Golden Ratio & Fibonacci Spiral
Build Fibonacci squares, compare consecutive ratios, and see how the recurrence approaches the golden ratio.
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 x^2 = 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.
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