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A rough idea of why the Fibonaccis and the golden ratio are related: in making a golden rectangle, you take an existing rectangle, attach a compass across its long end, and use that compass to construct a square. You add the square to the golden rectangle to get... another golden rectangle. If you wanted to repeat this, there is also a "flipping" of the rectangle by 90 degrees. Mathematically, the process takes a rectangle (a, b) (where b > a) and produces a rectangle (b, b + a). But that's just the Fibonacci recursion relation, so starting from (0, 1) or (1, 1) you'll generate all the Fibonacci rectangles as you 'goldenize' them further and further. An actual proof: Since φ and 1 − φ are the solutions to the quadratic x² = x + 1, they also solve xⁿ = xⁿ⁻¹ + xⁿ⁻² and thereby form a basis for all solutions of the Fibonacci recurrence relation: in other words, you can solve the system of equations {A + B = F₀ ; A φ + B (1 − φ) = F₁} for parameters A and B, and then the recurrence relation guarantees A φⁿ + B (1 − φ)ⁿ = Fₙ for all remaining Fibonaccis. (For F₀ = 0, F₁ = 1 this gives A = -B = 1/sqrt(5).) Since |1 − φ| < 1, this decays geometrically to 0 and the dominant term is simply Fₙ ≈ A φⁿ. The article thankfully does explain that the golden ratio is the most irrational number: because a continued fraction expansion gives the best rational approximations, large numbers in the continued fraction expansion make for extra-good rational approximations, and the golden ratio has the smallest numbers possible -- they're all 1. What it doesn't quite explain is that if you're growing a spiral by spitting out dots and you turn by an angle 2π · α each time you spit out a new dot, those spirals often seem to "line up" in lines given by the denominators of the best rational approximations. So if you have a rational number like α = 22/7, you would literally see just 7 straight spokes coming out of the center; if you use α = π you will see a region where there seem to be seven spirals due to the fact that 22/7 is a disproportionately good approximation to π, off not by ~ 1/14 but rather by ~1/790. Anyway this is the more concrete reason why, if you take a sunflower and count the spirals, you always seem to "magically" get Fibonacci numbers; they're denominators in the best rational approximations. You may wish to experiment with a spreadsheet to see all of this dynamically for yourself, a-la http://tmp.drostie.org/sunflower.png . |
