Golden ratio calculator
Phi is the only positive number where φ = 1 + 1/φ. Take a golden rectangle, cut off a square from one end, and what remains is another golden rectangle — the ratio is unchanged. Repeating that produces the spiral everyone has seen. It also means φ² = φ + 1, which is why the Fibonacci sequence converges on it: each term is the sum of the two before, and the ratio between consecutive terms settles on 1.618.
The golden ratio φ is (1 + √5) ÷ 2, about 1.6180339887. A golden rectangle with a short side of 100 has a long side of 161.8. Its defining property is that removing a square leaves another golden rectangle.
How to use the golden ratio
The golden ratio deserves its mathematical fame and rather less of its design mythology. The claims that the Parthenon, the Great Pyramid and the Mona Lisa were deliberately built on it do not survive measurement — the fits require choosing convenient reference points. Where it genuinely appears is in phyllotaxis, the arrangement of leaves and seeds, because being the most irrational number makes it the optimal packing angle. As a design proportion it is perfectly pleasant and no more magical than 3:2 or the square root of two.
Questions
(1 + √5) ÷ 2, about 1.618. It is the ratio where the whole is to the larger part as the larger is to the smaller.