Crystallography & Symmetry
patterns derived from their groups
Concept

Transfer matrix — where it appears

A way of counting arrangements one layer at a time, carrying forward how many partial arrangements end in each possible layer. It turns a count over an enormous set into a product of small steps, and it accepts a constraint on one cell, which is how an exact conditional count is taken.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The allowed energies of the Fibonacci chain, level by level. The set of energies at which a wave neither grows nor decays, for periodic approximants of the Fibonacci chain of 5, 8, 13, 21, 34 and 55 sites. Each row has exactly one band per site, and each band splits into smaller ones at the next level rather than growing. Nothing in the picture converges to an interval: the gaps opened at one level survive at every level after it, and the limit is a Cantor set — closed, containing no interval at all, and of measure zero, which is a theorem of Sütő's rather than something these six rows prove.

A spectrum that is a Cantor set

A wave in a periodic chain has bands with gaps between them. A wave in the Fibonacci chain has gaps inside the gaps, at every scale — and the traces that decide where they are obey a recursion with a quantity it cannot change.

aperiodic · Quasicrystals 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.

aperiodic · Entropy 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 both halves: almost all of the rise is the freedom to choose which vertices break, and with that removed the first defects subtract rather than add.

aperiodic · Entropy 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.

aperiodic · Entropy 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.

aperiodic · Entropy

Named alongside it

The objects these essays reach for when they reach for this one.

CensusEntropyLocal rulesCountingEnumerationHeight functionResidual entropyEquilibrium shapeApproximantCantor setConserved invariantDefect

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