Crystallography & Symmetry
patterns derived from their groups
Generator

The substitution, 6 generations

The substitution, 6 generations
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 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.

8 essays call fibonacci-chain. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

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. Order without repetition

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.

 The chain as a cut through a periodic pattern. A periodic pattern in two dimensions: one atomic surface through each lattice point, drawn as the curve x = n + A·sin(2πy). The physical chain is the cut along the line y = qx with q = 0.211, and the atoms are where that line meets the curves — plotted along the bottom. Every cut meets every curve exactly once, so every cut gives a chain with the same 9 atoms and the same lattice, differently displaced. That is the difference from cut-and-project, where the atomic surfaces are intervals with ends and moving the cut adds and removes points: here the extra coordinate is a phase, and shifting it is a symmetry of the material rather than a different material. Order without repetition

The extra dimension that makes it periodic

A structure with no cell in three dimensions can be a slice through one that has a cell in four. The construction is cut-and-project with continuous atomic surfaces instead of intervals — and that single difference is what separates an incommensurate crystal from a quasicrystal.

 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. Order without repetition

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.

 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. Lattices

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.

 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. Order without repetition

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.

 two chains, periods 1 and 1.62. Two interpenetrating chains of atoms with periods 1 and 1.62, whose ratio is irrational, so no length is a whole number of both. Each chain is displaced from its own lattice by a wave with the other's period — the short ticks show each atom's displacement from where an unmodulated chain would put it — which is what makes this one crystal rather than two side by side. Nothing here is a unit cell: any length chosen contains a whole number of one chain's atoms and a fractional number of the other's. Order without repetition

Two lattices, one crystal, and no cell at all

A modulated crystal has a lattice and a wave running through it. A composite has neither host nor guest: two interpenetrating substructures with periods that share no common multiple, each modulating the other. Every reflection needs an index from both, and the unit cell a diffractometer reports belongs to whichever half scattered harder.

 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. Order without repetition

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.

 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. Order without repetition

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.

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