Crystallography & Symmetry
patterns derived from their groups
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The theme: The lattice forbids — page 3

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.
How densely each shape packs, by translation alone. The densest lattice packing of each shape, as its area over the critical determinant of its difference body. The two that tile the plane by translation reach one and must, which is a check on the search rather than a result of it. The triangle reaches exactly two thirds because its difference body is a hexagon. The many-sided approximation to a circle reaches π/√12, which this collection computes a completely different way. And the pentagon is the worst of them, which is where the search is doing work nobody could do by inspection. Symmetry at work

The densest packing of a shape that is not a disc

Which lattice packs equal discs most densely has a proof that finishes. Replace the disc with a pentagon and the same question has no closed form, but it does have a reduction: translates overlap exactly when the difference of their positions lies inside the shape minus itself, so the question becomes the smallest determinant a lattice can have while avoiding one convex body — and that is a search with a resolution attached.

 The sphere fixes a count; the torus fixes only a difference. Euler's relation for a trivalent net gives Σ (6 − n) pₙ = 6χ, so the surface fixes one linear combination of the face counts and nothing else. On a sphere that combination is twelve, which with no face smaller than a pentagon forces exactly twelve pentagons. On a torus it is zero, which permits any number of pentagons provided as many heptagons pay for them — and permits none at all, which is the plain hexagonal net. On a surface of two holes it is minus twelve, so heptagons become compulsory instead. What a lattice forbids

As many heptagons as pentagons

A trivalent net on a sphere must have exactly twelve pentagons. The same three lines of arithmetic on a torus give zero — which does not forbid pentagons, it makes them pay: every pentagon has to be balanced by a heptagon, and the counts are otherwise free. One rotated bond in a wrapped honeycomb makes two of each and changes nothing else.

 The parity argument loses 36 pairs it had won alone. The argument that refutes ten of the twenty-one species walks round a polygon of odd size: the ring of polygons about it is a closed walk of odd length in a graph the species decides, and a bipartite graph has no such walk. With two species at a vertex the flanking pairs come from the union of two graphs, and a union of bipartite graphs need not be bipartite — so the walk stops being constrained. The fourth row is the cost: pairs whose members the argument kills on their own and which it cannot kill together. The classification

The argument that closes eleven

Twenty-one vertex species satisfy the angle equation; a parity argument kills ten before anything is drawn, and the eleven survivors are all built. Asking the same question of tilings with two kinds of vertex, the parity argument evaporates — it constrains a walk in a graph one species decides, and two species decide the union of two graphs, which need not be bipartite. What is left is a search, and a search cannot close a count.

 The cross, from the selection rule alone. The layer lines of a helix with the first maximum of each marked on both sides. Nothing here is a picture of a photograph: each mark is at the radius where the Bessel function of the lowest order the selection rule permits on that layer line first peaks, and that radius is proportional to the order. The order rises by one per layer line until the middle of the repeat, so the maxima lie on two straight lines through the origin — the X — and the larger marks are the layer lines that reach the axis. The classification

What a thread scatters

A helix with ten subunits in a turn is not a screw axis a crystal may have, and nothing about its diffraction pattern is lawless. The pattern lies on layer lines, and on each one only certain angular orders may contribute — a selection rule as hard as any extinction condition. The lowest permitted order rises by one per layer line, a Bessel function of order n does nothing until its argument is about n, and the maxima therefore lie on two straight lines through the origin.

 The same terms, added in two shapes. Partial sums of the alternating 1/r sum over the simple cubic lattice, taken over expanding cubes and over expanding spheres. The terms are identical and only the order differs. The cubes creep towards 1.747565 — 1.7258 by the last point drawn — and the spheres do not settle at all, landing at -3.527 after passing through values on both sides of it. A conditionally convergent sum has no value until the order is named. Lattices

The sum whose answer depends on the shape

Give the points of a cubic lattice alternating signs and add up one over the distance. Added over expanding cubes the total creeps towards 1.747565; added over expanding spheres it does not converge at all, landing on both sides of that number and never settling. The terms are identical and only the order differs. Splitting the sum in two with the theta transformation gives it a value — ten decimal places from a few thousand terms.

 Three variants, and not one undistorted plane. The three tetragonal variants a cubic parent produces, with the principal stretches of each. Every one of them has the same three numbers in a different order, and the middle one is not one — so none of the three leaves any plane undistorted, and none of them can meet the parent phase across an interface. That is the difficulty the whole of the crystallographic theory of martensite exists to resolve, and it is visible in one column. Symmetry at work

The plane a deformation leaves alone

Two differently deformed regions can meet across a plane only if that plane is deformed identically from both sides — which forces the two deformations to differ by a rank-one term. Multiplying each side by its own transpose removes the rotation and leaves a condition on a signature: one positive eigenvalue, one negative, one exactly zero. In that form the classical rule that the middle principal stretch must be one is not quoted but derived, and it says that no single variant of a cubic-to-tetragonal transition can meet its parent at all.

 Why a tetrahedron is not a cube cut up. The two invariants side by side. A cube's twelve right angles are each a rational part of a turn and contribute nothing; a regular tetrahedron's six edges each contribute one α, giving six. Cutting a polyhedron and rearranging the pieces cannot change the invariant, so no dissection takes one to the other however the volumes are matched. That is Hilbert's third problem, and the whole of it is one angle. The classification

The angle that is not a fraction of a turn

Any two polygons of equal area can be cut into pieces that rearrange into each other. In space that fails, and the obstruction is a sum over edges of length against dihedral angle — zero for anything that fills space, and not zero for a regular tetrahedron. The whole argument reduces to one claim about one angle, and that claim is an integer computation: a sequence that is never divisible by three, when it would have to be.

 The five, as generator counts. Each of the five convex bodies that tile space by translation, built as the set of combinations of a handful of vectors with coefficients between zero and one. Three generators give a cube, four give either a hexagonal prism or a rhombic dodecahedron depending on whether three of them are coplanar, five give the elongated dodecahedron and six the truncated octahedron. The last column is what the same number of generators would give in general position, and the shortfall is the number of faces lost to coplanarity. The classification

Every parallelohedron is a shadow of a cube

Take a few vectors and form every combination of them with coefficients between zero and one. All five of the convex bodies that tile space by translation come out of that recipe, from three vectors, four, four, five and six — and since the recipe is exactly the image of a cube of that many dimensions, the truncated octahedron is a three-dimensional shadow of a six-dimensional cube. The five are not the generic answers: they are the degenerate ones, and the degeneracy is what the tiling demands.

 Five solids, twice each. Each Platonic solid as a framework of rods hinged at the corners, and again with its faces made rigid by adding their diagonals. The rank of the rigidity matrix reaches 3V − 6 exactly when the framework cannot move; the shortfall counts the ways it can. Three of the five are rigid as rods and all five are rigid as plates, which is Cauchy's theorem in the form a rank computation can see. Symmetry at work

The polyhedra that can flex

A cube of rods folds and a cube of cardboard does not, and the difference is a rank. Cauchy proved in 1813 that a convex polyhedron with rigid faces is rigid; the rank of a rigidity matrix sees it directly, and it also sees where the hypothesis is doing the work. Drop convexity and an octahedron flexes — followed here for forty steps with every edge length held to five parts in a thousand million million.

 The three minima of seven lattices. Every lattice scaled to covolume one, with the smallest radius at which a ball holds one, two and three independent lattice vectors. The last column is the shortest vector as a fraction of the longest any lattice of this volume can have — Hermite's constant — and only the face-centred cubic lattice reaches it. The fifth column is the product of the three, which is capped whatever the lattice. Lattices

A lattice cannot have all its vectors long

The three successive minima are the radii at which a ball first holds one, two and three independent lattice vectors. Nothing bounds any of them above on its own — a cell can be flattened without limit — but Minkowski's second theorem caps their product, so pushing one up forces another down. That is why every crystal has a shortest direction worth naming, and why a very anisotropic cell has a very short one.

 How far apart points on a sphere can be kept. For each number of points, the largest smallest angle a search could find between any two of them. Two unit spheres touching a third do not overlap exactly when their contact points are 60° or more apart, so the largest count whose best arrangement still clears 60° is the kissing number. Twelve clears it with three degrees to spare and thirteen falls short by more than three. The circle column is the same problem in the plane, where the answer is exactly 360/n and needs no search at all. Symmetry at work

The room a thirteenth sphere would need

Twelve equal spheres touch one, and whether a thirteenth could was argued in 1694 and settled in 1953. The reason it took so long is measurable: the twelve leave three and a half degrees of slack, which is enough room to look promising and not enough to use — and in the plane, where the same question has no slack at all, nobody ever argued.

 One crossing or the other, and never both. Two events on a square patch of lattice: a path of occupied sites crossing from left to right, and a path of vacant sites crossing from top to bottom. On the triangular lattice exactly one of them happens in every configuration tested — the claim is combinatorial rather than statistical, so one counterexample would end it. On the square lattice both can fail at once, and do, in more than a quarter of the configurations. That difference is the whole of what follows. How it is known

The threshold a symmetry pins down

Occupy sites at random and somewhere the occupied ones first join up across the crystal. For almost every lattice that occupancy is known only to a few digits. For the triangular lattice it is exactly a half, and the reason is that on a lattice whose faces are all triangles an occupied path and a vacant path cannot slip past each other — a statement about one configuration at a time, with no probability in it.

 The same accounting, at every coordination number. One row per number of edges at a vertex. The bill a sphere charges is 2dχ; the face worth nothing is 2d/(d − 2), which is a whole number at three, four and six and is 10/3 at five; the faces that can pay are those with fewer sides than that; and the last column is every way of paying the whole bill with faces of a single size. At three edges a vertex there are three such ways and twelve pentagons is one of them. At six there are none, which is the statement that six-fold coordination belongs to the plane and to no closed surface at all. What a lattice forbids

The twelve belongs to the vertex

Twelve pentagons is read as a fact about closing a surface. It is not: it is a fact about three edges meeting at a point. Let four edges meet instead and the sphere charges eight triangles; let five meet and it charges twenty; let six meet and it cannot be paid at all.

 Two turns and their undoing leave a slide. A turn g by 90° about the point c and a turn h by 60° about d. The marked point p is carried back 60° about d, back 90° about c, forward 60° about d and forward 90° about c, and does not return: it arrives displaced by a vector of length 2.371, which is 4·sin 45°·sin 30°·|c − d|. Two other points put through the same four motions move by the same vector, drawn beside them, because the commutator g h g⁻¹ h⁻¹ of two rotations of the plane is a translation — (I − A)(I − B)(c − d) exactly — whatever the angles and the centres. What a lattice forbids

What forces a lattice

Every enumeration here starts from a lattice of translations, and the lattice is usually taken as given. It need not be. A group of motions that is discrete, and leaves no point far from an orbit, has to contain one — in the plane by an argument four lines long, each line a picture, and in space by an inequality whose threshold turns out to be the six-fold rotation.

 The rectangle a (4, 2) tube is rolled from. A patch of honeycomb turned so that the rolling vector C = 4a₁ + 2a₂ lies along the page. C has length √28 ≈ 5.292; the shortest lattice vector perpendicular to it, T, has length 4.583; and the rectangle on the two holds 28 hexagons and 56 atoms. Rolling the rectangle so that its left and right edges meet makes one repeat of the tube. The two lines through the corner are the sheet's mirror directions nearest C: the zigzag direction along a₁ and the armchair direction thirty degrees from it. C makes an angle of 19.11° with the first and lies on neither. What a lattice forbids

The tube has a screw no lattice allows

Roll a honeycomb along one of its lattice vectors and the tube turns and climbs with a screw of order 14, 98 or 794 — orders the flat sheet could never have. The rolling keeps the sheet's translations and spends them on turns, and it keeps the sheet's mirrors only along two directions, which is why almost every carbon nanotube comes in two hands.

 Whether a rolled sheet ever comes back round. Three plane lattices, each with the same rolling vector C = 3a₁ + a₂ drawn from the origin and the line through the origin perpendicular to it. A translation of the rolled pattern straight up the tube, with no turn, is a lattice vector on that line. The square lattice has one, marked T, and the tube repeats every 10 turns. The general rectangular lattice has none in this direction — only along its cell edges — and the general oblique lattice has none in any direction at all, so its rolled pattern climbs forever without returning to the same angle. What a lattice forbids

Most sheets roll into a tube that never repeats

Rolling the honeycomb along a lattice vector always gives a tube with a repeat, and that is a property of the honeycomb rather than of rolling. Over the seventeen plane groups, 567 of 1,008 rolling directions give a tube with no translation along its axis at all — and every direction of an oblique pattern is one of them.

 What lies between a group and its copies at index 25. For each group, the lattices carried to themselves by its point group that contain a copy of the group at index 25, arranged by index, with lines for containment; each lattice is labelled by the Gaussian or Eisenstein integer that generates it. Copies are the bottom row, drawn large when nothing lies between them and the whole lattice. p4: 3 copies at index 25, 0 maximal, with invariant lattices between of index 5; p4m: 1 copies at index 25, 1 maximal, with invariant lattices between of index none. Into space

The primes a cell can grow by

A plane group contains copies of itself in bigger cells, and the International Tables list the ones that are maximal — the copies nothing else sits between. For p4 they come at 2, at 5, 13, 17 and 29 twice each, and at 9 and 49 once, and the list is the list of primes of the Gaussian integers. Put mirrors on the pattern and the copies at 5 and 13 vanish while 25 becomes maximal: a mirror cannot keep one factor of a prime without the other.

 Free going down, two conditions going up. The edge between p2 and p4 in the diagram of maximal translationengleiche relations, read in both directions. Downwards it costs nothing: the quarter-turns are discarded and the lattice is exactly the lattice that was there, so every p4 pattern contains a p2 pattern. Upwards the added quarter-turns must carry the lattice onto itself, which forces the cell to have equal edges at a right angle — two conditions on a general oblique cell, which has only two parameters to give. So a p2 structure has a p4 supergroup exactly when its measured cell happens to be square, and the question is about the metric rather than about the group. Operations

Going up costs the cell a parameter

The usual asymmetry — finitely many maximal subgroups below, infinitely many minimal supergroups above — is false in both halves for a plane group. Both directions are infinite and equinumerous index by index. The real asymmetry is that 17 of the 31 edges cost the lattice a parameter going up and nothing going down.

 Fifty-four of the seventy-five rod groups are a rolled plane pattern. Every rod group, one dot each, grouped by its crystal class. A dot is filled when some plane pattern rolled along some lattice vector has exactly that group, and the 106 crystallographic rollings of the seventeen plane groups fill 54 of them. The eighteen improper classes are full: every one of the 43 achiral rod groups is reached. The nine proper classes are not, and the twenty-one groups named on the right are what is missing — the sixteen whose screw is one of a left- and right-handed pair, and the five bare axes with no climb at all, p1, p112, p3, p4 and p6. What a lattice forbids

A rolled sheet is never one of a pair

Rolled up along every lattice vector that gives a crystallographic tube, the seventeen plane groups reach fifty-four of the seventy-five rod groups: every achiral one and eleven of the chiral. Not one of the sixteen screws that come in left- and right-handed pairs is among them, and the reason is a single fact about how far a rolled lattice can climb.

 Identical layers, each turned by an angle no number of turns undoes. Plan views of 4 layers of a stack. Each layer is the same square lattice with one cell shaded and one direction drawn, and each is turned from the one below through the angle whose cosine is three fifths, about 53.13 degrees. That angle is not a rational part of a full turn, so no number of layers brings the drawn direction back to where it started. A tiling of space with this structure has a symmetry that turns one layer onto the next and climbs one layer, and it has no translation. Order without repetition

Aperiodic is two words in space

A tile is aperiodic when none of its tilings is periodic, and periodic has been read two ways: a tiling with a translation, or a tiling with infinitely many symmetries. In the plane those are one condition, provably. In space they come apart, and a prism found in 1988 sits exactly in the gap.

 The fewest contacts twelve pentagons can have, by size. For every cage of pentagons and hexagons up to forty-four atoms, the number of pairs of pentagons sharing a bond. The lower line is the fewest any cage of that size achieves — 30, 24, 21, 18, 17, 15, 14, 12, 11, 10, 9, 8 — the upper line the most, and the dashed line the bound that counting edges gives: the twelve pentagons carry sixty edges between them, a contact uses two and an edge to a hexagon uses one, so the contacts cannot fall below 30 − 3h with h hexagons. The bound is attained while the hexagons are few and goes loose at five, after which each extra hexagon removes about one contact rather than three. The number of cages at each size is printed beneath, and it is the least rather than the average that the bound is about. What a lattice forbids

How close the twelve must be

The charge fixes twelve pentagons and says nothing about where they go, because it is a sum over faces and cannot see which face touches which. What it cannot see is a graph on twelve points, and the fewest edges that graph can have falls from thirty to eight over the cages a census reaches — then keeps falling at a rate that puts its first zero exactly where the truncated icosahedron is.

 An orbit on a parabola, discrete and cocompact. The images of the origin under the group generated by two commuting affine maps of the plane: A slides one step along x and lifts y by the x it started at plus a half, and B is the translation by one in y. The images are the points with whole-number first coordinate and second coordinate a whole number above half the square of it, so the large dots lie on the dashed parabola and the small ones are the rest of the orbit. No two distinct images come closer than 1.000, and no point of the square between the axes lies farther than 0.610 from one — so the action is discrete and its quotient is compact, which is exactly what Bieberbach's first theorem asks for. What a lattice forbids

Straight lines, and no distances

Every finiteness met so far rests on the motions preserving a metric, because the trick that produces one is an average and an average needs something to average over. Keep the straight lines and drop the distances, and Bieberbach's first theorem is false in the plane — by an example two lines long, whose group is the plane's own translations and whose translations have rank one.

 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.

 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.

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