Crystallography & Symmetry
patterns derived from their groups
Concept

Substitution — where it appears

A rule replacing each tile by a word in the tiles, which repeated produces a self-similar structure that need never repeat. Its matrix's dominant eigenvalue is the inflation factor, and its rotations are invisible to that matrix.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

One tile becomes 130 in 3 inflations. One tile subdivided into smaller copies of the same two shapes, repeatedly. The rule is local and deterministic, and the pattern it builds has long-range order without any repeating cell. The panel counts are 10, 20, 50, 130, and each is the one before it multiplied by the substitution matrix — an identity in whole numbers, checked every time the figure is drawn rather than quoted.

Inflation, and where the golden ratio comes from

A Penrose tiling is not laid out tile by tile. It is grown by cutting every tile into smaller ones and rescaling, and the growth matrix's dominant eigenvalue is the golden ratio squared.

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 Which inflation factors a tiling may have. Every distinct inflation factor produced by a two-letter substitution whose matrix has entries up to 4, plotted against its algebraic conjugate. The two grey lines are the unit circle, which in the quadratic case is the pair of values ±1. A factor whose conjugate lies strictly inside is a Pisot number and the chain it grows has sharp Bragg peaks; 39 of the 77 factors here lie outside and cannot. The golden ratio is the smallest of them all, which is the arithmetic reason it turns up in every quasicrystal anybody has drawn.

Which inflation factors exist

A tiling grown by substitution has an inflation factor, and it is an eigenvalue of an integer matrix — so it is an algebraic integer, and sharp diffraction demands that its conjugates be small. That condition is an inequality between two integers, and it explains why the golden ratio turns up in every quasicrystal anybody has drawn.

aperiodic · Aperiodic 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 625 tiles, 32 directions. The subdivision applied 4 times to one right triangle with legs 1 and 2, giving 625 tiles of one shape and size. They point in 32 distinct directions — the tint follows the direction — and the count grows every time the rule is applied, without bound.

The tiling that points every way

A Penrose tiling never repeats and its tiles still point in only ten directions, which is why its diffraction pattern has ten-fold symmetry. One triangle, cut into five copies of itself, breaks that — and the difference between it and a tiling with eight directions is which diagonal of one small rectangle gets drawn.

aperiodic · Aperiodic A peak that grows, and not fast enough. The strongest peak of three chains, divided by the square of the number of letters, as each chain is lengthened. A Bragg reflection is a sum of terms in phase, so its intensity grows as the square of the count and this number settles: the Fibonacci chain and the period-doubling chain both do, at exponents of about two. The Thue–Morse chain does neither — its strongest peak grows, so it is not diffuse scattering, and it grows more slowly than the square, so it is not a Bragg peak. The fitted exponents are printed beside each curve and no threshold enters the comparison.

Neither a peak nor a bump

A chain whose strongest reflection grows as the length to the power one and a half. A Bragg peak grows as the square and a diffuse bump grows as the length itself, so this is neither — and the essay that ruled out the first possibility could only say so by quoting a theorem.

aperiodic · Complexity 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 The window of a three-letter chain. The cut-and-project window of the tribonacci chain: every prefix of the chain, projected onto the plane spanned by the two complex roots of x³ = x² + x + 1, and coloured by the letter that follows it. 223,317 points. It is a bounded region in three pieces whose areas are 0.542, 0.296, 0.162 of the whole, which are the frequencies of the three letters; it fills 71.5 per cent of its bounding box, and a straight cut across it meets up to 6 separate pieces. A window for a two-letter chain is an interval.

A window that is not an interval

The usual cut-and-project construction takes a strip through a lattice and keeps the points falling within an interval. Add a third letter and the window stops being an interval: the tribonacci chain's window is a fractal in three pieces, and a straight cut across it meets up to six.

aperiodic · Aperiodic Mirrors everywhere in a chain with none. Thirty-four tiles of the Fibonacci chain, long tiles L and short tiles S, with an arc over every window of at least 5 tiles that reads the same backwards, drawn from its own centre, and a tick at the centre where its mirror stands. The chain as a whole has no mirror, since no palindrome in it is infinite, yet mirrors of every size stand all along it, the long ones spaced further apart than the short ones, and every one of them a local symmetry the chain lacks globally.

A chain with no mirror has mirrors everywhere

The Fibonacci chain has no mirror: nowhere can it be reflected onto itself. Yet it holds exactly one window of every even length that reads the same backwards and exactly two of every odd length, each recurring all along the chain, and no aperiodic chain that is not of its kind manages that count. Every letter of it creates a palindrome never seen before, and a mirror of any size is never further away than a fixed multiple of that size.

aperiodic · Complexity One invariant for every invertible chain. The quantity x² + y² + z² − 2xyz − 1, built from the half-traces of the transfer matrices of the two words a substitution produces at each level and of their product, for six substitutions at one energy, logarithmic. For the golden, silver and bronze chains and for the golden rule reversed it is V²/4 = 0.09 at every level — the same number for all four, computed in exact integers. For period-doubling and Thue–Morse it wanders by orders of magnitude from one level to the next.

One invariant for every chain that can be undone

The Fibonacci chain's transfer matrices obey a recursion with a quantity it cannot change, and the natural guess is that every quasiperiodic chain has its own. It does not. The conserved quantity is the trace of the commutator of the two tiles' matrices, so every chain a reversible substitution builds from the same two tiles — golden, silver, bronze — conserves the same number, exactly, at every length and every energy. Period-doubling and Thue–Morse, whose rules cannot be undone, conserve nothing.

aperiodic · Quasicrystals Tribonacci keeps one number at every length, and its neighbours do not. The size of tr(ABC) − tr(ACB), computed in exact integers on three two-site tiles, at each application of four rules, on a logarithmic scale. Under tribonacci it stays 24 while the traces themselves reach dozens of digits. Under the plastic rule and the rule x³ = x² + 1, both of which can be undone, it wanders and grows. Under a rule that cannot be undone it passes through zero and runs off the scale.

Three tiles keep a number that asks which way they face

Two tiles always conserve something: every reversible rule keeps the trace of their commutator. Three letters were expected to conserve nothing, and most rules do conserve nothing. The tribonacci rule keeps exactly one number, the difference between a product of three matrices and the same product reversed. It is zero for any tile that reads the same both ways, so it measures direction rather than difference, and one rule in fifty keeps it, for a reason that can be read off the words.

aperiodic · Quasicrystals

Named alongside it

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

Inflation factorThe Fibonacci chainGolden ratioFactor complexityInflationWindowAperiodicityAutomorphismBragg peakCommutatorCut-and-projectInvariant

All concepts