Crystallography & Symmetry
patterns derived from their groups
Concept

Continued fraction — where it appears

The expansion of a number as a nested sequence of reciprocals, whose partial quotients decide how well rationals approximate it. It bounds how far apart two copies of a patch can be, and it is also the length of a lattice reduction.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

13 of one on 12 of the other. Two rows of atoms whose spacings are in the ratio 1.042. Every 12 cells of the substrate come to within 3.97 per cent of 13 cells of the film, so the two are nearly in register at those points and out of register between them. There is no exact coincidence anywhere, and there cannot be: exact coincidence needs the ratio to be rational, and no measured ratio is.

Two different lattices never coincide, and the question becomes how nearly

Grow one crystal on another and their spacings are in a ratio that no measurement ever makes rational, so exact coincidence is unavailable in principle. What is left is the best rational approximation inside a tolerable repeat — a quantity that jumps rather than drifts as the ratio changes, and whose acceptability is decided by elasticity rather than by arithmetic.

applied · Interfaces The region, and its copies. Words in S and T up to length 4, each carrying the region somewhere else. The copies do not overlap and they do not leave gaps: the upper half-plane is tiled by them, one copy per change of basis. That is the whole content of the claim that reduction picks a canonical basis — every basis of every lattice is in exactly one copy, and reduction is the walk back to the shaded one.

Two moves reach every basis

A lattice has infinitely many bases and reduction picks one. Why it can is a fact about a group with two generators and two relations — and the fundamental region tiles the plane with its own copies, one per basis, which is what makes the walk home finite.

lattices · Moduli Every window returns within 3.0 n. For each window length, the largest distance between two consecutive occurrences of the same window, measured over 46,368 tiles. The gaps are Fibonacci numbers, and the ratio to the window length stays below 3.00 — the chain is linearly repetitive. That is a strong statement of uniformity: there is no stretch of the chain, however far out, in which a given patch fails to occur within a bounded multiple of its own size.

Every patch comes back

A chain that never repeats still repeats everything in it. Every block of tiles occurs again, and again, within a bounded multiple of its own length — and how large that multiple is turns out to be a fact about the continued fraction of a slope.

aperiodic · Complexity incommensurate: "dense on a line". The shortest non-zero vector a subgroup contains, as the search widens, against the square lattice drawn flat behind it as a control. For a lattice the answer is constant: the shortest vector is the shortest vector, and looking further finds nothing nearer. For a subgroup that is not a lattice it falls without limit, because the convergents of a continued fraction give integers making the combination arbitrarily small. This one falls from 0.414 to 1.2e-2 over bounds 1 to 64, which is the verdict "dense on a line" arrived at by measurement rather than by reading a definition. Nothing here is decided by asking whether a ratio is rational; the ratio is a float and the question would be undecidable of one.

Discrete, or dense, and nothing between

Every count in this collection rests on a hypothesis nobody states, because it is built into the word lattice: the translations of a pattern form a discrete subgroup of the plane. Drop it and the counts do not become larger — they stop existing, because the object stops being a lattice. A subgroup of the plane is one of five things, and only two of them are lattices.

lattices · Lattice 21 points, 3 gap lengths. The first 21 multiples of 377/610, marked on a circle of circumference one, together with the point at zero. The 22 gaps between neighbours take 3 distinct lengths — 13/610 (1 of them), 21/610 (9 of them), 34/610 (12 of them). The largest is the sum of the other two: 13 + 21 = 34. Every quantity here is a whole number over the denominator, so nothing is measured.

Three gaps, and never four

Mark the points α, 2α, 3α round a circle of circumference one. They look scattered. The gaps between neighbouring points do not: for every angle and every number of points there are at most three distinct gap lengths, and when there are three the largest is the sum of the other two. That is where a chain with exactly two tile lengths comes from.

aperiodic · Complexity 9 approximants, period 2 to 89. The approximants of the Fibonacci chain: the n-th Fibonacci word taken as a unit cell and repeated. Each is a perfectly ordinary periodic crystal — it has a lattice, a cell and a space group — and each has the composition of the quasicrystal to the accuracy a ratio of Fibonacci numbers can manage, since its long and short tiles are consecutive Fibonacci numbers and their ratio is a convergent of the golden ratio. The last column is where the approximant stops agreeing with the infinite chain letter for letter: always past its own period, because the infinite word begins with every finite Fibonacci word, and never for ever. At order 9 the error in the composition is -3.87e-4, and it falls by a factor of τ² at every step up the sequence.

The crystal you get by rounding τ off

Everything aperiodic about a Fibonacci chain comes from one irrational number in the slope of a cut. Replace it by a fraction and the whole construction survives: the same lattice, the same strip, the same rule, and a chain that is periodic — agreeing with the quasicrystal for a length that grows with the denominator.

aperiodic · Quasicrystals How often each block of 5 occurs. Every block of length 5 in the fibonacci chain, with its frequency from the Perron eigenvector of the block substitution and again from a count over a chain of 46368 letters. The two share nothing: one is a linear algebra problem over a matrix of integers, the other a loop over a string. The eigenvalue of the block matrix is the inflation factor of the letter matrix, which is a second check and a stronger one — a chain inflates at one rate whatever length of window is being counted.

How often each patch occurs

That a patch has a frequency at all is the ergodic theorem. What the frequency is turns out to be an eigenvector: the substitution acts on blocks as well as on letters, the block matrix has a Perron vector, and its entries are the frequencies exactly. For the Fibonacci chain those entries take three values at every length, and the three values are the three gaps of a rotation.

aperiodic · Complexity 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 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.

restriction · Finiteness

Named alongside it

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

Golden ratioFactor complexityThe Fibonacci chainIrrational slopeBalanceCut-and-projectDiscretenessQuasiperiodicSturmianThree distance theoremTranslation groupApproximant

All concepts