φ (phi) is the positive solution to x² = x + 1. This equation has a geometric meaning: if you divide a line segment so that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part, that ratio is φ. No other number has this self-similar property.
Table of Fibonacci ratios converging to phi
| Fib pair | ratio | distance to φ |
|---|---|---|
| 1, 1 | 1.000 | 0.618 |
| 2, 3 | 1.500 | 0.118 |
| 8, 13 | 1.625 | 0.007 |
| 55, 89 | 1.61818… | 0.00015 |
| → ∞ | 1.61803… | 0 |
The golden ratio appears in the regular pentagon and pentagram, where diagonals intersect each other in the golden ratio. Every Fibonacci number divided by the previous one approaches φ. The continued fraction [1; 1, 1, 1, …] is the simplest infinite continued fraction: all 1s. This makes φ the hardest number to approximate by fractions, earning it the title "most irrational number".
Cut a square from a golden rectangle. The remaining piece is another golden rectangle, smaller by factor 1/φ. Repeat forever. The arc traces the golden spiral seen in shells and galaxies.
φ satisfies φ² = φ + 1, so φ = 1 + 1/φ. Substituting repeatedly: φ = 1 + 1/(1 + 1/(1 + …)). This infinite continued fraction of all 1s is both the definition and the reason for its "most irrational" status. Computed to full precision: 1.61803398874989484820…
In a regular pentagon with side length 1, every diagonal has length φ ≈ 1.618. The diagonals also divide each other in the golden ratio. Draw all five diagonals and you get a pentagram: itself full of golden proportions.
The golden ratio phi is approximately 1.61803398874989484820. It is the positive solution to x² = x + 1. Phi is irrational, algebraic, and the limiting ratio of consecutive Fibonacci numbers. It appears in the regular pentagon and icosahedron, in sunflower seed spirals, and in the proportions studied since ancient Greece. Its continued fraction [1; 1, 1, 1, ...] makes it the hardest real number to approximate with fractions, which is why phyllotaxis uses the golden angle derived from phi.
Golden Ratio φ is irrational. Its decimal expansion never ends and never repeats. Every digit shown below is computed from the quadratic formula.