Crystallography & Symmetry
patterns derived from their groups
Series

Centring — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. One lattice, two cells, and the absences the choice creates. The same set of points described on a centred rectangular cell and on its primitive rhombic cell, with what the centred description scatters. Half the reflections vanish, and they vanish because of how the cell was drawn rather than because of anything the crystal does.

    Centring, and why cm is not pm

    A centred cell has a lattice point in the middle and twice the area it needs, and crystallography prefers it anyway. The preference has a price, and the price is paid in reflections that vanish for reasons that have nothing to do with the crystal.

    part 1 · lattices
  2. Centring the five lattices. Each of the five plane lattices with the midpoint of every cell added, and the type of lattice that results — read off the reduced basis of the new point set rather than looked up. Every centring halves the cell area, so the original lattice is a sublattice of index two in the centred one, and every centred lattice is again one of the five. Two of the five come back as themselves and are therefore no richer for being centred. The rectangular and rhombic lattices exchange, which is what makes them one family under two descriptions. And the hexagonal lattice centred is rectangular — its holohedry falls from 12 to 4, so centring destroys the symmetry it was meant to display.

    Centring, counted as a sublattice

    Adding the centre of every cell to a lattice produces another lattice, containing the first with index two. Doing it to each of the five in turn shows why the list is five rather than ten, and why only one of the five has a centred description worth keeping.

    part 2 · lattices
  3. One lattice, two cells, and the absences the choice creates. The same set of points described on a centred rectangular cell and on its primitive rhombic cell, with what the centred description scatters. Half the reflections vanish, and they vanish because of how the cell was drawn rather than because of anything the crystal does.

    Why the bigger cell wins

    A centred cell has twice the area it needs and crystallography prefers it anyway. The preference is not conservatism — it buys operations that read as whole numbers along the axes, and the price is a set of reflections that vanish for reasons having nothing to do with the crystal.

    part 3 · lattices
  4. The centrings each holohedry allows, prime by prime. For each of the seven holohedries on its conventional lattice and each prime from 2 to 7, how many of the lines and planes of the lattice reduced modulo that prime the holohedry leaves invariant, and which of them give a lattice other than the one being centred. At 2 they give C for the monoclinic system, C, I and F for the orthorhombic, I for the tetragonal and I and F for the cubic; at 3 they give the rhombohedral lattice from the hexagonal one; at 5 and 7 nothing new.

    Only two and three can centre a lattice

    The centring letters P, A, B, C, I, F and R are usually a list brought to the Bravais lattices. They can be derived instead: a centring is a subspace of the lattice reduced modulo a prime, and a centring that keeps a crystal system's symmetry is a subspace its holohedry leaves invariant. Enumerating every one for the seven holohedries and the primes up to seven gives exactly the fourteen, with new lattices only at two and three. The reason is the crystallographic restriction, one step removed: every holohedry has an order made of twos and threes.

    part 4 · lattices
  5. The centrings six four-dimensional lattice groups allow, prime by prime. For each of the six four-dimensional lattices whose symmetry groups are maximal, with the group's order, and each prime from 2 to 7: the subspaces of the lattice modulo that prime the group keeps, sorted by what centring by each one makes. New Bravais types appear only at 2, three of them, all over the cube-and-line lattice, and at 5, where A₄ centred by fifths is A₄*. Both centrings of the hypercubic lattice by halves make D₄, which has three times the symmetry.

    The body-centred hypercube is the checkerboard lattice

    In three dimensions the centring letters come out as the subspaces a holohedry keeps modulo a prime, and only two and three make new lattices. In four dimensions the same census, run on the six lattices whose symmetry groups are maximal, finds a prime that the three-dimensional argument never could: five centres A₄ into its dual. It also finds that the most familiar centring does not survive the extra dimension. The body-centred hypercubic lattice is not a centred hypercube at all: its body centre is exactly as far away as an edge, and it is D₄, with three times the symmetry.

    part 5 · lattices
  6. Every four-dimensional lattice with a five-fold rotation, on one interval. The lattices with a five-fold rotation in four dimensions are one ring, ℤ[ζ₅], with the form Tr(a·x·x̄), and the number t = log(a/σ(a)) / 2 log φ, folded into the interval from 0 to 1, says which lattice a is. The dots are the members with a = u + v√5 for u up to 9, 8 distinct lattices. The two ends are A₄* and A₄, with 240 symmetries each; every point between has 20. The arcs join each member to the one a centring by fifths makes of it, its reflection t ↦ 1 − t.

    A five-fold lattice has one free number

    In four dimensions a lattice can have a five-fold rotation, and A₄ and its dual both do. Every other lattice with one turns out to be the same ring, the integers of the fifth roots of unity, with a single number deciding its shape. Folded onto an interval, the number runs from A₄* at one end to A₄ at the other, and every lattice strictly between has twenty symmetries. Centring by fifths does not make a new lattice. It reflects the interval end to end, exchanging A₄ with A₄* and pairing every other lattice with a partner, and no lattice sits at the middle to be its own.

    part 6 · lattices
  7. 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.

    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.

    part 7 · lattices
  8. 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.

    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.

    part 8 · lattices
  9. 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.

    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.

    part 9 · lattices
  10. 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.

    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.

    part 10 · lattices
  11. 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.

    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.

    part 11 · lattices

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