The collection

Every essay — page 22

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

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 first order of stress that can magnetise each magnetic class. The ninety magnetic point groups that are not grey, in four columns by the lowest power of a stress that can give them a magnetisation: 31 have one at rest, 35 acquire one in proportion to the stress, 3 only in proportion to its square, and 21 under no power of it at all. The three that need the square are the ordinary cubic classes 432, 4̅3m and m3̅m, and the twenty-one that no stress magnetises are exactly the classes holding the inversion combined with time reversal.

A squeeze magnetises a cubic crystal across itself

Three ordinary cubic classes forbid a magnetisation proportional to a stress. At the square of the stress they each admit exactly one constant, and nothing else joins, at that order or any higher: sixty-nine classes is the final list. The one constant has a shape no table of coefficients shows. A squeeze along any direction magnetises the crystal at right angles to the squeeze, and along each of the cube's twenty-six rotation axes it magnetises nothing, which is the least a field on a sphere is allowed to vanish.

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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.

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.

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.

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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.

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.

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What symmetry decides

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.

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.

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Where the two-fold axis sits decides what the coloured optic axes keep. Interference-figure positions of the two optic axes from 400 to 700 nm for four model crystals seen down the acute bisectrix. Orthorhombic: both mirrors survive and only the angle between the axes changes. Two-fold axis along the optic normal: the axes stay on one line and slide unequally (inclined dispersion), one mirror along that line. Along the obtuse bisectrix: the axial plane stays parallel and shifts (horizontal), one mirror across it. Along the acute bisectrix: the axial plane turns about the centre (crossed), a half-turn and no mirror.

Three places for the two-fold axis

A monoclinic crystal fixes one direction of its indicatrix, the two-fold axis, and an ellipsoid has three principal directions for it to be. Which one it is decides what an interference figure keeps in white light: a single mirror along the plane of the optic axes, a single mirror across it, or a half-turn and no mirror at all. Mineralogy calls the three inclined, horizontal and crossed dispersion. They are one subgroup computed three times, and in light of a single colour all three disappear.

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What a lattice forbids

Only two-, three-, four- and six-fold rotations are compatible with periodicity. The proof takes one line and the exception took a Nobel Prize.

Seventeen ways to have order thirty in eight dimensions, and four on E₈. Every product of cyclotomic polynomials of total degree eight whose order is thirty: 17 of them. E₈'s symmetry group holds 4: the single block Φ₃₀ and three split kinds, each with a one-dimensional piece of each sign or a doubled minus one. The other thirteen fit the dimension and have no element on this lattice.

Thirty as one block lives on two lattices

Eight dimensions is where an operation of order thirty can finally be a single irreducible block. It was predicted to sit alone on E₈ with the split kinds elsewhere, as the twelves do in four dimensions. It does not: E₈ holds three split thirties beside it, and exactly three of the sixteen that fit, because a five-fold block leaves nothing but a permutation of five things. The block also lives on a second lattice, A₂ ⊗ A₄, and there it really is alone.

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What symmetry decides

Down one optic axis, at three colours, for two crystals. Interference figures of two model monoclinic crystals, inclined above and crossed below, each a plate cut across the optic axis at the sodium line, at 450, 589 and 650 nm. In the inclined crystal the melatope slides along the horizontal mirror and the figure stays symmetric across it. In the crossed crystal the melatope leaves the line away from sodium, and the pattern loses the mirror.

The mirror a plate keeps in one colour

Cut a plate of a crystal across one optic axis, and in the light the plate was cut for its interference figure has a perfect mirror, whatever the crystal's class. Change the colour and the mirror survives only if the crystal really has one. Computed ring by ring, five model crystals give the prediction exactly, including the part the eye would miss: the mirror does not come back when the line is moved to follow the melatope.

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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.

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.

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.

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What a lattice forbids

Only two-, three-, four- and six-fold rotations are compatible with periodicity. The proof takes one line and the exception took a Nobel Prize.

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.

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.

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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.

Eight of the thirty-two classes allow a gradient term at the zone centre. The thirty-two crystal classes, marked by whether any order parameter at the centre of the zone has a term linear in its gradient. In the plane none does. In space eight do: 3, 4, 6, 32, 422, 622, 23 and 432. All eight are among the eleven classes with no operation that reverses hand, and the other three, 1, 2 and 222, have only one-dimensional order parameters.

A screw keeps a line and a glide keeps a plane

The census of the seventeen zones left two predictions for space. A transition at the zone boundary should meet a gradient term where one screw or glide is active, along that operation's direction, and a second should remove it. Across all 219 types, the first half holds without exception in the monoclinic groups and the direction is wrong: a glide's term fills its plane. The second half fails from the orthorhombic groups on. And the zone centre, empty in the plane, is not empty in space.

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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.

The count of reflected hats is never one out

In a hat tiling one tile in about eight is reflected, and that ratio was only ever quoted here. Four clusters and a table of small integers give it exactly: seven plus three root five over two, the fourth power of the golden ratio. The same table says something the ratio does not. Grow a cluster to two trillion hats and its reflected count is still within half a tile of the exact share, where the same share chosen at random would be out by nearly half a million.

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Operations

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

1,694 positions, 1,182 sets on the general cell, and 1,128 on the best. Every affine type of space group by crystal system: its Wyckoff positions, the sets those fall into under the normaliser on the cell the group requires, on the best cell it may sit on, and under the whole affine normaliser. The total is 1694 positions, 1731 with enantiomorphs counted twice, 1182 sets on the general cell and 1128 on the best, which equals the affine 1128 in every system. The two columns differ only in the orthorhombic system and in one monoclinic type.

One cell reaches every Wyckoff set

In the plane, every identification the affine normaliser makes between two positions is made by an isometry of some cell, and the essay that found this warned against carrying it into space, where changes of basis can mix axes without limit. It carries. The 1,731 positions of the 230 groups were found from stabilisers and the sets from normalisers, and on one cell for each type the two partitions agree. Past the monoclinic system that is a one-line average. Below it, the infinite normaliser turns out to have nothing left to join.

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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.

530 printed rows, 499 descriptions, by crystal system. For each crystal system: the number of space group types, the number of distinct descriptions up to an origin shift, the number of rows in the printed list, and the difference. All thirty-one extra rows are monoclinic or orthorhombic.

Five hundred and thirty rows, four hundred and ninety-nine descriptions

The International Tables list 530 settings of the 230 space groups, and that number has long been quoted as a measure of how much naming a group can take. Counted from the operations, a setting being a description up to a shift of origin, the answer is 499. Thirty-one printed rows name a description already on the list, all of them in six types, and the rule that gives a group its second origin finds twenty-three of the twenty-four and misses one.

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The classification

Exactly seventeen ways to fill the plane with a repeating pattern. Not a fact to memorise: a theorem, with a finite argument and no eighteenth case.

Two positions in three can be lifted, and more than one in three can lose their cell. All 1694 Wyckoff positions of the 219 affine types by crystal system. On 1132 a motif with more symmetry than its site can raise the group of the crystal, and on 606 of those, across 126 types, some such motif makes the true cell smaller than the group's.

In space a motif can lift two positions in three

A molecule with more symmetry than its site can raise the group of the whole crystal, or not, and which one is decided by the site. In the plane twenty-nine positions of seventy-two allow it. In space it is 1,132 of 1,694, and 606 of them allow a lift that shrinks the cell, the case a structure solution meets as a warning. General positions can be lifted only in polar groups, as predicted, but eleven polar groups refuse: a mirror turns their screws over.

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Operations

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

The subgroups of index two, and what joins them. Every affine type of space group by crystal system: its subgroups of index two, 1813 in all; classes under the origin shifts alone, 1392; classes under the whole affine normaliser, 1159; and the fewest classes one cell can give, 1161. Best cell and affine agree in every system except the triclinic.

No cell joins the seven halves

Every identification the normaliser makes between Wyckoff positions is made by an isometry of some cell, and it was natural to expect the same of subgroups. It nearly is. Across all 219 types and their 1,813 subgroups of index two, one cell does everything the normaliser does, except in the two triclinic groups. P1 has seven ways to be halved, and its normaliser makes them all one. No cell can, because seven is the number the crystallographic restriction forbids.

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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.

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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The classification

Exactly seventeen ways to fill the plane with a repeating pattern. Not a fact to memorise: a theorem, with a finite argument and no eighteenth case.

One tile, two lattices. The smallest polyomino on exactly two lattices with no slide between them, of 5 cells. On the left its copies sit on one lattice and on the right on another, and neither tiling can be turned into the other by moving rows.

One tile, two lattices, and a row that slides

A crystal chooses its lattice, and it is natural to think a tile does too. Among all polyominoes to thirteen cells, more than five hundred that are not rectangles lie on more than one lattice, and some on seven. They fall into at most two families, and a family is a row of copies that can slide against the next row, which is exactly what makes polytypes possible in a real crystal. A tile with a prime number of cells has no such row unless it is a bar, and so it lies on at most two lattices.

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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.

Completeness against sweep, for four spindles. Completeness against rotation range for one tetragonal crystal, class 4/mmm, on four different spindles. The four-fold spindle rises fastest at first, then stops at 88.9 per cent at forty degrees and never improves. The two-fold spindles reach completion near 87 degrees. The general spindle is highest at forty-five degrees and reaches completion at the same 87.

The same measurements, and a hole that stays

How much of the sphere must be swept for a given redundancy? The question has two answers, because it has two parts. The mean number of observations depends only on the sweep, to within two per cent whichever axis the crystal turns on. Completeness depends entirely on the axis, and on one kind of axis no sweep ever finishes: when the Laue group keeps the spindle's line, every copy of a blind reflection is blind too, and fifteen of a tetragonal crystal's 135 reflections are never measured.

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