Fibonacci quasicrystals

A Fibonacci quasicrystal is a one-dimensional pattern with long-range order but no translation period. Its finite patches recur, but no nonzero shift reproduces the entire infinite pattern. It is not random: one global rule determines every tile.

The familiar construction is almost absurdly small:

L → LS
S → L

Here L and S denote long and short tiles. Apply both replacements to every letter at once, starting with L:

L
LS
LSL
LSLLS
LSLLSLSL
…

Each word is a finite prefix of the one-sided infinite Fibonacci word, which extends to the right. In an arbitrary common length unit, give S length 1 and L length the golden ratio, φ = (1 + √5) / 2 ≈ 1.618. The symbol counts explain why this length ratio is natural; the count ratio does not depend on the chosen tile lengths.

Let ℓₖ and sₖ count the L and S symbols after k substitutions. Generation 0 is the single letter L. Each old L produces one L and one S, while each old S produces one L, so:

(ℓ₀, s₀) = (1, 0)
ℓₖ₊₁ = ℓₖ + sₖ
sₖ₊₁ = ℓₖ

Define Fibonacci numbers by F₀ = 0, F₁ = 1, and Fₙ₊₂ = Fₙ₊₁ + Fₙ. The recurrences give ℓₖ = Fₖ₊₁ and sₖ = Fₖ, so for k ≥ 1

ℓₖ / sₖ = Fₖ₊₁ / Fₖ  →  φ

The arrow means the ratios approach φ as k grows; a finite generation need not equal the limit. One way to check the limit is the exact solution Fₖ = (φ^k − (−1/φ)^k) / √5. A power with base z satisfies the recurrence when z² = z + 1, as both bases here do. Their difference divided by √5 gives the stated initial values. Since |−1/φ| < 1 < φ, the second power becomes negligible, giving the ratio limit. Dividing each count by ℓₖ + sₖ then gives limiting tile fractions φ/(φ + 1) = 1/φ long and 1/(φ + 1) = 1/φ² short. These are count frequencies, not fractions of the physical line's length. With the chosen lengths, the same φ also makes the substitution an inflation: φ² = φ + 1, so scaling an S by φ gives length φ (one L), while scaling an L gives length φ² (one L plus one S).

The small rule and the large order

The deterministic substitution produces order at every scale. Patches recur, and the tile frequencies approach fixed limits. In the ideal infinite Fibonacci chain, equally weighted boundary points have sharp diffraction peaks: the mathematical scattering intensity consists of point peaks, called Bragg peaks, with no continuous diffuse component. This does not mean the peak positions form an isolated, evenly spaced lattice. The statement concerns the ideal chain, not a measurement on a particular sample.

A conventional crystal repeats by a translation. Here the tile choices come from a prescribed substitution, rather than independent random choices. The Fibonacci chain has no translation period and no finite unit cell that can be copied indefinitely.

An irrational step around a circle

A useful way to describe non-repetition is to rotate around a unit circle. Write the phase after integer step n as

e^(2πinα),    n ∈ ℤ

Here n is an integer step index, α is the number of turns per step, i is the imaginary unit (i² = −1), and ℤ is the integers. The complex exponential places each phase on the circle. Equivalently, record the fractional parts 0, α, 2α, 3α, … (mod 1), where “mod 1” means that phases differing by an integer number of turns are identified.

Two integer steps n and m land at the same phase exactly when (n − m)α is an integer. If α = p/q in lowest terms, the orbit repeats after q steps and visits finitely many phases. If α is irrational, no positive integer step returns it to its starting phase; moreover, the standard irrational-rotation result says the orbit is dense, meaning it comes arbitrarily close to every point on the circle.

For the discrete rotation, choose α = φ. The identity φ² = φ + 1 implies φ − 1 = 1/φ, so its fractional part is 1/φ ≈ 0.618. Thus φ and 1/φ describe the same integer-step rotation modulo 1. The particular tile coding below uses another increment, β = 1/φ² ≈ 0.382. Since 1/φ + 1/φ² = 1, β = 1 − α for α = 1/φ: the coding advances in the reverse direction around the circle. It is related to, but is not the same numerical slope as, the preceding rotation.

The rule that turns the rotation into tiles

Fix the mark at phase 0 on a circle of circumference 1, and set β = 1/φ². To match the indexing below, begin recording at phase β, not phase 0. For tile n, advance from phase nβ to (n + 1)β, reducing phases modulo 1. Write S if that interval crosses the marked point (equivalently, crosses an integer before reduction); otherwise write L.

The same rule is expressed without a picture by the floor function. ⌊x⌋ is the greatest integer no larger than x; aₙ is the binary code for tile n:

aₙ = ⌊(n + 1)β⌋ − ⌊nβ⌋,    n = 1, 2, 3, …

aₙ = 0  →  L
aₙ = 1  →  S

Because 0 < β < 1, each difference is either 0 or 1. For example, using β ≈ 0.382, the first two intervals are:

`n`	`nβ`	`(n + 1)β`	floor difference	tile
1	0.382	0.764	0	L
2	0.764	1.146	1	S

The first interval does not cross an integer; the second does. Continuing gives L S L L S L S L L S …, the displayed Fibonacci prefix. The standard rotation description of the Fibonacci word identifies this entire one-sided word with the substitution limit, not just its first few letters. The index starts at 1, so the first recorded crossing test is from β to 2β. Recording every step starting at 0 would add an initial L and shift this displayed convention.

The word from this specific crossing rule is not periodic. To see why, let N be a positive prefix length and let N_S(N) count its short tiles. Summing the floor differences cancels the intermediate terms:

N_S(N) = a₁ + … + a_N
       = ⌊(N + 1)β⌋ − ⌊β⌋
       = ⌊(N + 1)β⌋
N_S(N) / N  →  β

The last equality uses 0 < β < 1, so ⌊β⌋ = 0; the limit follows because a floor differs from its argument by less than 1. A periodic word with p tiles per period and r short tiles in that period would have short-tile frequency r/p, a rational number. An eventually periodic word has the same rational limit, since a finite initial prefix does not change the limiting frequency. Here β = 1/φ² is irrational, so this word is neither periodic nor eventually periodic. This argument is specific to the crossing code. An irrational orbit alone does not guarantee that every possible symbolic observation is nonperiodic: a rule that labels every phase L would produce a constant word.

Assigning lengths to the letters turns the word into a tiling; placing a point at each boundary gives a point-set model. One may then attach weights to those already defined positions—for example density, scattering strength, or occupancy—to model an observable. The weights describe a field on the arrangement; they do not define the arrangement. A quasiperiodic function can supply weights sampled at those positions. This does not identify the point set itself with a continuous quasiperiodic function. With equal boundary weights, the ideal point set has the diffraction property described above; arbitrary added weights need not preserve it. These are mathematical models, not reports about a particular physical sample.

Why the torus is the better picture

For a continuous, dimensionless real parameter x, a single phase e^(2πiαx) with nonzero α travels around the circle and repeats after 1/|α|. This is different from the integer-indexed orbit above: discrete rational increments visit only finitely many phases.

To keep two phases with no common period, follow the continuous path

x ↦ (e^(2πix), e^(2πiαx)).

Both coordinates are phases, and here α is any fixed irrational number. The pair lies on a two-dimensional torus, the product of two circles. A return after a nonzero parameter interval T would require both T and αT to be integers. No such T exists for irrational α. To see density, fix the first phase at t, with 0 ≤ t < 1, and take x = m + t for integer m. The second phase then follows the dense irrational rotation αm + αt modulo 1. Thus the continuous path comes arbitrarily close to every phase pair. For rational α, it instead closes after a finite interval. Unlike discrete samples, continuous frequencies cannot generally be replaced by their fractional parts without changing the path.

A continuous function on this torus is periodic in each angle. Restricting such a function to the path produces a one-dimensional signal built from the two periodic phases. This is a common way to make a quasiperiodic function, though the result need not be nonperiodic: an observable can ignore one phase. For example, with u = x mod 1 and v = αx mod 1,

G(u, v) = cos(2πu)             →  g(x) = cos(2πx)
H(u, v) = cos(2πu) + cos(2πv) →  h(x) = cos(2πx) + cos(2παx)

Here G and H are torus functions, while g and h are their restrictions to the path. g is periodic with period 1. For h, note that h(0) = 2. A period T would require h(T) = 2; since each cosine is at most 1, both would have to equal 1. This forces T and αT to be integers, impossible for nonzero T when α is irrational. Thus h is nonperiodic. The orbit's density alone does not decide whether a particular observable repeats; that also depends on what the function measures.

The higher-dimensional picture explains one source of quasiperiodic order: a periodic object in two phase coordinates can yield a nonrepeating one-dimensional observable along an irrational path. To construct a quasicrystal point set by projection, one selects lattice points inside an appropriate strip and projects them onto the line; merely intersecting a line with a lattice is not that construction. The Fibonacci tile rule above gives a direct one-dimensional construction.

From Fibonacci counting to quantum information

The name Fibonacci also appears in a model of non-Abelian anyons. Anyons are quasiparticles—collective excitations that behave like particles in a two-dimensional system—whose exchanges can act on a multi-state quantum space. “Non-Abelian” means that these exchange operations need not commute. In the Fibonacci model, the charge types are 1 (vacuum or trivial charge) and τ (nontrivial charge). These labels classify fusion behavior, not ordinary electric charge. Fusion combines anyons and asks for their resulting total charge. The relevant rules are

1 × τ = τ
τ × τ = 1 + τ

Here × means fusion of charge types, and + lists allowed alternative outcomes; it is not ordinary addition and does not specify probabilities. A fusion tree records which groups of anyons are combined first; a fusion path labels its intermediate charges consistently with the rules. We count paths in one chosen tree for anyons in the plane, with no extra degrees of freedom from handles or additional boundaries. The count depends on the number of τ anyons and their fixed total charge.

For clarity, let dₙ(1) and dₙ(τ) count fusion paths for n identical τ anyons ending in total charge 1 or τ. Start with one anyon: d₁(1) = 0, d₁(τ) = 1. Adding one more τ and applying the rules gives

dₙ₊₁(1) = dₙ(τ)
dₙ₊₁(τ) = dₙ(1) + dₙ(τ)

Only a previous total τ can fuse with the added τ to give total 1. Either previous total can give total τ. With F₀ = 0, F₁ = 1, and Fₙ₊₂ = Fₙ₊₁ + Fₙ, these counts are dₙ(1) = Fₙ₋₁ and dₙ(τ) = Fₙ. For instance:

Number of `τ` anyons	Total `1` paths	Total `τ` paths	Combined paths
1	0	1	1
2	1	1	2
3	1	2	3
4	2	3	5
5	3	5	8

The combined sequence 1, 2, 3, 5, 8, … sums different total-charge sectors; it is not the dimension of one fixed-charge space. In the fixed-τ sector, the sequence is 1, 1, 2, 3, 5, …. These are basis-path counts, not a claim that every path is measured and recorded during computation.

With three τ anyons and total charge fixed to τ, the first pair can fuse to 1 or to τ; either intermediate choice can then combine with the last τ to give total τ. These two fusion paths span a two-dimensional state space. Encoding quantum information in that selected two-dimensional subspace gives a logical qubit. Braiding means exchanging the anyons along paths without fusing them. It transforms the fusion-path states; because the transformations need not commute, changing the exchange order can change the operation. A two-dimensional space and braid operations alone do not establish protection or fault tolerance, which also requires handling faults in the physical operations without losing the encoded information.

In the cited tiling proposal, a history records the long/short tile type around a chosen position across successive inflation or deflation levels. Those entries describe different scales, not neighboring tiles in one chain. The allowed transitions are L → L or S and S → L: a short entry must be followed by a long one. At the level of these paths, identify L with cumulative charge τ and S with cumulative charge 1. The transitions then match adding one τ anyon. For example, an n-entry history starting and ending with L matches a fusion path starting with one τ and ending with total τ, giving Fₙ paths under these endpoint conditions.

This explains the constrained-history comparison without identifying fusion dimensions with all patches of the Fibonacci word. The proposal concerns a specified encoding between spaces; matching counts alone does not supply physical braid operations or make a tiling a fault-tolerant quantum computer.

A hypothetical diamond architecture

One possible hardware story uses diamond colour centres—crystal defects whose spins can encode qubits—in a two-dimensional network. This explicitly hypothetical architecture would realize a Fibonacci string-net code, also called a Fibonacci Turaev–Viro code. Each edge of a graph with three edges at every vertex would carry an encoded two-state degree of freedom labelled 1 or τ. A string-net state can superpose such labelled networks. The labels are basis-state labels in a spin encoding, not a claim that physical Fibonacci anyons already exist in diamond.

The ideal vertex rule allows the unordered triples (1,1,1), (1,τ,τ), and (τ,τ,τ); permutations are also allowed. (1,1,τ) is forbidden because two trivial charges fuse to 1, not τ. A vertex measurement tests the allowed subspace; its passing outcome projects onto that subspace, while a failing outcome detects a constraint violation.

These local label tests alone do not make the full code. For each face, a quantum projector forms a weighted combination of adding a closed 1 or τ loop and absorbing it into the surrounding edges using the model's fusion data. This operation can mix edge configurations coherently; it is not a classical comparison of labels. A code state must be left unchanged by each face projector as well as pass every vertex check. In the ideal mathematical model, these projectors commute; their shared passing subspace is the code space.

The proposed logical information would be stored in global degrees of freedom that survive all those local checks. Which logical degrees of freedom exist depends on the network and its boundaries; “global patterns” alone does not determine a qubit count. A diamond implementation would additionally need state preparation, multi-spin checks, logical operations, and reliable syndrome readout: measurement outcomes identifying constraint violations. Those requirements are hypothetical assumptions, not capabilities established by the existence of addressable spins.

The cited 2022 diamond experiment demonstrated a five-data-qubit stabilizer code. Such a code uses commuting multi-qubit checks to detect errors without reading the encoded logical state. The experiment used seven spins: five nearby carbon-13 nuclear-spin data qubits, a nitrogen-vacancy (NV) electron-spin measurement ancilla (an auxiliary qubit), and a nitrogen nuclear-spin flag that detects dangerous error propagation from the ancilla. It was not a Fibonacci Turaev–Viro code or the proposed two-dimensional colour-centre network. The experiment also reported that its logical fidelities—the measures of agreement with intended logical states—did not yet outperform the constituent physical qubits. These results demonstrate related diamond control and error-correction ingredients, not Fibonacci anyon operations or the architecture described here.