Crystallography & Symmetry
patterns derived from their groups
Concept

Repetitivity — where it appears

How far apart two occurrences of the same patch can be, bounded by a constant times the patch's own size in a linearly repetitive structure. A chain that never repeats still repeats everything in it, and how soon is a fact about a continued fraction.

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

A Penrose tiling, 4 inflations. Two rhombs, subdivided into smaller copies of themselves over and over. The result covers the plane, has five-fold symmetry about its centre, and never repeats — there is no translation that maps it to itself.

Penrose tilings

Two rhombi, a rule about how their edges may meet, and a tiling that covers the plane completely and never repeats. The five-fold symmetry a lattice forbids, obtained by giving up the lattice.

aperiodic · Aperiodic 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 One patch of radius 2. A Penrose patch of 476 vertices, with the vertices within 2 edge lengths of one of them marked and the circle drawn. That marked set, written in coordinates relative to its centre, is what the census compares: two vertices have the same patch when their marked sets agree. Every vertex of the tiling is the centre of one such patch, and the question is how many different ones there are.

How many patches of each size

A periodic tiling has one kind of neighbourhood however far out you look. Random points have as many kinds as neighbourhoods. A Penrose tiling has a number in between that never stops growing and never catches up — and the count is a measurement rather than a theorem.

aperiodic · Complexity 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

Named alongside it

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

Factor complexitySturmianAperiodicityThe Fibonacci chainGolden ratioLocal indistinguishabilityLong-range orderPenrose tilingsBalanceContinued fractionCut-and-projectDiscreteness

All concepts