Concept

Inflation — where it appears

The rescaling that carries a substitution-generated structure onto itself, by a factor that is an eigenvalue of an integer matrix. Sharp diffraction requires that factor's algebraic conjugates be small, which is a condition on a number rather than on a picture.

Named by 4 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

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

66 squares and 106 rhombs. The Ammann–Beenker tiling, built by keeping the points of a four-dimensional lattice whose companion image falls inside an octagon and projecting them into the plane. Every tile has the same edge length; the squares and the forty-five degree rhombs are told apart by their diagonals. Nothing was placed — the faces were found among the projected points.

Eight-fold, with the golden ratio taken out

The familiar quasicrystals are all built on five: Penrose's rhombs, the Fibonacci chain, the ten-fold pattern Shechtman measured. A method that works only on the golden ratio is a method tuned to its answer — so here is the same construction run on eight, where the irrational is √2 and nothing else changes.

aperiodic · Quasicrystals

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

Named alongside it

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

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