The theme: Exactly this many — page 16
Into space
Where a gradient term survives at the edge of the zone
A transition that freezes a mode at the edge of the zone doubles the cell, but only if the energy has no term linear in the mode's gradient. That condition was stated here for the zone boundary and never computed. Computed for every order parameter at every special point of the seventeen zones, it fails in exactly ten places: the edge points of the four glide groups where one glide acts and no second glide crosses it, and the zone corner of three hexagonal groups that have no glide at all.
Operations
A cubic cell has fourteen chances to fail
In the plane, deciding whether a measured cell is square meant two steps up a short diagram, and dividing the error rate between them fixed the test. In space, a primitive cubic cell sits on ten strata joined by fourteen steps on ten different paths, and a walk that tests every step at five per cent calls more than a quarter of true cubic cells something less. Dividing the level by the steps of one path is not enough and dividing it by all fourteen is too much. Measured directly, the right level is two-thirds of a per cent a step, and calibrated that way the walk notices a tetragonal distortion more often than the single test it replaces.
![Every six-dimensional lattice with a seven-fold rotation, on one torus. The lattices with a rotation of order seven in six dimensions are the ring ℤ[ζ₇] with three positive weights, one on each complex line it spans. Up to scale and isometry the weights live on a torus, drawn as a sixty-degree rhombus with opposite sides glued, shaded by how well each lattice packs spheres. The Galois automorphism turns the torus a third of a turn about three points, on the long diagonal: A₆* with 10080 symmetries, A₆ with 10080, and the ideal (1 − ζ)² with 672, the densest lattice in the family. The small dots are integral members, each with 28 symmetries.](/f/every-six-dimensional-lattice-with-a-seven-fold-rotation-on-one-torus-551a6153.svg)
Lattices
A seven-fold lattice lives on a folded torus
In six dimensions a lattice can have a seven-fold rotation, and every one that does is the integers of the seventh roots of unity with three weights on it. Up to scale the weights live on a hexagonal torus that a Galois symmetry folds by a third-turn, and the fold has three fixed points: A₆*, A₆, and a third lattice that packs better than either. Centring by sevenths is no longer a reflection, as fifths was in four dimensions. It moves every lattice a third of the way along the torus and carries the three special ones round in a cycle.
How it is known
The wavelengths that phase best do not find the edge
A MAD experiment refines the anomalous atom's scattering factors so that it can solve for the phases, and those same factors report where the absorption edge sits and how tall its white line stands, which is chemistry. Scored for the edge instead of the phase, the best triple of wavelengths moves: two straddling the step and one far below it, where the phase-best triple put two on the step and one far above. No triple is best at both, and the textbook peak, inflection and remote turns out to be the compromise between them.
Operations
A three-fold axis puts a grid point on itself
A grid of wavevectors offset from the zone centre can keep a crystal's whole symmetry and touch no mirror, no axis and no point where time reversal folds the zone onto itself, and then its share of the zone is exactly one over the order of the group, with no boundary excess at all. Four Laue groups have such a grid, as small as the group itself. Every group with a three-fold axis has none, for a reason that fits in a line: the rotoinversion fixes one point of the plane across its axis, and above that point the three-fold rotation fixes the grid.
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.
Into space
A crystal with no hand holds rods of both hands
Rolling a flat pattern into a tube reaches fifty-four of the seventy-five rod groups and never a screw that comes in a pair of hands. Cutting a column out of a crystal reaches all seventy-five. Sixteen of them are handed screws, and half of those can be cut from a crystal that has a centre of symmetry, from a line that happens to be a screw of one hand, with its mirror image a short distance away. A handed thread does not need a handed crystal, only a handed place in one.
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.
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.
Into space
Two mirrors never make a crystal
Sixty-four space groups are generated by two of their own operations. Which two kinds can do it is a finite question, and it has been settled for all sixty-four and all fifteen pairs of kinds. A screw and a glide make thirty-two of them. Two mirrors make none, and neither does a rotation with a mirror, for reasons that need no computation: two operations that undo themselves generate a dihedral group, and an axis and a plane always share a point or a direction. Two plain rotations make six groups, every one of them cubic and without a mirror.
What symmetry decides
Two classes go dark far from absorption
The second-harmonic tensor of a crystal has as many components as its piezoelectric tensor, and far from any absorption it gains a symmetry of its own that removes eight of the eighteen. What it removes is not arbitrary: it transforms as a polar vector plus the traceless part of the tensor that carries optical activity. So the classes that lose the most are the polar ones and the handed ones, and two of them, 422 and 622, lose everything.
Operations
Thirteen sites obey Friedel's law without a centre
At five and seven sites the earlier census found that a three-species structure has a contrast that darkens a spot exactly when it has a centre, and asked whether that is a theorem. It is not. The two phase sums are parallel exactly when the A–B pairs are as common at every separation forwards as backwards, and exactly when no anomalous scatterer can produce a Bijvoet difference anywhere in the pattern. At 5, 7 and 11 sites only a centre achieves that. At 13 sites one arrangement does it without one.
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.
How it is known
Two leads need their vector to half a step
With one heavy atom in a cell with no symmetry, charge flipping did not care where the atom was subtracted: it rebuilt the structure round whatever place it was given. With two heavy atoms that freedom runs out, as predicted, because no translation can put both under wrong references at once. Subtract one of them a whole grid step from where it belongs and the solver never succeeds. But the tolerance is five times looser than the estimate it was predicted from: up to a quarter of a step the rate does not move, at half a step it halves, and the residual the solver absorbs is three carbons' worth, not one.
What symmetry decides
A crystal in 422 needs two resonances
Kleinman's rule silences classes 422 and 622 far from absorption, and the obvious guess is that dispersion brings them back near it. In the simplest model of dispersion, one charged oscillator with an anharmonic potential, it does not: they stay dark at every frequency. Two oscillators with different resonances light them, with a response growing as the square of the frequency. 432 needs three, and then only mixes two different frequencies, growing as the sixth power. The number of resonances a class needs is a count of invariants, and it sets how fast the forbidden part can grow.
Into space
Three mirrors never make a crystal either
Ninety-three space groups need three of their own operations to generate them, and which three kinds can do it is a finite question with 3,255 cases. All but two are now decided. Three mirrors make none of the ninety-three, as a theorem about reflection groups requires. A rotation with two mirrors makes exactly three, all cubic. Three plain rotations make thirteen, every one without a mirror. One group can be made thirty ways and three can be made only one.
Operations
No Petersen graph in the plane
A finite graph can look the same from every vertex and still be the picture of no group; the Petersen graph is the smallest that does. A periodic net in the plane cannot. Whether one is a Cayley graph depends only on where its vertices sit, and they sit on one of the fifty-five special positions of the seventeen plane groups or on a general one. Every one of the fifty-five has a subgroup that carries any of its points to any other in exactly one way, and four of them have exactly one.
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.
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.
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.
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.
What symmetry decides
Thirty-five classes a squeeze can magnetise
A magnetisation proportional to a stress is permitted in sixty-six of the hundred and twenty-two magnetic classes. Thirty-one are the classes that may be magnetised already; the other thirty-five may have no moment at rest and acquire one when squeezed, and in them a stress tells apart two domains that differ only by the direction of time. Among the ninety classes that are not grey, it is forbidden only by a primed centre and in the three ordinary cubic classes with four-fold operations.
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.
Lattices
The centring survives the limit
Stretch a body-centred tetragonal lattice until its axes are equal and it becomes body-centred cubic; stretch it further and it becomes face-centred cubic. No stretching of a primitive tetragonal lattice reaches either. Which of the fourteen lattices can be deformed into which is decided by the subgroups of each lattice's own group, and the answer keeps track of centring in a way the familiar chain of crystal systems does not.