Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Symmetry is decidable — page 16

Unusually for a physical science, the central questions here have exact answers computable in integer arithmetic. There is no tolerance to choose and no residual to interpret.
Which of the fourteen lattices is a limit of which. The fourteen Bravais types, with an edge from one type up to another when the higher is a limit of the lower and no third type lies between: nineteen edges, computed from the subgroups of each lattice's own group. Primitive cubic is reached through primitive tetragonal and the rhombohedral type; body-centred and face-centred cubic through body-centred tetragonal and the rhombohedral type; the hexagonal lattice only through the C-centred orthorhombic one. 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.

 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. What symmetry decides

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.

 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.

 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. What symmetry decides

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.

 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. What a lattice forbids

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.

 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. What symmetry decides

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.

 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. Operations

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.

 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. Into space

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.

 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. The classification

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.

 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. Operations

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.

 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. The classification

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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