Crystallography & Symmetry
patterns derived from their groups
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The theme: The lattice forbids — page 5

Periodicity is a strong constraint. It rules out five-fold rotations, most rotation orders above six, and a great many patterns that look perfectly reasonable until the arithmetic is done.
Whole stars of the rhombohedral zone, and the boxes that hold them. For eight wavevectors of the rhombohedral zone, the number of cells in the smallest periodic box holding one arm of the star, in the smallest holding every arm, and, where it costs more, in the better of the two regular grids: the rhombohedral cell repeated n times along each edge, or the hexagonal triple cell repeated n by n by m. The axis end T is held whole in two cells against six, the face centre F in four against eight, and the in-plane third K₃ in nine against twenty-seven; L needs eight either way. Every smallest box for a whole star keeps all twelve operations of 3̅m. The classification

A rhombohedral box saves a factor of two the hexagon never offered

A rhombohedral crystal is drawn on two cells, the small rhombohedron and the hexagonal cell three times its size, and a periodic box is usually one of them repeated. The smallest box that holds a whole star of wavevectors is the star's own lattice instead, and across every star with denominators up to eight it beats the better of the two grids by one, two or three, never more. Three was the hexagon's number. Two is new, and it is the face centre's. And the boxes that keep the three-fold axis and lose the mirrors still come in mirror pairs, now at nine cells as well.

 Which plane groups can be coloured with five, twenty-five, seven and forty-nine colours. For each of the seventeen plane groups, its colourings with 5, 25, 7 and 49 colours, as subgroups of that index. At p² colours the count splits into those whose translations meet the lattice in the whole plane pL and those that meet it in a cyclic sublattice. The five groups with a mirror across a rotation, shaded, have none at 5 or 7 and p² of them at 25 and 49, all on pL. In all, 104, 1449, 172 and 4197 colourings. The classification

Twenty-five colours, and the mirrors come back

With a prime number of colours from five up, five of the seventeen plane groups can never be coloured at all: the ones with a mirror across a rotation. With the square of a prime they all can. A colouring with twenty-five colours may meet the lattice in the whole plane modulo five, which every group keeps, and the colours then form a plane the point group acts on by its matrices. The designs come out as the prime's count again, with one more for each of the seventeen groups.

 The window in which A₄ and its dual are both minima. The exact second derivative of the Gaussian sum at A₄ and at A₄ along their five-dimensional family of changes of shape, for widths within six hundred-thousandths of one. A₄'s turns positive at t = 0.99998110 and A₄'s turns negative at 1.00001890, the reciprocal. Between them both are positive, so both lattices are local minima; at t = 1 itself both equal 8.59e-6. Lattices

The self-dual width is not flat

A search over four-dimensional lattices found that A₄ and its dual stop being traps for the Gaussian sum almost exactly at the self-dual width, and read that as a flat of the sum sitting on the width itself. Computed exactly rather than by finite differences, the curvature there is not zero. It is six parts in a million of the other curvature, positive, and the same for both lattices, because duality makes their Hessians one matrix at that width. Both are minima, in a window four parts in a hundred thousand wide, with basins a few millionths across.

 Every stacking reflection of a warm stack of sheets falls as a power of its size. The Bragg peak per site squared per layer squared of the stacking reflections (0, 0, l), l = 1 to 5, of a stack of flapping square-lattice sheets with bending stiffness 1 and coupling 0.1 between neighbouring layers, at kT/k₁ = 0.02, against the stack's width, with the number of layers growing as the width squared. Each order falls as a fixed power of the size, and the powers grow as the square of the order: 0.20, 0.80, 1.79 and 3.18 for the first four. The fifth falls as the fourth power, as fast as the crystal's volume grows, which is what a reflection with no peak at all does. How it is known

A stack lends each sheet a tension

A single warm sheet wanders so far out of its plane that every reflection with a component along its normal is lost. Stack the sheets and couple each to its neighbours, and inside the stack each sheet wanders only as a sheet under tension does, logarithmically. The stacking reflections survive as peaks that fall as powers of the crystal's size, and the power grows as the square of the order, so the fifth order of the stack computed here has no peak at all. A stack of finitely many sheets keeps even the first only out to a width growing as the square root of its thickness.

 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. Lattices

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.

 Every stratum a primitive cubic cell lies on, and the steps between them. The ten lattice strata a primitive cubic cell lies on, from triclinic, with no condition, to cubic, with five, each with the number of linear conditions it imposes on the cell's six parameters, joined wherever one adds conditions to the other with nothing between. There are 14 steps and 10 paths from the bottom to the top: nine of four steps, through a monoclinic, the orthorhombic and a tetragonal stratum, and one of two, through the rhombohedral. 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. 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.

 Which Laue groups have a sampling grid that touches no special point. Every Laue group on every Bravais lattice it occurs on, twenty-four pairs, with the smallest grid of wavevectors that keeps the whole group and has no point fixed by any operation but the identity. Such a grid exists for 1̅, 2/m, mmm and 4/m, at exactly as many points as the group has operations. It does not exist, as far as the search reaches, for 4/mmm or for any group with a three-fold axis: 3̅, 3̅m, 6/m, 6/mmm, m3̅ and m3̅m. 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.

 Where the four-fold screws of one crystal cross a plane. A space group of the body-centred class I4/m that has a centre of symmetry, seen along the direction of its four-fold axes: the points where the special lines along that direction cross a plane, over four cells of the projection. A rod cut along the large filled lines is p4₃, a four-fold screw of one hand; along the other large lines, p4₁, its mirror image. The crystal's centre of symmetry carries each kind onto the other, so both are there and neither hand is favoured. 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.

 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. 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.

 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. 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.

 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. 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.

 Copies of a plane group of itself, counted to index ninety-six. For eight of the seventeen plane groups, the number of subgroups isomorphic to the group, up to conjugacy, of index at most N, on logarithmic axes. The curves fall into four tiers by their slopes: groups with no operation but a half-turn grow as N², groups with mirrors and nothing of order three or more as N log N, groups with a rotation of order three or more and no mirror as N, and groups with both as √N. By index 96 p2 has 13446 copies and p6m 13. 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.

 Every square-symmetric grid of wavevectors has a point on a mirror. One cell of the square reciprocal lattice, centred on the zone centre, with the mirror lines of 4mm through it and three grids the four-fold rotation keeps with an offset off the axis. The quarter turn allows that offset at one place only, the second four-fold centre of the grid's own lattice, ringed. On an aligned grid of odd side the rotation fixes a point; on an aligned grid of even side the offset lies on a diagonal mirror; on a grid turned by 45 degrees it lies on an axial mirror. No grid escapes all three. Operations

The offset a quarter turn allows lies on a mirror

A census of sampling grids found free grids for the tetragonal group 4/m and none for 4/mmm, to a hundred and twenty-eight points, with no reason. The reason is two lines long. A quarter turn lets a symmetric grid be offset to exactly one place off its axis, the second four-fold centre of the grid's own lattice, and a square lattice that a mirror keeps can only be aligned with its mirrors or turned by half a right angle. Either way that centre lies on a mirror line.

 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. 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.

 Sublattices a space group's point group keeps, counted to index sixty-four. For six of the seventy-three arithmetic classes, the number of sublattices of ℤ³ of index at most N that every operation of the point group carries to itself, on logarithmic axes. The six curves have six slopes: N³ for a triclinic class, N² for a monoclinic one, N log² N for an orthorhombic one, N log N for a tetragonal class with no two-fold axis or mirror through its four-fold axis, N for one with them, and the cube root of N for a cubic class. By index 64 P1̅ keeps 176874 sublattices and Pm3̅m 9. 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.

 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.

 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. Into space

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.

 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.

 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. What a lattice forbids

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.

 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.

 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. How it is known

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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