is the fourteenth, the rhombohedral one, which already appears on the list under its own axes. It is the only row in the enumeration where losing symmetry and being a duplicate are the same verdict.

 Two-colourings of the seventeen. How many ways each of these 17 plane groups can be two-coloured so that every symmetry either preserves the colours or exchanges them. 74 in all, each one a subgroup of index two enumerated by trying every assignment of colours to a generating set and keeping the assignments that turn out to be consistent. p3 admits none: a homomorphism onto a group of order two has nothing to send a three-fold rotation to but the identity, and once the rotation and its conjugates are killed nothing is left to reverse the colours. pmm admits the most, with 15. Every count is one less than a power of two because the homomorphisms of a group onto the two-element group are the non-zero elements of a vector space over that field. The classification

Two colours, and a symmetry that swaps them

A chessboard and a grid of identical squares have the same group, which is plainly not what anybody sees. Admitting the colour swap as an operation gives a finer classification — and one of the seventeen turns out to admit no two-colouring at all.

 Every property count, in every class. For each of these 32 classes, how many independent components a property may have: piezoelectric moduli from 18 down to 0, 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

Permitted is not present

A symmetry argument says which components of a property may be non-zero. It is routinely read as saying they are — and the gap between the two is where every practical use of this table either works or quietly fails.

 Which inflation factors a tiling may have. Every distinct inflation factor produced by a two-letter substitution whose matrix has entries up to 4, plotted against its algebraic conjugate. The two grey lines are the unit circle, which in the quadratic case is the pair of values ±1. A factor whose conjugate lies strictly inside is a Pisot number and the chain it grows has sharp Bragg peaks; 39 of the 77 factors here lie outside and cannot. The golden ratio is the smallest of them all, which is the arithmetic reason it turns up in every quasicrystal anybody has drawn. Order without repetition

Which inflation factors exist

A tiling grown by substitution has an inflation factor, and it is an eigenvalue of an integer matrix — so it is an algebraic integer, and sharp diffraction demands that its conjugates be small. That condition is an inequality between two integers, and it explains why the golden ratio turns up in every quasicrystal anybody has drawn.

 Sublattices of index n in the plane. For each index up to 12: the number of sublattices found by building every Hermite normal form of that determinant, and the number the Dirichlet series ζ(s)ζ(s−1) predicts — the sum of the divisors in the plane, and a longer sum in space. The two columns are computed by routines that share no code, and the figure does not appear at all if any row disagrees. Lattices

How many ways there are to thin a lattice

A sublattice of index n keeps one lattice point in n, and there is never only one way to do it. In the plane the number of them is the sum of the divisors of n; in space it is a longer sum; and both are counted here by writing every one of them down.

 Each arithmetic class holds exactly one symmorphic group. The arithmetic crystal classes this site enumerates in full, with the number of space groups each produces and the number of those that are symmorphic — that is, that have an origin at which every operation's translation part vanishes. The right-hand column is one in every row, over 25 groups in all, and the figure asserts it rather than reporting it. That is the bijection behind the number 73: there are seventy-three arithmetic crystal classes in three dimensions and seventy-three symmorphic space groups, and the correspondence is this one, class by class. Into space

One symmorphic group per class

Every arithmetic crystal class holds exactly one space group in which some origin clears every translation part at once. That bijection is why there are seventy-three symmorphic space groups and seventy-three arithmetic classes, and it is checked here class by class rather than counted.

 How often a dot gives the wrong group. Every position on a grid inside the cell, tried as a single-dot motif for each of the seventeen groups. The bar is how often the resulting pattern turned out to have more symmetry than the group it was made with — so the caption would have been wrong and nothing about the picture would have shown it. The classification

The Alhambra question

Textbooks say the Alhambra contains all seventeen wallpaper groups. Careful analysts of the same building have counted eleven, thirteen, fourteen and seventeen — and the disagreement is not about the mathematics but about what counts as an instance.

 A spontaneous vector in 222. Averaging each of the three axes over the 4 operations of 222 leaves 0 independent components. No direction survives, so the class permits no spontaneous polarisation at all — which is a statement about what is forbidden, not about any measurement. What symmetry decides

The ten with a direction of their own

A crystal has a spontaneous electric polarisation only if some direction is left completely alone by every one of its symmetry operations. Ten of the thirty-two classes have such a direction, and the same ten are computed here twice by routes that share nothing but the group.

 How many ways each class can lose symmetry. Every crystal class, with the number of distinct classes it can descend to — 247 parent-and-child pairs in all across the thirty-two, counted up to conjugacy in the parent, which is the equivalence that says two descents differing only by which axis was chosen are one transition. The count rises steeply with the order of the parent, which is why the cubic and hexagonal holohedries dominate the list of materials with rich domain structures. Symmetry at work

Two hundred and forty-seven descents, or two hundred and twelve

How many distinct ways can a crystal lose symmetry? Counting parent-and-child pairs up to conjugacy in the parent gives 247. The standard enumeration in the ferroics literature gives 212, and the operation that merges the extra thirty-five turns out to be a rotation through forty-five degrees — which no lattice may have, and which no integer matrix in a lattice basis can therefore express.

 Six classifications, and which are enumerated here. The families of symmetry groups by how many directions they repeat in and how many they live in. The thirty-two crystal classes, the seven friezes and the seventeen plane groups are each built from their own operations and counted. The seventy-five rod groups, the eighty layer groups and the two hundred and thirty space groups are numbers from the literature, marked as such wherever they appear: reaching them needs the translation extensions and their equivalences in full, which is the content of the classification rather than an application of it. The subperiodic cases sit exactly between the two halves, which is why they are so easy to assume are already known. The classification

A layer is not a wallpaper

A sheet repeats in two directions and lives in three, and its symmetry group is not one of the seventeen. There are eighty of them, the difference between one and another is a single sign per operation, and the arithmetic that supplies those signs is the arithmetic of a two-coloured pattern.

 Σ5: two square lattices at 36.87°. Two square lattices, one turned through 36.87° about a shared point. At this angle one point in 5 lands exactly on a point of the other lattice — 29 of the 149 drawn — and those shared points are themselves a lattice, the coincidence site lattice, of index 5. The angle comes from tan(θ/2) = 1/3, and Σ is the odd part of 3² + 1² = 10. Nothing here is measured: whether a point is shared is decided by an integer congruence. Symmetry at work

Turn a lattice against itself and almost nothing lines up

Two copies of one lattice rotated about a shared point share that point and, at almost every angle, no other. At a discrete set of angles they share a whole sublattice — one point in three, or five, or seven — and a grain boundary built on such an orientation costs a fraction of what a general one costs, because a fraction of the atoms are already where both sides want them.

 Which indices have a square sublattice. For each index up to 26: how many sublattices of the square lattice are themselves square, found by testing whether the quarter-turn maps each one onto itself; the same count as a sum over divisors, +1 for each divisor one more than a multiple of four and −1 for each one less; and the ways of writing the index as a sum of two squares. The three agree at every row, which is Fermat's theorem — and it says that 3, 7 and 11 have no square sublattice at all while 5, 13 and 17 have two. Lattices

The sublattices that stay square

A sublattice of the square lattice is itself square exactly when its index is a sum of two squares — so index five has two and index seven has none, and which superstructures a surface can form is decided by a theorem of Fermat's about primes.

 What the inversion does to three sums in 4/mmm. The 16 operations of 4/mmm come in pairs — every operation together with its own negative, because the class contains the inversion — and each row pairs the two terms they contribute. For an even-rank polar property the pair is two equal bars: the inversion changes an even number of indices and the character cannot see it, so the average is whatever it was before the inversion was added. For an odd-rank polar property and for an axial one the pair is a bar and its reflection, and the sum is exactly zero — which is why piezoelectric moduli and gyration tensor are forbidden here outright rather than merely small. Neither statement is about this class: the cancellation is checked over all 11 centrosymmetric classes and all four odd or axial properties every time this figure is drawn. What symmetry decides

Twenty of the twenty-one

Twenty-one crystal classes have no centre of symmetry, and twenty of them permit piezoelectricity. The exception is 432, which has twenty-four operations, no inversion, and a character sum that cancels to nothing — and the reason it fails is not that it has too much symmetry in any ordinary sense.

 Every coincidence index is odd. The rotations that bring a cubic lattice into coincidence with itself, up to index 25: 17 distinct relations across 12 indices, each found by enumerating integer quaternions and each index computed as the size of a sublattice rather than from the usual formula — the two are then required to agree. 5 of the 12 indices carry more than one relation, which is why the tables write 13a and 13b. Every index is odd. That is not a feature of this range: a rational orthogonal matrix written in lowest terms has an odd denominator, so the factors of two always cancel. Symmetry at work

Every coincidence index is odd, and in the plane most of them do not exist

The indices at which two copies of a cubic lattice share points are 3, 5, 7, 9, 11 and every odd number after them. There is a two-line proof that no even index can occur. Ask the same question about a square lattice and the answer is a different list entirely, governed by which numbers are sums of two squares — so Σ3, which is the commonest boundary in every metal, has no plane analogue at all.

 May be piezoelectric, against may be optically active. Two questions asked of all thirty-two classes, and the classes where the answers part company. 14 classes are in both lists, 6 in only the first, 1 in only the second and 11 in neither. Both lists are computed from the same character sum with a different tensor, so a class appearing in one and not the other is a statement about which representation survives rather than about anything measured. Every entry is a permission: a class in a column is a class whose symmetry fails to forbid the effect, which is a weaker statement than it is usually read as. What symmetry decides

Each permits what the other forbids

432 and 4̅3m are both cubic, both of order twenty-four, both without a centre. One of them can be piezoelectric and the other can be optically active, and it is not the same one — which is as clean a demonstration as the subject offers that "amount of symmetry" is not a quantity.

 The Wigner–Seitz cell of the hexagonal lattice. Every point closer to the central lattice point than to any other. The faint lines run to the 6 neighbours whose perpendicular bisectors bound the region; every other lattice point is cut off by one of them. The cell has exactly the area of a unit cell — asserted while the figure is drawn, against √det G computed from the metric — and it carries all 12 of the lattice's symmetries, which a conventional cell need not. Nothing was chosen to build it: no basis, no axes, no convention. Two people who agree about the lattice cannot disagree about this cell. Lattices

The cell nobody chose

Every unit cell on this site is a convention, and one construction escapes the warning entirely: the region of the plane closer to one lattice point than to any other. It needs no basis, no axes and no rule — and its combinatorics are decided in integers, with the square roots confined to drawing it.

 Subgroups of index 3, across the seventeen. Every subgroup of index 3 with cyclic quotient in each of the seventeen plane groups, sorted into the two kinds: 4 keep all the translations and lose operations, 22 keep all the operations and lose translations, and the total is 26. The split is decided by whether the homomorphism onto ℤ3 kills the two lattice translations, which is a property of the kernel and not a judgement. Every one of them is found by enumeration inside the finite quotient by 3Λ, and the count for the whole classification is a measurement. The classification

Three colours, and why most patterns cannot have them

Seventy-four of the seventeen plane groups' subgroups have index two, and every group but one has at least one. At index three there are twenty-six, and ten of the seventeen have none at all — because a symmetry of order two cannot survive being asked to permute three colours.

 The seventeen signatures, and the seventeen groups. Every combination of features costing exactly two, beside the plane group each one names. The left column is produced by an accounting identity that has never heard of a lattice; the right by reading seventeen groups' own operations — their rotation centres and orders, which of those lie on mirrors, and how many closed curves the mirror lines make once equivalent lines are identified. The map between the two lists is a bijection, and the figure does not appear unless it is one — in both directions. A signature with no group and a group whose signature is not on the list are both refused, and so is the failure that actually happens: two groups deriving one signature, which costs exactly two and passes every check but injectivity. The classification

Seventeen dollars

Conway's magic theorem prices the features a folded-up pattern can have — a handle costs two, a mirror boundary one, a cone point of order n almost one — and requires the total to come to exactly two. There are seventeen ways to pay, and the classification falls out of an accounting identity that never mentions a lattice.

 Two species on one lattice, ordered at index 2. Every position is a lattice point of the parent and none of them has moved. What has changed is which atom sits where: the larger marks are a sublattice of index 2, the smaller ones its other 1 coset, and the outlined cell is the new repeat. The lattice of positions is untouched and the repeat of the contents is 2 times as large, which is the whole of what an ordering transition does and the reason its signature is in reciprocal space rather than in the positions. Lattices

The reflections a superlattice adds

Centring a lattice makes reflections vanish. Ordering two kinds of atom onto a sublattice makes new ones appear, exactly n − 1 of them per parent cell, and their intensity is a difference rather than a sum — which is why an ordered alloy of two neighbouring elements can be invisible to X-rays and obvious to neutrons.

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