Crystallography & Symmetry
patterns derived from their groups
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The theme: Exactly this many — page 13

Seven friezes, seventeen wallpapers, fourteen lattices, two hundred and thirty space groups. Each number is a theorem with a finite proof, and the word that matters in every one of them is 'exactly'.
Seven indices in 40, and one lattice at each. For every index to 40, how many sublattices of a cubic lattice there are and how many of them the full cubic point group carries to itself. The first number runs into the hundreds; the second is nought at almost every index and one at 1, 2, 4, 8, 16, 27, 32. A cubic point group is forty-eight conditions on a sublattice, and forty-eight conditions leave very little. The richest point group in three dimensions has the poorest arithmetic of copies, and the two are the same fact. Into space

The richest group has the poorest arithmetic

A plane group with no symmetry has seven hundred and sixty-two sublattices to grow by; one with a six-fold axis and mirrors has eight. Take the trend to its end in space and a cubic group has six to index forty — one at every cube, one at twice a cube, one at four times a cube, and nothing anywhere else. Every operation of a point group is a condition, and forty-eight conditions leave almost nothing.

 Six matrices and eight: a monoclinic normaliser modulo two. For P2/m and C2/m, every 3 × 3 matrix over the field of two elements that preserves the reductions of the two eigen-sublattices of the class's two-fold or mirror — six on the primitive lattice and eight on the centred one — each drawn as a tile of nine squares filled where the entry is one. The whole infinite normaliser reduces into this set, and a dot under each tile marks that some normaliser element with entries −1, 0 and 1 reduces to it. Every tile is realised, and the same holds in all six monoclinic classes. What a lattice forbids

The normaliser only acts modulo two

The space groups were counted as orbits of a small, finite set of changes of basis — only those with entries −1, 0 and 1 — out of normalisers that are infinite in the monoclinic classes. That the counts came out right was evidence, not proof. It becomes proof in two steps. A missing change of basis can only leave the count too high, never too low, so the published totals already force every class. And in the monoclinic classes the infinite normaliser acts on the cohomology only through its reduction modulo two, where the small set already reaches everything.

 The closure is a lattice exactly when the orders allow one. Twelve pairs of rotation orders, with a centre of each order placed one unit apart and the group they generate closed out to words of length 6. The linear parts reached are exactly the least common multiple of the two orders, every time — two rotations generate rotations, and the angles they generate are the multiples of the smaller of two fractions of a turn. A lattice admits rotations of order one, two, three, four and six and no others, so the closure can be a plane group exactly when that multiple is one of those five. The pairs where it is not are the pairs where the translations keep getting shorter. Operations

Closing the plane from two centres

Put two rotation centres down and close under composition: the result is a plane group or is not discrete, and nothing in between. What decides it is the least common multiple of the two orders, because two rotations generate rotations and the angles add — so the crystallographic restriction arrives as a condition on a closure rather than as one on a lattice.

 Which four-dimensional lattices carry which rotations of order twelve. Six four-dimensional lattices, the order of each one's symmetry group, and how many of its symmetries are of each of the three kinds of order twelve: the single block Φ₁₂, the third-turn with a quarter-turn Φ₃Φ₄, and the quarter-turn with a sixth-turn Φ₄Φ₆. D₄ has 96 of the first among 1,152 symmetries, the two hexagonal planes 24 among 288, and the hexagonal-and-square sum 4 of each of the other two among 96. No lattice has all three, and none has the first with either other. What a lattice forbids

No lattice carries all three twelves

Four-dimensional lattices admit three different operations of order twelve: one that turns a single irreducible block by a twelfth, and two that turn two planes, by a third and a quarter or by a quarter and a sixth. Computing the symmetry group of every kind of four-dimensional lattice that can have a maximal one shows that no lattice carries all three, and none carries the first with either of the others. D₄, the most symmetric of them, has only the first.

 A screw out of two rotations that have none. Four pairs of located rotations of space, composed, with the result read back as a screw: its angle, and its pitch, which is the part of its translation lying along its own axis. Axes that meet give a rotation and no translation at all, because the point where they meet is fixed by both. Parallel half-turns give a translation. Skew axes give a screw — a motion with a translation in it, out of two motions with none — and the translation is twice the distance between the two axes. Nothing in either factor moves anything along the product's axis, and the product does. Operations

The axis a product lies on

Two rotations of space about axes that do not meet compose to a screw — a motion with a translation in it, out of two that have none. The translation is twice the distance between the axes and the angle twice the angle between them, and the screw's own axis is not somewhere arbitrary: it lies on the two axes' common perpendicular, at a place the arithmetic gives.

 The eighty layers by what rolling does to their two faces. The eighty layer groups sorted by how the two tubes a sheet makes, one with each face outside, are related. Seventeen one-sided layers make two different tubes in every direction. Forty-six make the same tube in every direction, because they have a mirror or glide in the sheet or a centre of inversion. Sixteen relate the two tubes only along two, four or six special directions, and there as mirror images. One, the layer with a four-fold rotoinversion and nothing else that turns it over, exchanges its faces and still makes two different tubes in every direction. What a lattice forbids

A sheet with two faces rolls into two tubes

Roll a sheet whose two faces differ and every operation that turns it over dies, as expected: a tube's inside is not its outside. But the sheet can be rolled with either face out, and the operations that died are exactly the congruences between the two tubes that result. Over the eighty layer groups that settles, direction by direction, whether the two are one tube, a pair of mirror images or two different objects, and one layer exchanges its faces and still never rolls the same tube twice.

 The turns that keep the join discrete. Two copies of p4 on one square lattice, one turned against the other, with the shortest translation their union generates. At a turn whose cosine and sine are both rational the translations are a lattice, and its shortest vector is one over the square root of Σ — where Σ is the odd part of p² + q² for the rational point (p, q) — which the measurement reproduces to six places at every one tried. At a whole number of degrees other than a multiple of ninety there is no such point, and the search finds shorter translations the further it runs. Operations

Two patterns laid over one another

Compose two plane groups rather than two operations. The group generated by both is one of the seventeen or is not discrete at all, with nothing between — and it takes two conditions, one on the rotation orders and one on the turn between the lattices. The turns that work have rational cosines, which by a theorem of Niven's means none of them is a whole number of degrees.

 The fewest pentagon contacts, from twenty atoms to sixty. The least number of pairs of pentagons sharing a bond, over every cage of pentagons and hexagons at each size from 20 to 60 atoms, with two lower bounds: 30 − 3h from counting edges, and 20 − h from counting the pentagons round each hexagon, which is also the prediction of one fewer contact for each hexagon added. The measured values past 44 atoms are 8, 8, 7, 5, 5, 4, 4, 3, 0; the bound is met at 20, 24, 26, 28, 36, 38, 40, 42, 44, 50, 60 atoms and nowhere else. What a lattice forbids

The last three contacts go at once

The fewest pentagon contacts a cage can have fell by one for every hexagon added over eight sizes up to forty-four atoms, which pointed straight at the truncated icosahedron at sixty. Carried to sixty by a faster search, the fall does not continue evenly. It pauses, jumps, pauses again, and loses its last three contacts in a single step. The steady fall was not a trend. It was a lower bound, twenty minus the number of hexagons, which a short count round each hexagon proves, and the census now shows where cages meet it and where they cannot.

 One group without an axis, and as many as the order with one. For a rotation of each order that an integer matrix can have in a small dimension, the number of space groups its arithmetic class admits — computed from the cohomology rather than enumerated. A rotation acting on the smallest lattice that will hold it fixes no direction and admits exactly one group: the symmorphic one, with no screw. Add a direction it leaves alone and the count becomes the order of the rotation, and the extra groups are its screws. The four-fold with an axis gives four, which are P4, P4₁, P4₂ and P4₃; the five-fold with an axis gives five, in five dimensions, where no published table exists to check it against. The classification

The screw a dimension does not have

The extension count is a machine that runs in any dimension, and the seventeen were the case where every step could be checked against a list arrived at four other ways. Run on a cyclic point group it has a closed form two lines long — and it says a five-fold screw axis does not exist in four dimensions, which is a prediction rather than a check.

 One sheet with two faces rolled into two scrolls. The cross-section of a sheet with different atoms on its two faces, rolled three turns into an Archimedean spiral, once with face A outward and once with face B outward. Unlike a tube, a scroll has an inner edge and an outer edge and a radius that grows with every turn, so an isometry of space can neither turn it about its axis nor reverse the direction along the spiral: it can only slide it along the axis, reflect it across the axis, or reflect the spiral itself. What a lattice forbids

A scroll cannot be turned end for end

A sheet with two different faces rolls into two tubes, face A out and face B out, and for forty-six of the eighty layer groups the two tubes are one object whichever way the sheet is rolled. Roll the same sheets into scrolls, spirals with an inner edge and an outer one, and only seventeen keep that. A spiral cannot be turned about its axis or run backwards, so the half-turns and screws that joined the tubes have nothing to act on. What survives is a mirror lying in the sheet, and a two-fold axis lying along the roll.

 Tight where there is an axis and vacuous where there is not. The order of a point group annihilates its cohomology, so every intrinsic translation is a multiple of one over that order — the bound every account of the subject quotes. What occurs is one over the exponent of the cohomology, which divides the bound. For a rotation with a direction it fixes the two agree exactly: a four-fold screw does need quarters and a six-fold sixths. For a rotation acting with no fixed direction the exponent is one — the cohomology is trivial and no fraction occurs at all — so the bound is slack by the whole order. The same bound is sharp and useless in the same table. The classification

The denominator a group actually needs

The order of a point group annihilates its cohomology, so every intrinsic translation is a multiple of one over that order — a bound every account of the subject quotes. What occurs is one over the exponent, which divides it. The same bound turns out to be attained exactly and to be slack by its whole size, in two rows of one table, and what decides which is whether the rotation fixes a direction.

 Every finite normaliser of a lattice's group, found whole. The fourteen Bravais lattices on primitive bases, with the number of primitive vectors along the axes of their rotations, the order of the lattice's group, and the order of its normaliser in the integer matrices, found by trying every assignment of axis vectors to three independent ones. On the three lattices up to monoclinic the axis vectors lie on one line or none and the normaliser is infinite. On the other eleven it is finite (48, 16, 48, 48, 16, 16, 24, 12, 48, 48, 48), its largest entry is one, and it is exactly the set of matrices with entries −1, 0 and 1 that the space-group census used. What a lattice forbids

A normaliser must permute the axes

The space groups were counted with a small set of changes of basis, and past the monoclinic system nothing had shown that set was the whole normaliser. It is, and the reason is geometric: a change of basis that carries a lattice's group to itself must carry each rotation axis to a rotation axis, so it permutes a finite set of vectors, and past the monoclinic system three of them pin it down. Every such permutation tried, the eleven finite normalisers turn out to be exactly the matrices with entries −1, 0 and 1, and 219 and 230 stand without any table behind them.

 The sign turns when the cross terms go weak. Every quartet among the strong reflections of a small structure, sorted by the mean of its three cross terms, with the mean cosine of the phase sum in each bin. Where the cross terms are strong the quartet behaves like a triplet and the cosine is near one. Where they are weakest it is -0.698 — negative — and 93 per cent of those quartets have a cosine below nought. The four reflections of the quartet are equally strong in every bin; what changes is three reflections that are not in the sum at all. How it is known

The relation that can say no

Every phase relation before this one pushes a sum towards zero, so none of them can contradict another — a phase set satisfying all of them badly is still satisfying them in the same direction. A quartet can be estimated at π instead, and it is when its three cross terms are weak, so the information arrives from the reflections nobody would have thought worth measuring.

 An equality, with a known factor in it. Sayre's identity says the structure factor is proportional to the convolution of the structure factors with themselves — an equality, not a probability — with a factor that depends on the atoms and the resolution and on nothing else. Computed for a structure of Gaussian atoms, the ratio of the two sides falls with resolution exactly as predicted: the logarithm of the ratio is a straight line in the squared index, its slope is minus half the width in the atomic factor, and the ratio is very nearly real and positive, so the identity relates the phases and not only the amplitudes. How it is known

The relation that is an equality

Every relation of this kind so far is a probability — right nine times in ten, useless applied once and decisive applied ten thousand times. One is not. For a structure of equal, resolved atoms the squared density has peaks in the same places, so every structure factor is exactly a convolution of all the others, with a factor that depends only on the atom. It is exact, and on its own it is useless.

 How often a broken vertex carries four. Of all the broken vertices in all the arrangements with a given number of them, the fraction whose four arrows all point the same way — on tori four, five and six vertices across. If the vertices were independent that fraction would be one in five at every count. With only two defects it is far lower, 0.085, 0.075 and 0.066 on the three tori, and falling with the size; as the count rises it climbs, oscillating, to one in five and stays there. The odd counts sit above the even ones because an odd number of defects cannot be made of twos alone — three defects must hold exactly one four, which is a share of one third. Order without repetition

A four is not two twos

A vertex of square ice can break the rule with three arrows one way or with all four, and a census that counts the two apart finds them behaving as charges of two and four should: the fours are rare while defects are few, a pair of fours is squeezed about four times as hard as a pair of twos, and the dip an earlier census reported turns out to be nothing but the first count that cannot be made without a four.

 Pentagon contacts as pieces of the icosahedron. The icosahedron drawn with one vertex at the centre, two rings of five, and the twelfth vertex as the outer circle; each vertex is a pentagon and each edge a possible contact. Left: the dodecahedron, whose twelve pentagons all touch five others, uses every edge. Centre: the commonest contact graph among the 5,770 cages to sixty atoms, 6 contacts in clusters of 3 + 3 + 2 + 2 + 1 + 1, carried by 179 cages, placed on the icosahedron with every contact on an edge. Right: the one cage with no contacts at all. All but two of the 340 contact graphs can be placed this way. What a lattice forbids

Three hundred and forty ways for twelve pentagons to touch

Every cage of pentagons and hexagons has twelve pentagons, and which of them share a bond is a graph on twelve points. Over all 5,770 cages from twenty atoms to sixty there are only 340 such graphs, and the graph almost never tells one cage from another: the commonest is shared by 179 cages and one recurs at thirteen different sizes. And every one of them except two, both from small cages, fits inside a single graph, the icosahedron's, which is what the dodecahedron's twelve pentagons make when every one touches five others.

 Every arrangement on two tori, by how it winds. The ice arrangements on tori eight and nine vertices across, each placed at the pair of numbers it winds by — how much the height on the faces gains going once round the torus across and once round it down — with a disc whose area grows with the number of arrangements there. On the even torus every winding is even and the arrangements crowd the centre; on the odd torus every winding is odd and there is no centre to crowd, so they sit at plus and minus one. Solid discs are the sectors whose windings are both multiples of three, which are exactly the arrangements that are also three-colourings. On the even torus that is exactly a third of everything; on the odd torus it is the four sectors at plus and minus three, and they hold about one arrangement in eighty. Order without repetition

A winding does not dilute

Three-colourings of an odd torus are a small and shrinking fraction of the ice arrangements on it, and the natural guess is that the fraction goes to nothing. It does not. It settles near one in twenty-eight, because the windings that decide it are a property of the whole surface rather than of its edge — and on even tori the same windings put exactly a third of the arrangements in one class, which in the limit is Jacobi's theta identity.

 Twelve kinds of operation six dimensions adds, and the lattices that carry them. The twelve rational kinds of operation whose orders first appear in five or six dimensions — 7, 9, 14, 15, 18, 20 (two kinds), 24 (two) and 30 (three) — against six lattices whose symmetry groups were computed element by element. Every kind is carried by at least one. All three kinds of order thirty are carried by A₄ ⊕ A₂, forty-eight symmetries each; the single-block kinds Φ₇ and Φ₁₄ only by A₆, and Φ₉ and Φ₁₈ by E₆ and by three hexagonal planes. What a lattice forbids

The thirties share a lattice

Four dimensions has three kinds of operation of order twelve and no lattice carries all three. Six dimensions has three kinds of order thirty, and one lattice carries every one of them, forty-eight times each. The difference is not the order. One of the twelves is a single block that cannot be split, and no thirty is: thirty needs eight dimensions to be irreducible, so in six it is always a five-fold piece beside a three-fold one, and the lattice that is both pieces holds every combination.

 How many locked phases an experiment of a given resolution can see. For three convex repulsions, the number of steps of the staircase wider than a given share of the chemical-potential axis, as that share is made smaller by factors of ten. A resolution of one part in a million shows 199 steps for the short exponential, 829 for the long one and 41 for the inverse sixth power. Every count is exactly the number of fractions with denominator up to some bound, because every step with one denominator has one width. The short exponential and the inverse power cross: the power law shows fewer steps at every practical resolution and more at one part in a hundred thousand million. Order without repetition

An experiment sees a Farey sequence

A devil's staircase has a step at every fraction, and no experiment sees them all. Which ones it sees is not a matter of luck: every step with the same denominator has the same width, so a finite resolution shows every fraction up to one denominator and nothing beyond it — a Farey sequence, whose length is a sum of Euler's totient. How that denominator grows with the resolution is the one thing the interaction decides, and what is left unseen is steps, never an incommensurate phase.

 A structure that is its own negative, and one that is not. Two arrangements of two species on a four-by-four block with p4m symmetry, each with eight sites of each species, beside the arrangement that exchanging the species produces. On the left the exchanged arrangement is the original one moved by an operation of the group, so the two are one structure: it is self-complementary, and its balance of species is forced by a symmetry. On the right the exchanged arrangement is a genuinely different structure, and the balance is a coincidence of count. Counting structures with the species interchangeable merges the pair on the right into one and leaves the one on the left as it was, which is why the count does not halve. Operations

Swapping the species does not halve the count

Let the two species on a lattice be interchangeable and every structure pairs with its negative, so the count should halve. It does not, because some structures are their own negative: an operation of the group carries them onto the arrangement with the species exchanged. Those are counted by the operations whose every cycle is even, they all sit at exactly half composition, and read as spins they are the arrangements whose zero magnetisation a symmetry enforces.

 Mirrors everywhere in a chain with none. Thirty-four tiles of the Fibonacci chain, long tiles L and short tiles S, with an arc over every window of at least 5 tiles that reads the same backwards, drawn from its own centre, and a tick at the centre where its mirror stands. The chain as a whole has no mirror, since no palindrome in it is infinite, yet mirrors of every size stand all along it, the long ones spaced further apart than the short ones, and every one of them a local symmetry the chain lacks globally. Order without repetition

A chain with no mirror has mirrors everywhere

The Fibonacci chain has no mirror: nowhere can it be reflected onto itself. Yet it holds exactly one window of every even length that reads the same backwards and exactly two of every odd length, each recurring all along the chain, and no aperiodic chain that is not of its kind manages that count. Every letter of it creates a palindrome never seen before, and a mirror of any size is never further away than a fixed multiple of that size.

 Three rules for 'square' are three shapes, and the cell falls between them. A measured cell placed by how far it is from each of the two conditions a square cell needs — equal edges across, a right angle up — each in its own standard uncertainties, so the square cells are the origin. Three rules for accepting the cell as square at the five per cent level are three regions. The two-condition χ² accepts inside the larger circle, of radius 2.45; testing each condition separately accepts inside the square of half-side 1.96; reading the distance as if it were one condition accepts inside the smaller circle, of radius 1.96. The measured cell, 2.22 and 0.75 from the two conditions, is inside the first region and outside the other two. Operations

Two conditions are not one condition twice

Whether a plane structure has a higher-symmetry supergroup is decided by its measured cell, and the obvious measure — how many standard uncertainties the cell sits from a right angle, or from equal edges — cannot rank the candidates on its own. A square needs two conditions and a rectangle one, so a single distance means different things on each; counted properly, a square can score above the equal-edged lattices that contain it, and walking up the diagram one condition at a time is the reading that keeps the ranking coherent, at a price that has to be paid in its error rate.

 Three mirrors of a 30–30–120 triangle, closed. A triangle with angles of thirty, thirty and a hundred and twenty degrees, shaded, and every image of it under the group its three side-mirrors generate, within a window. The images alternate between the triangle and its mirror image, and they tile the plane — so the closure is discrete. But the group it produces has a mirror along the triangle's own altitude, drawn dashed, that is none of the three given: the triangle is two copies of the thirty–sixty–ninety triangle, and the group is that triangle's, p6m. The closure found a mirror the generators did not name. Operations

Four groups are made of mirrors

Close the plane from located mirrors instead of rotation centres and the closure is discrete only for four triangles of mirrors — and one of those four manufactures a mirror along its own altitude. Of the ten plane groups that contain a reflection, exactly four are generated by their mirrors; three hold a kaleidoscope at index two, three and four, which are the numbers in front of the star in their orbifold symbols; and three have mirrors in one direction only, which generate nothing along themselves.

 One invariant for every invertible chain. The quantity x² + y² + z² − 2xyz − 1, built from the half-traces of the transfer matrices of the two words a substitution produces at each level and of their product, for six substitutions at one energy, logarithmic. For the golden, silver and bronze chains and for the golden rule reversed it is V²/4 = 0.09 at every level — the same number for all four, computed in exact integers. For period-doubling and Thue–Morse it wanders by orders of magnitude from one level to the next. Order without repetition

One invariant for every chain that can be undone

The Fibonacci chain's transfer matrices obey a recursion with a quantity it cannot change, and the natural guess is that every quasiperiodic chain has its own. It does not. The conserved quantity is the trace of the commutator of the two tiles' matrices, so every chain a reversible substitution builds from the same two tiles — golden, silver, bronze — conserves the same number, exactly, at every length and every energy. Period-doubling and Thue–Morse, whose rules cannot be undone, conserve nothing.

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