Golden ratio calculator

Two multiplications, and the arithmetic is the least interesting thing on the page. φ is the only positive number satisfying φ = 1 + 1/φ, which is the same as saying φ² = φ + 1, and every property people find remarkable follows from that one line. Its reciprocal is 0.6180339887, the same digits after the point. Its square is 2.6180339887, the same digits again.

Fibonacci, and how quickly it arrives

Each Fibonacci number is the sum of the two before it, so the ratio between neighbours obeys the same recurrence as φ = 1 + 1/φ and converges on it. What surprises people is the speed. 21/13 is 1.615385, already right to two decimals. Twelve steps in, 377/233 is 1.6180258, within 8 in the millionth place. The seeds barely matter: begin with 4 and 7 rather than 1 and 1 and the ratio still walks to the same place, because the convergence belongs to the recurrence rather than to the numbers you started from.

The design mythology mostly does not survive measurement

The Parthenon, the Great Pyramid, Notre-Dame, the Mona Lisa and the proportions of the human body are all routinely presented as built on φ. Markowsky went through the standard claims in the College Mathematics Journal in 1992 and found the usual pattern behind each: the fit is obtained by choosing where to put the ends of the measurement, ignoring the fact that a rectangle drawn generously enough will match 1.618 as easily as it matches 1.6 or 1.5, and quoting a figure to more precision than the ruined stone supports. Nothing in the surviving documentary record from any of these projects mentions the ratio, and no measurement of them singles it out from its neighbours.

What is left is still worth having. As a proportion for a layout it is pleasant, and it is pleasant in the same undramatic way that 3:2 and √2 are pleasant. The A-series paper sizes use √2 precisely because halving an A4 sheet preserves the proportion, an argument from a real requirement rather than from aesthetics; the golden rectangle has the analogous property under removing a square instead of halving.

Where it does turn up, and why

Phyllotaxis is the case that holds. New leaves or florets placed at a fixed angle around a growing stem overlap badly if that angle is a rational fraction of a turn, because after a few placements the pattern repeats and everything lines up in spokes. The angle that avoids repetition longest is the one hardest to approximate by a fraction, and φ is exactly that number: its continued fraction is all ones, which is the slowest possible convergence. Divide a full turn by φ² and you get 137.5078 degrees, the golden angle, and that is the spacing you find measured in sunflower heads and pine cones. The mechanism is a packing constraint, not a preference for a nice number.