Crystallography & Symmetry
patterns derived from their groups
Concept

The Fibonacci chain — where it appears

The one-dimensional quasicrystal built by replacing each long tile with a long and a short and each short with a long. Three completely different constructions produce it and are required here to agree, and its diffraction needs two integers per peak.

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

Cut and project. A square lattice, a strip along a line of the given slope, and the shadow on that line of every lattice point inside the strip. The shadow has two gap lengths; whether their order repeats depends entirely on whether the slope is rational.

Cut and project

Take a periodic lattice, cut a strip through it at an irrational angle, and keep the shadow of what falls inside. The result never repeats, has exactly two spacings, and is a quasicrystal — built from something perfectly periodic that is simply not where anybody was looking.

aperiodic · Aperiodic The substitution, 6 generations. The rule "every long tile becomes a long and a short, every short tile becomes a long", applied 6 times from a single tile. Each generation is as long as the previous two together, so the tile counts are Fibonacci numbers — 13 long and 8 short at the last row — and their ratio is 1.62500 against the golden ratio's 1.61803. The sequence never repeats and every finite piece of it recurs infinitely often, which is order without periodicity in its smallest form.

The smallest quasicrystal

Two tile lengths on a line, in the golden ratio, in a sequence that never repeats. Three completely different constructions produce it, they are required here to agree, and its diffraction needs two integers per peak where a periodic chain needs one.

aperiodic · Quasicrystals The Fibonacci chain: p(n) = n + 1. The number of distinct windows of each length in the Fibonacci chain, measured by sliding a window along 46,368 tiles. Every count is checked against the same count on half the chain, and only lengths where the two agree are drawn — a factor count on a finite word is otherwise a lower bound wearing the clothes of an answer.

n plus one, and no fewer

Slide a window along a chain and count what it can show. A periodic chain runs out of new views; an aperiodic one never does; and the fewest an aperiodic chain can manage is one more than the window's length — which is exactly what the Fibonacci chain manages.

aperiodic · Complexity 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 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 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 How fast a window's answer settles: 1/L on the chain, 1/√L on a shuffle. The largest error a window of each length makes about a block's frequency, over every position the window can take, on logarithmic axes. The upper line is a shuffle of the chain's own letters — same frequencies, no order — and its slope is close to −½, which is the random walk a sequence with no structure produces. The lower line is the Fibonacci chain itself and its slope is close to −1. The frequency of a block in the chain is therefore something a finite window measures rather than approaches: to know it to a part in a thousand needs a window of a thousand tiles, not a million.

The average is the same wherever it is taken

A measurement is made on a window somewhere, and the question is whether the answer belongs to the chain or to the window. For the Fibonacci chain the error falls as one over the window's length; for a shuffle of the same letters it falls as one over the square root, and the two exponents are fitted rather than asserted.

aperiodic · Complexity

Named alongside it

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

Cut-and-projectGolden ratioBalanceContinued fractionFactor complexityIrrational slopeLong-range orderQuasiperiodicSturmianWindowApproximantSubstitution

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