The collection

Every essay — page 23

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Operations

Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.

Which plane groups can run every operation backwards, kind by kind. For each of the seventeen plane groups and each kind of operation it has, whether every operation of that kind in a window of 5 × 5 cells is conjugate in the group to its own inverse, decided exactly. The groups in which every kind passes, shaded, are p2, pmm, cmm, p4m, p6m. The last column is the answer the point group gives, which is yes for fourteen; it is wrong for nine of them.

Five groups can run every motion backwards

Every symmetry of a pattern has an inverse, the same motion run backwards. Sometimes the pattern's own group contains a third motion that turns the one into the other, so that a motion and its reverse are the same symmetry seen from somewhere else. In five of the seventeen plane groups that is true of every operation, and the group's point group, which is where anyone would look, says it is true of fourteen. The difference is nine groups, and every one of them fails on something the point group cannot see: a translation, a glide's slide, or a rotation centre with no mirror through it.

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Lattices

The repeating scaffold underneath every crystal. Five in the plane, fourteen in space, and no others — a classification with a short proof.

The fibres of a cubic, read off a conic. Over the field of 17 elements, the points of the conic x² + xy + y² = 14, which is −β for β = 3. Each point (x, y) with x, y and −x − y all different is one ordering of a set of three roots on which t³ + 3t takes a single value; the 12 such points fall into 2 sets of six. The 6 dashed points lie on the lines where two roots coincide.

The third power's shortest vectors lie on a conic

Craig's lattices reach their bound when a polynomial over a prime field takes one value at k points and another at k more, and each shortest vector is such a pair. Counting them for the third power turns out to be counting points on a conic, and the answer is a closed formula: thirty-two orbits at seventeen, as a sixteen-dimensional search had found without explanation. The fourth power is a polynomial in p at exactly the primes that do not split in the field of √−2, and at the others it is not.

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What symmetry decides

Thirty-two classes, enumerated rather than listed — and then the one thing they are for outside crystallography's own bookkeeping: which components of a physical property a crystal is permitted to have before anybody measures it.

The rank at which a crystal's free angle first shows, with and without time reversal. For the thirteen ordinary crystal classes whose frame can be turned without moving any operation, and the 17 black-and-white magnetic classes over them (bold), the lowest rank of tensor, polar or axial, whose generic member a turn of the axes moves: for time-even tensors (open dots) and time-odd ones (filled). A time-even tensor never notices the primes. A time-odd one shows the frame at half the rank in 9 classes, exactly those whose highest proper rotation is primed: 2′, 4′ and 6′ with or without a mirror.

A primed axis shows its frame at half the rank

An n-fold axis hides a crystal's free angle from every property below rank n, so a six-fold crystal shows it in nothing anybody measures. Combine the axis with time reversal and a magnetic property sees each operation with an extra sign, which moves the parts the axis keeps by half a period. In 6′ the angle shows at rank three, in the piezomagnetic tensor; in 4′ at rank two, in the magnetoelectric tensor; in 2′ in the magnetisation itself. Nine of the seventeen black-and-white classes with a free turn halve the rank, exactly those whose highest rotation is primed, and no property that is even under time reversal ever notices.

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How it is known

A crystal's structure is read from the pattern it scatters. The reciprocal lattice, systematic absences, and what an experiment can and cannot see.

How long a second sweep must be, against how far its axis leans. After a full turn about a kept axis, the least arc about a second axis, tilted from the first by the angle on the horizontal axis, that reaches every orbit the first sweep never saw, for 4/mmm about c, mmm about c, 2/m about b at 1.4 Å. Below about twenty degrees of tilt some missing orbits stay blind whatever the arc; just past that threshold the arc is near eighty degrees, and it falls to under twenty at a right angle.

The second spindle must lean past one Bragg angle

A crystal turned about an axis its Laue group keeps leaves a hole no amount of turning fills: on the four-fold axis of a tetragonal crystal, one reflection class in nine is never measured. A second axis fills it, but only if it leans far enough, and the threshold is not a rule of thumb. It is the Bragg angle of the outermost reflection lying on the first axis itself, twenty-one degrees at 1.4 ångström. Past it the second sweep shortens quickly, to nineteen degrees at a right angle.

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Into space

Seventeen becomes two hundred and thirty. Screw axes, glide planes, and the translations that no choice of origin can remove — the operations a flat surface has no room for.

Two glides always share a line

At a point on the edge of the Brillouin zone where two screws or glides are active, an order parameter carries a gradient term exactly when the two keep a common axis. It holds at all 104 such points of the orthorhombic groups, and the reason is plain geometry: two glide planes always meet in a line, two screws on different axes never share one, and a glide and a screw share one exactly when the screw lies in the plane. Where more operations are active the rule breaks sixteen times, every time on an order parameter of four dimensions. In two dimensions one determinant decides everything.

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How it is known

A crystal's structure is read from the pattern it scatters. The reciprocal lattice, systematic absences, and what an experiment can and cannot see.

$A twin confines every pair of partner reflections to a wedge. Each of 300 pairs of reflections related by the twin law (h, k) → (k, h), for a structure of 24 atoms in a square cell with no symmetry of its own, plotted as one intensity against its partner's, in units of the mean (the few pairs beyond four times the mean are off the scale). Left, untwinned: the pairs fill the quadrant. Right, the same structure twinned at a fraction of 0.3: every pair lies between the two lines of slope 0.429 and 2.333, because twinning keeps each pair's sum and multiplies its difference by exactly 1 − 2α. Nothing statistical confines them; the wedge is arithmetic.$

A twin leaves an edge that nothing else can

Compare each reflection with its partner under the twin law and twinning turns out to keep their sum and multiply their difference by exactly 1 − 2α. So no pair of a crystal twinned at thirty per cent can differ by more than forty per cent of its sum, and the distribution of contrasts stops dead at that edge. A structure that merely resembles its own twin image lowers the average contrast as a twin does, and leaves no edge. A pseudo-translation that hides a twin from the moment test leaves the pairs reading the fraction to three places.

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Order without repetition

Penrose tilings and quasicrystals — patterns with long-range order, five-fold symmetry, and no repeating cell at all.

How many of the four triple satellites the cube strengthens. Over 160 random sets of three incommensurate wavevectors, each kept only when its triple satellites stand clear of every other combination, the number of the four triple satellites at which the cubic term adds to the rest of the amplitude rather than taking away from it: 0 in 1, 1 in 10, 2 in 73, 3 in 73, 4 in 3. Two and three are commonest, and all four occurs.

No phase chooses which triples the cube strengthens

In a chain modulated by three waves, the cubic part of the size effect reaches the triply mixed satellites, and at most of them it arrives against the curvature's contribution rather than with it. The obvious hope was that shifting the waves' phases would turn the cancellations into reinforcements. It cannot: every path to a satellite carries the same phase factor, so the phases rotate the amplitude and change nothing. What decides the split is the wavevectors, and the sign of the cubic coefficient reverses it. Over random wavevectors the four triples split every possible way, two and two most often, all four together three times in 160.

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How it is known

A crystal's structure is read from the pattern it scatters. The reciprocal lattice, systematic absences, and what an experiment can and cannot see.

$Undoing a twin with too large a fraction pushes pairs past zero. 300 pairs of law-related reflections of a crystal twinned at 0.3, measured without error, after the twin has been undone by inverting its mixing: each pair's sum kept and its difference divided by 1 − 2a. With a = 0.3 (left) every pair returns to its untwinned intensities and all lie in the positive quadrant. With a = 0.35 (right) the differences are divided by too small a number and 73 pairs are stretched past an axis (red, in the shaded region): one member negative, which no intensity can be. Intensities in units of the mean.$

Undoing a twin shows only one of its two mistakes

A twin can be undone exactly: keep each pair's sum and divide its difference by 1 − 2α. Divide by too small a number and some intensities come out negative, in a share of pairs that rises in a straight line from the true fraction. Divide by too large a one and nothing looks wrong at all; what is left is a smaller twin, which only the pair test can see. Noise always pushes the estimate toward the side that cannot be seen, and the division multiplies every error, fiftyfold in counting time at α = 0.45.

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Order without repetition

Penrose tilings and quasicrystals — patterns with long-range order, five-fold symmetry, and no repeating cell at all.

An aperiodic chain made of periodic ones. The first 40 letters of the period-doubling chain, a → ab, b → aa, with b dark, and below it the same sites sorted into layers by the power of two in n + 1. Layer j repeats with period 2ʲ⁺¹ and carries a when j is even and b when j is odd. Each layer is perfectly periodic; the chain is their union, and since no single period contains every layer, the chain has no period at all.

An aperiodic chain whose peaks are all at rational positions

The period-doubling chain, a → ab and b → aa, never repeats, and it diffracts in sharp peaks. Where those peaks are is the surprise. Every one sits at a fraction m/2ʳ, every peak with the same denominator has the same height, (2/3)·2⁻ʳ, and together they carry all the intensity there is. The chain is a stack of periodic chains of period 2, 4, 8 and so on, and its diffraction is theirs. Unlike the Fibonacci chain, it is the slice of no lattice in any number of dimensions.

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The period-doubling chain with its deep layers randomised two ways. The first 48 letters of the period-doubling chain (b dark), and below it the same chain with every site at depth 2 or more redrawn at random: in the middle row each deep site independently, with b at the deep layers' own density, and in the bottom row one letter per deep layer, the same for every site of that layer. The deep sites, every 4th, are outlined; a dot marks a letter that differs from the perfect chain. The shallow layers are untouched in both. The middle row is no longer built of periodic layers; the bottom row still is.

A fault in the deep layers costs exactly the fine peaks

Redraw the period-doubling chain's deep layers at random, site by site, and every peak finer than the fault's depth vanishes while every coarser one stays exactly where and as high as it was — provided the random letters keep the layers' own density, which is a third below an even layer and two thirds below an odd one. The lost intensity, (2/9)·2⁻ʳ, comes back as a flat floor to the last digit. Redraw them once per layer instead and nothing is lost at all: every peak survives, with a height drawn by lottery.

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