Crystallography & Symmetry
patterns derived from their groups
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The theme: The same arithmetic, renamed — page 9

A crystal form is an orbit. A twin law is a coset. The domain states left by a phase transition are the cosets of the low-symmetry group in the high-symmetry one. Four subjects that grew up in different centuries and different departments, doing one piece of arithmetic under four names.
The fifty-eight black-and-white classes, sorted by reflection class. Every black-and-white magnetic point group, grouped by the smallest reflection class containing its underlying crystal class, with the degrees of the mirror products that generate its relative invariants for a polar and for an axial time-odd quantity. Every class's two series are exactly those sums of mirror products, and the number of generators is the class's index in its reflection class: one, two or four. What symmetry decides

Every magnetic class counts its properties by mirrors

The invariants of every crystal class turned out to be products of mirrors of a reflection class, times that reflection class's own free invariants. A magnetic class asks for polynomials that some of its operations negate, which is harder in principle. It is not harder in fact: all fifty-eight black-and-white classes, for a time-odd quantity that is polar and for one that is axial, are counted by mirror products — provided the reflection class contains the whole class and not only its unprimed part, which for thirty of them it does not.

 The square net drawn as the Cayley graph of four different groups. The same square net four times. In each panel the edges are coloured by the group operation that spells them, read from one regular subgroup of the net's symmetry group: the lattice of translations, a p2 whose half-turn centres sit at the midpoints of edges, a pmm whose four mirrors run between rows of vertices, and a p4 whose quarter-turn centres sit in the middle of the squares. No symmetry element drawn passes through a vertex, which is what makes each group act without fixing any, and the large dot is the vertex named by the identity. Operations

The square net draws eleven groups

The Cayley graph of a wallpaper group is a periodic net. Asked the other way, the question is which nets are Cayley graphs, and of what. A net is one exactly when some group of its symmetries carries every vertex to every other and fixes none. All eleven uniform nets pass. The square net is the Cayley graph of eleven of the seventeen groups, the kagome net of exactly one, and between them the eleven nets use all seventeen.

 The five special eleven-fold lattices in ten dimensions. The five lattices at the fixed points of the Galois rotation of order five on the 4-torus of eleven-fold lattices: A₁₀*, A₁₀ and the ideals (1 − ζ)², (1 − ζ)³ and (1 − ζ)⁴ with the trace form. For each, where it sits on the torus's diagonal, its symmetry group (the root lattices' by identification, the others' by exhaustive search: 2,640 each), its kissing number, its centre density at determinant one (0.0020, 0.0094, 0.0274, 0.0189, 0.0221), and which of the five its dual is. The densest, (1 − ζ)², is shaded; the one duality fixes is (1 − ζ)³. Lattices

The densest eleven-fold lattice is not its own dual

Every ten-dimensional lattice with an eleven-fold rotation is one ring with five weights, and its shapes form a four-dimensional torus folded by a Galois rotation of order five. The seven-fold essay predicted five special lattices, the powers of 1 − ζ, cycled by centring, and all of that holds. What does not carry over is its best result. In six dimensions the densest seven-fold lattice was the one duality fixes. In ten, duality fixes (1 − ζ)³, and the densest is (1 − ζ)², whose dual is (1 − ζ)⁴.

 Four flipping rules on four cells. The best success rate of the flipping solver, pooled over five arrangements and eight starts, on four cells: twenty carbons, a quarter sulphur, one sulphur among twenty and one lead among twenty. Four rules for what is flipped: below a number of standard deviations of the map, below a height set by the lightest atom's weight, the same height above the map's mean, and the original rule with the known heavy atom's density removed first. On the lead cell only the last solves anything, 55% against nothing; on both sulphur cells it raises the rate to near certainty; on carbon alone it is the original rule. How it is known

The lead taken out solves what no threshold could

Charge flipping never solved a cell of nineteen carbons and one lead, and two repairs were proposed: a threshold set by the composition rather than by the map's spread, and the lead's own density removed before each flip. The threshold does nothing for the lead cell at any setting, measured from zero or from the map's mean. Removing the lead takes it from nought to fifty-five per cent, and lifts both cells with sulphur in them to near certainty. The trouble was never where the threshold sat.

 The six special thirteen-fold lattices in twelve dimensions. The six lattices at the fixed points of the Galois rotation of order six on the 5-torus of thirteen-fold lattices: A₁₂*, A₁₂ and the ideals (1 − ζ)² to (1 − ζ)⁵ with the trace form. For each, where it sits on the line through the fixed points, the symmetries known for it, its kissing number (26, 156, 390, 312, 78, 26), its centre density at determinant one (0.0005, 0.0043, 0.0213, 0.0187, 0.0081, 0.0071), and which of the six its dual is. None is its own dual. The densest, (1 − ζ)², is shaded; the densest known twelve-dimensional lattice, Coxeter and Todd's, is given below for comparison. Lattices

Thirteen-fold, and nothing is its own dual

Every twelve-dimensional lattice with a thirteen-fold rotation is one ring with six weights, and its shapes form a five-dimensional torus folded six ways. The fold has six fixed points, the powers of 1 − ζ, and duality fixes none of them, as in four dimensions and unlike six and ten: it pairs them off. The eleven-fold essay asked whether a longer cycle would let a higher power take over as the densest. It does not. (1 − ζ)² is the densest member at seven, at eleven and at thirteen, whatever duality does.

 The 8 special 17-fold lattices in 16 dimensions. The 8 lattices at the fixed points of the Galois fold on the torus of 17-fold lattices: A16*, A16 and the ideals (1 − ζ)² to (1 − ζ)⁷ with the trace form. For each, its place on the torus, its minimum where it reaches the bound 2kp, its kissing number (34, 272, 952, 1088, 476, 816, 306, 34), its centre density at determinant one (0.0000, 0.0009, 0.0143, 0.0215, 0.0126, 0.0191, 0.0112, 0.0007), and which of them its dual is. The densest, (1 − ζ)³, is shaded: the third power, where the seven-, eleven- and thirteen-fold families had the second. Lattices

At seventeen the densest power of 1 − ζ moves

Three primes running, the densest lattice with a p-fold rotation sat at the second power of 1 − ζ. At seventeen, in sixteen dimensions, and at nineteen, in eighteen, it sits at the third. The move is not a surprise once the minima are read: every power up to the fourth reaches Craig's bound exactly, which turns each density into a formula and the competition between powers into an inequality. The second power wins exactly while (3/2) raised to half of p − 1 stays below p, and the formula's next prediction, the fourth power at twenty-three, comes true by three per cent.

 Photoelastic constants of the eleven Laue classes, split into four parts. The number of independent photoelastic constants each Laue class permits, from 36 for 1̅ to 3 for m3̅m, drawn as squares and split into the elastic tensor's part, which is symmetric under exchanging the two index pairs, and the extra fifteen-dimensional part, which divides into an axial vector, a second quadrupole and an octupole. The axial vector survives only in 1̅, 2/m, 4/m, 3̅ and 6/m; the octupole is what separates m3̅ from m3̅m; and in m3̅m nothing extra survives, so the two counts agree at three. What symmetry decides

Fifteen constants an energy would have forbidden

A strain changes a crystal's refractive index through a fourth-rank tensor with the elastic tensor's index pairs and without its one remaining symmetry, because no energy is differentiated twice to make it. The missing symmetry is worth fifteen components. They split into an axial vector, a quadrupole and an octupole, and each does something the elastic count cannot: the octupole gives 23 and m3̅ a fourth constant, and the axial vector is a part of the tensor no turn of the axes removes.

 Copies of a plane group of itself, counted to index ninety-six. For eight of the seventeen plane groups, the number of subgroups isomorphic to the group, up to conjugacy, of index at most N, on logarithmic axes. The curves fall into four tiers by their slopes: groups with no operation but a half-turn grow as N², groups with mirrors and nothing of order three or more as N log N, groups with a rotation of order three or more and no mirror as N, and groups with both as √N. By index 96 p2 has 13446 copies and p6m 13. Into space

The copies a group keeps of itself grow at four speeds

A plane group's copies of itself sit on the sublattices all its operations keep, several to a lattice. A richer point group keeps fewer lattices and puts more copies on each, and the two trends run in opposite directions. Counted to index ninety-six they do not balance. The copies per lattice never exceed sixteen, while the lattices grow as N², N log N, N or √N according to the point group, so a richer group's early lead always ends, at index ten, eleven or twenty-seven, and never returns.

 Every square-symmetric grid of wavevectors has a point on a mirror. One cell of the square reciprocal lattice, centred on the zone centre, with the mirror lines of 4mm through it and three grids the four-fold rotation keeps with an offset off the axis. The quarter turn allows that offset at one place only, the second four-fold centre of the grid's own lattice, ringed. On an aligned grid of odd side the rotation fixes a point; on an aligned grid of even side the offset lies on a diagonal mirror; on a grid turned by 45 degrees it lies on an axial mirror. No grid escapes all three. Operations

The offset a quarter turn allows lies on a mirror

A census of sampling grids found free grids for the tetragonal group 4/m and none for 4/mmm, to a hundred and twenty-eight points, with no reason. The reason is two lines long. A quarter turn lets a symmetric grid be offset to exactly one place off its axis, the second four-fold centre of the grid's own lattice, and a square lattice that a mirror keeps can only be aligned with its mirrors or turned by half a right angle. Either way that centre lies on a mirror line.

 A shortest vector of a sixteen-dimensional lattice, as a polynomial with two full fibres. The values of f(t) = t(t − 1)(t − 2)(t − 3) at every t in the field of 17 elements, value against t. It takes the value 0 at the four points 0, 1, 2, 3 and the value 12 at the four points 3, 5, 12, 14. Those two fibres are two sets of four with equal first, second and third power sums modulo 17, so the vector with +1 on one set and −1 on the other lies in (1 − ζ)⁴, with squared length 8 in the trace form's units of 17: Craig's bound, attained. Lattices

The shortest vector is a polynomial with two full fibres

Whether Craig's lattices reach their bound, minimum 2kp for the ideal (1 − ζ)ᵏ, looked like a question for shortest-vector searches in twenty-eight to thirty-six dimensions. It is a question about polynomials over the field of p elements. The bound is reached exactly when some polynomial of degree k takes two values k times each, and every divisor of p − 1 supplies one. Read that way, every power at 29, 31 and 37 is settled, the densest are the fourth, fourth and fifth as predicted, and past the middle of each cycle the minimum climbs until it meets the vector of quadratic residues and stops.

 Sublattices a space group's point group keeps, counted to index sixty-four. For six of the seventy-three arithmetic classes, the number of sublattices of ℤ³ of index at most N that every operation of the point group carries to itself, on logarithmic axes. The six curves have six slopes: N³ for a triclinic class, N² for a monoclinic one, N log² N for an orthorhombic one, N log N for a tetragonal class with no two-fold axis or mirror through its four-fold axis, N for one with them, and the cube root of N for a cubic class. By index 64 P1̅ keeps 176874 sublattices and Pm3̅m 9. Into space

In space the sublattices a group keeps grow at six speeds

The plane's seventeen groups kept their sublattices at four rates. Counted to index sixty-four for all seventy-three arithmetic classes, space has six: the cube of the index, its square, the index times two logarithms, the index times one, the index, and its cube root. Four systems keep one rate each. The tetragonal, hexagonal and rhombohedral classes are cut in two, by whether anything turns their axis over. And the richer groups' lead lasts much longer than in the plane, because they carry up to five hundred and twelve subgroups on a single lattice.

 Where the two crossings the census could not reach fall. The ratio of cumulative subgroup counts with the full point group, poorer class over richer, against the largest index. P1̅ passes Pmmm at 20, as the census to sixty-four found. P2/m passes Pmmm at index 81 and stays ahead to 1,023. P4/m passes P4/mmm at index 12,104 and stays ahead to 65,536. Every count comes from the values at prime powers, multiplied. Into space

The last two crossings fall at eighty-one and twelve thousand

A census to index sixty-four found that P2/m and P4/m must eventually overtake Pmmm and P4/mmm in the number of subgroups they keep of themselves, and could not see either happen. The count turns out to be multiplicative: a sublattice splits into pieces at the primes dividing its index, and so does the cohomology on it, which lives only at two. Every odd prime is then a closed form, two is a short computation, and the crossings fall at index 81 and at index 12,104.

 Discriminant −23 at the prime 2: the crater and three levels down. The lattices of discriminant minus 23 and of 4, 16 and 64 times that, joined by index-two sublattices. Three lattices form the crater, a cycle in which each has two neighbours of its own discriminant. Each crater lattice has one sublattice going down, and every lattice below has one edge up and two down. The levels hold 3, 3, 6, 12 lattices. Lattices

One way up and p ways down

Every plane lattice has exactly p + 1 sublattices of prime index p. Restrict an integer form to each and the discriminant can do only three things, so the lattices become the vertices of a graph with a definite shape: a ring of lattices at the top, and below it a tree in which every lattice has one way up and p ways down. Whether the ring exists at all is decided by how p factors, which is why the square lattice has two copies of itself at index five and none at index three.

 The constants a turn of the axes removes, class by class and property by property. The thirteen crystal classes whose operations all commute with some turn of the axes, against six properties: pyroelectric, dielectric, optical activity, piezoelectric, elastic and photoelastic, and all six together. Each cell gives the printed number of constants, and where a turn removes some, the number left. Triclinic classes have three turns and lose up to three; the others have one and lose at most one. Classes 6 and 6/m lose nothing in any property, and the last column loses the turns once, however many properties could spend them. What symmetry decides

A six-fold crystal hides its frame until rank six

Thirteen crystal classes can be turned without moving any of their operations, and the tables print each property's constants in one frame among the many such a turn allows. Spending the turn removes a constant only where a property has a part that turns with the frame, and an n-fold axis keeps such a part only at rank n or above. So a monoclinic crystal's free angle shows in its dielectric tensor, a tetragonal one's first in its elastic constants, and a hexagonal one's in nothing anybody measures. And a crystal has one frame, not one per property: the angle is spent once.

 Tribonacci keeps one number at every length, and its neighbours do not. The size of tr(ABC) − tr(ACB), computed in exact integers on three two-site tiles, at each application of four rules, on a logarithmic scale. Under tribonacci it stays 24 while the traces themselves reach dozens of digits. Under the plastic rule and the rule x³ = x² + 1, both of which can be undone, it wanders and grows. Under a rule that cannot be undone it passes through zero and runs off the scale. Order without repetition

Three tiles keep a number that asks which way they face

Two tiles always conserve something: every reversible rule keeps the trace of their commutator. Three letters were expected to conserve nothing, and most rules do conserve nothing. The tribonacci rule keeps exactly one number, the difference between a product of three matrices and the same product reversed. It is zero for any tile that reads the same both ways, so it measures direction rather than difference, and one rule in fifty keeps it, for a reason that can be read off the words.

 Every form tried, by its symmetries and by how well it packs. 5902 totally positive forms of the ring of fifteenth roots of unity, sorted by the order of their symmetry group into five rows. Forms with 60 and 120 symmetries form scattered bands. Two lattices have 1,440, the tensor products of A2 with A4 and with its dual. One has 14,400, a member of the icosian line; one has 28,800, the icosian ring with its plain trace; and one is E8. The highest Hermite number among forms with only 60 symmetries is above that of the 28,800 lattice. What a lattice forbids

One ring is E8 and a lattice with no mirror

An operation of order thirty as one block in eight dimensions preserves a whole family of forms, and E₈ was one point of it. Walking the family finds sixty symmetries almost everywhere and two lines above that. One is tensor products. The other is the icosian ring, on which E₈ is famously built, under a sliding choice of form: 14,400 symmetries all along it, 28,800 and not one reflection at one end, and at the other end E₈, where the group jumps by a factor of nearly fifty thousand.

 Fitting the outer shells makes it worse, and correcting the fit recovers it. The substructure's weaker peak as a share of its height with no scaling, on data with no error, with the scale error each scaling builds in, for five falloffs and three scalings. Fitting everything loses 42% with no falloff and less as the falloff grows. Fitting only the outer shells loses more, and at the steepest falloffs builds in a large error of the opposite sign. The fit corrected by the ratio the composition expects recovers the whole peak at every falloff, within a few per cent. How it is known

Fit the scale, then put the lean back

Scaling two wavelengths to each other builds in an error of the cancelling sign, because their means differ for a physical reason. The proposal was to fit against the part the edge does not touch, at high resolution. It does the opposite: f₀ falls with angle and f′ does not, so the outer shells lean furthest and fitting them costs more. Taking the scale from the composition alone reproduces the lean as well. What works is fitting as usual and then putting back the difference the composition expects, and that recovers the whole peak.

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