Golden Ratio & Fibonacci Spiral

Build Fibonacci squares, compare consecutive ratios, and see how the recurrence approaches the golden ratio.

Starting values — Golden Ratio & Fibonacci Spiral

Terms (N)	12

Change any control above and the results recompute from these.

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why phi is the fixed point 🖖

The Fibonacci recurrence keeps replacing a ratio with 1 plus its reciprocal: F(n)/F(n-1) = 1 + F(n-2)/F(n-1). Once that ratio stabilizes at x, it must satisfy x = 1 + 1/x, so x² = x + 1 and the positive solution is phi. The spiral is a geometric picture of that same self-similarity: each larger square contains the previous rectangle plus one new square.

The ratio zigzags in toward phi 🖖

Watch the bar chart: consecutive Fibonacci ratios land alternately just above and just below φ, and the gap shrinks at every step. By 21/13 = 1.615 you are already within 0.003 of φ, and a few terms later the ratio matches φ to as many decimals as the display can show. Convergence is fast, and no single overshoot ever stays far off for long.

The most irrational number 🖖

Written as a continued fraction, φ = [1; 1, 1, 1, …] — nothing but ones, the smallest coefficients possible. That makes φ the hardest number to approximate well with any fraction, and its best approximations are exactly the Fibonacci ratios this tool plots. Nature exploits this: seeds placed 360°/φ² ≈ 137.5° apart never line up into gaps, giving sunflowers and pinecones their tight, even packing.

Problem solved in full

Error measuring 144/89 = 1.617978 against φ = 1.618034 6 steps
144/89 = 1.617978 against φ = 1.618034. Predict that error before measuring it, then explain why φ is the worst number in mathematics to approximate.
1. The defining equation is not assumed, it is forced: if the ratio of consecutive terms settles anywhere, the recurrence says the limit satisfies this.
  
  $\frac{F _{n + 1}}{F _{n}} \to φ ⟹ φ = 1 + \frac{1}{φ} ⟹ φ^{2} - φ - 1 = 0$
2. Solve the quadratic and take the positive root.
  
  $φ = \frac{1 + 5}{2} = 1.618034$
3. The twelfth convergent, and the error you are asked to predict. It is negative, and the previous one was positive — the convergents alternate around φ.
  
  $\frac{F _{12}}{F _{11}} = \frac{144}{89} = 1.617978, error = - 5.65 \times 1 0^{- 5}$
4. Binet's formula gives the error in closed form. The conjugate root ψ is what the Fibonacci numbers throw away, and it is exactly what the approximation is missing.
  
  $ψ = \frac{1 - 5}{2} = - 0.618034, \frac{F _{n + 1}}{F _{n}} - φ = \frac{ψ ^{n}}{F _{n}} = \frac{- 0.005025}{89} = - 5.65 \times 1 0^{- 5}$
5. Compare against Hurwitz's theorem, which is the strongest statement possible about approximating any irrational by rationals. The error is not merely below the bound, it is on it.
  
  $Hurwitz : \frac{p}{q} - α < \frac{1}{5 q ^{2}} infinitely often; \frac{1}{5 \cdot 8 9 ^{2}} = 5.6459 \times 1 0^{- 5}$
6. The continued fraction says why. All ones is the slowest convergence available, and the golden angle is that fact turned into a growth rule.
  
  $φ = [1; 1, 1, 1, \dots] ⟹ golden angle = \frac{36 0 ^{\circ}}{φ ^{2}} = 137.50 8^{\circ}$
Answer

The error is ψ¹¹/F₁₁ = −5.6461 × 10⁻⁵, and it sits exactly on the floor no irrational can go under. Hurwitz proved that every irrational α has infinitely many rationals with |p/q − α| < 1/(√5 q²), and that the √5 cannot be improved. Here 1/(√5 × 89²) = 5.6459 × 10⁻⁵ — the actual error and the theoretical floor agree to one part in forty thousand. That is what "the most irrational number" means: not that φ is mysterious, but that its continued fraction [1;1,1,1,…] never offers a large digit, so no convergent ever gets a bargain and every one of them lands on the bound. Plants exploit it. Seeds placed 360°/φ² = 137.508° apart never fall into rows, because falling into rows means the angle is near a simple fraction of a turn — and φ is provably the number furthest from every simple fraction there is.

$\frac{ψ ^{11}}{F _{11}} = - 5.6461 \times 1 0^{- 5} against the Hurwitz floor \frac{1}{5 F _{11}^{2}} = 5.6459 \times 1 0^{- 5}$

References (1)

Insight block 3 — the divergence angle, and why it packs without gaps: H. Vogel, "A better way to construct the sunflower head." Mathematical Biosciences 44(3–4), 179–189, 1979 — the 137.5° model this block describes.

By the 42Math team · Updated 2026-07-20

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

early wobble - First 5 Fibonacci terms: ratio 5/3 = 1.667 — visibly above φ = 1.618
near phi - 10 terms: ratio 89/55 = 1.618181… — settling within 0.004% of φ
locked in - 16 terms: ratio 1597/987 = 1.6180339… — matches φ to 7 significant figures
Binet check - Binet's formula: F(20) = 6765 from closed-form expression using φ and ψ