Crystallography & Symmetry
patterns derived from their groups
Series

Entropy — the series

10 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Order 5: 32,768 arrangements. An Aztec diamond of order 5, with every possible dimer drawn at an opacity equal to the fraction of arrangements it appears in — a probability computed exactly, by counting the arrangements of the region with that dimer's two sites removed, rather than sampled. The four corners come out nearly certain and the middle nearly even, with a circle between them. The most certain dimer here occurs in 0.97 of the arrangements, which is 1 − 2⁻5 exactly, so nothing is frozen at any finite size.

    How many arrangements one rule allows

    Every count in this collection so far has been a count of symmetries, or of orbits under one. Here is a different count: the arrangements a purely local rule permits on a fixed lattice, with no symmetry quotient anywhere in it. The answers are enormous, they are exact, and the useful quantity is not the number but its growth per site.

    part 1 · aperiodic
  2. Arrangements per site, falling towards the exact value. The number of configurations of an L × L torus obeying the ice rule, taken to the power of one over the number of sites. The largest computed here is 4,484,823,396 configurations on a 7 × 7 torus. The values fall towards Lieb's exact 1.54 from above, and every one of them is above Pauling's estimate of 1.5 — which undercounts, because it treats the vertices as independent and they are not.

    The arrangements a crystal keeps at absolute zero

    Ice has a residual entropy, and the number a calorimeter measures is the logarithm of a count of arrangements. Pauling's one-line estimate of that count is out by two and a half per cent; the exact count in two dimensions is available, falls towards its limit from above, and the whole disorder is invisible to a diffraction experiment, which sees only the average.

    part 2 · aperiodic
  3. Four of the 980 piles in a three-cube box. A stack of unit cubes in the corner of a box, seen down the body diagonal. Every visible face is one of three rhombi and the picture is a tiling of one fixed hexagon — the same hexagon for every pile, because a pile in an a×b×c box always shows ab+bc+ca faces however it is stacked. The four here are taken at even intervals through the enumeration, from the empty box to the full one.

    A facet with no energy in it

    Stack cubes into the corner of a box and look down the body diagonal: the pile is a tiling of a hexagon by three rhombi, and the number of piles is a product MacMahon wrote down in 1916. Because the count is exact, so is the average pile — and the average has a flat corner meeting a rounded middle, which is the shape of an equilibrium crystal, arrived at by counting with no surface energy anywhere in the argument.

    part 3 · aperiodic
  4. Square ice scatters a pinch at the origin. The intensity scattered by the horizontal arrows of square ice, averaged over 100 configurations on a 32 by 32 torus, over the whole Brillouin zone with the origin at the centre; darker is more intense. Along the horizontal axis through the origin the intensity falls to zero — 2.9e-32 at the smallest wavevector — while along the vertical axis it stays near 1.75, so the two meet at the origin in a pinch.

    The ice rule is a conservation law

    Two arrows in and two out at every vertex is a statement that nothing flows in or out anywhere. That makes one half of the arrow field vanish identically, in every arrangement and not merely on average, and what is left scatters with a pinch at the origin: an intensity that approaches different values from different directions. Break the rule now and then and the pinch acquires a width Debye and Hückel predicted for a salt solution.

    part 4 · aperiodic
  5. Every arrangement on a torus 4 across, sorted by defects. The transfer matrix that counts ice arrangements chooses, at each vertex, the one horizontal arrow the rule permits. Enumerating both choices instead and carrying a polynomial that records how many vertices end up with three arrows in or three out gives the number of arrangements at every defect count at once. The first column, drawn solid, is the ice count — 2970 arrangements with no defect at all, which is the number the earlier transfer matrix gives and is checked against it. The second column is empty: no arrangement has exactly one defective vertex, because a defect carries a charge and the charges on a closed surface must cancel. The columns together add to two raised to the number of edges, which is every assignment of arrows whatever.

    What a defect costs the count

    Each broken vertex relaxes the rule and so adds arrangements — the question left standing was whether each adds a fixed amount or the cloud around it costs some back. The exact count at every defect number at once answers the first half outright: almost all of the rise is the freedom to choose which vertices break, and what is left once that is taken out exactly is small, uneven, and lowest at the first count that needs a four.

    part 5 · aperiodic
  6. One rule, one lattice, two entropies. The number of arrangements per vertex for square ice, counted two ways on the same lattice with the same rule. On a torus the count falls towards Lieb's exact value of 1.5396 from above. Inside a domain wall — every arrow on the top and bottom edges pointing in, every arrow on the left and right pointing out — the count rises towards 3√3/4, which is 1.2990, from below. A residual entropy is supposed to be a bulk quantity that forgets the boundary; these two differ by sixteen per cent and the only difference between them is the boundary.

    The count that depends on the edge

    A residual entropy is supposed to be a bulk number: so much per vertex, whatever surrounds the lattice. Square ice has two of them. On a torus the count per vertex heads for 1.5396 and inside a domain wall it heads for 1.2990, with the same rule on the same lattice — and the sixteen per cent between them is sitting in the corners.

    part 6 · aperiodic
  7. A colouring, and the arrows it writes. A proper three-colouring of the cells of a four-by-four torus — no two cells sharing an edge carry the same colour — with an arrow drawn on each shared edge by the difference of the two colours it separates. The difference is one or two modulo three, never nought, so every edge gets a direction. At each corner four cells meet and their four differences go round a cycle and add to nothing modulo three, which forces two of the arrows in and two out. That is the ice rule, arrived at from a colouring with no arrows in its statement.

    Three colours on a chessboard

    Colour the cells of a board in three colours so that no two sharing an edge agree. The number of ways is the number of ice arrangements on the same board — the same integer, to the last digit, at every even size — so a residual entropy a calorimeter reads is also the answer to a colouring problem with no physics in it at all. At odd sizes the two counts part company, and why they do is a condition on going round.

    part 7 · aperiodic
  8. 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.

    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.

    part 8 · aperiodic
  9. 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.

    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.

    part 9 · aperiodic
  10. The exact third on every even rectangle. Square ice on twelve tori with both sides even, eight of them rectangles that are not squares, up to 6 × 12 and 8 × 8: the number of ice arrangements, the number whose two windings are both divisible by three, and their ratio. It is one third exactly, as a ratio of integers, on every one.

    The third survives a rectangle

    On every even square torus, exactly a third of the square-ice arrangements have both windings divisible by three, and the explanation offered was a theta function that is its own dual. A test was proposed: on a torus twice as long as it is wide the two windings have different stiffnesses, the explanation gives a different number, and the exact third should fail. It does not fail on any of eight rectangles. But the explanation, worked through properly for a rectangle, never predicted that it would. The two directions of a rectangle are each other's duals, and their factors cancel to a third at every shape.

    part 10 · aperiodic

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