Golden Ratio Calculator
The golden ratio φ = (1 + √5) ÷ 2 = 1.6180339887… is the number that solves φ² = φ + 1, so 1÷φ = φ − 1 = 0.618… Pick a mode, then scroll on for φ to 1000 places, the Fibonacci connection, and where φ shows up in nature and the human face.
φ to 1000 decimal places
Computed exactly with integer square roots — every digit is correct, truncated (not rounded) at the last place.
Fibonacci → φ
Divide each Fibonacci number by the one before it — 1, 2, 3, 5, 8, 13, 21… — and the ratio zig-zags ever closer to φ, overshooting and undershooting by less each time. It is the simplest place φ falls out of plain counting.
| Ratio | Value | Distance from φ |
|---|
The golden spiral
Quarter-circles drawn inside nested golden rectangles trace a logarithmic spiral that grows by a factor of φ every quarter-turn — the same self-similar curve a nautilus shell, a fern frond, and a spiral galaxy's arms approximate.
φ in nature
φ turns up wherever growth packs things efficiently — it is the “most irrational” number, the hardest to approximate with fractions, which is exactly why it spaces things out without repeating:
- Sunflowers & pinecones: seeds and scales spiral in counts that are consecutive Fibonacci numbers (often 34 and 55), set at the golden angle of ~137.5° so each new seed lands in the largest remaining gap.
- Nautilus & snail shells: grow as logarithmic spirals, widening by a near-constant factor each turn — the same self-similar growth the golden spiral idealises.
- Plant stems (phyllotaxis): leaves emerge at the golden angle so higher leaves shade lower ones as little as possible.
- Galaxies & weather: spiral galaxy arms and hurricanes trace logarithmic spirals close in feel to the golden spiral.
A caveat worth keeping: many “φ in nature” claims are loose. The Fibonacci spiral counts in sunflowers are real and well studied; perfect golden rectangles overlaid on shells and faces are usually after-the-fact and approximate.
φ and the human face & body
The classic “divine proportion” idea is that pleasing faces carry φ in their measurements — total face length ÷ width, the mouth’s width ÷ the nose’s, the distance between the eyes versus the width of an eye, and so on hovering near 1.618. The Is your body golden? mode above uses the best-known body version: height ÷ navel-to-floor height. Treat all of this as play, not science — real faces and bodies vary widely, and the “ideal” ratios are picked loosely after the fact. It is a fun yard-stick, nothing more.
How it works
The larger partner of a number is that number × φ; the smaller is the number ÷ φ. A golden cut splits a length into a long and a short part whose ratio is φ : 1; the long part is the whole ÷ φ. A golden rectangle has its sides in the ratio φ : 1, so a width’s golden height is either width × φ (portrait) or width ÷ φ (landscape) — both are golden. To check two numbers we divide the larger by the smaller and report how far that sits from φ as a percentage.