Glossary entry

Golden Ratio (Phi, φ ≈ 1.618)

noun · / ˈɡəʊl.dən ˈreɪ.ʃiː.əʊ / · mathematical constant · also: phi, divine proportion, golden section

The unique positive number φ ≈ 1.6180339887 that satisfies the equation (a+b)/a = a/b. Geometrically, the division of a line such that the ratio of the whole to the larger part equals the ratio of the larger part to the smaller. Defined by Euclid in his Elements c. 300 BC under the name "extreme and mean ratio" and renamed "the divine proportion" by Luca Pacioli in 1509. Now the standard alternative to the rule of thirds in photography.

By Sarah Chen · Last updated 15 May 2026

Mathematics

The golden ratio is defined by the equation (a + b) / a = a / b. Solving gives φ = (1 + √5) / 2 ≈ 1.6180339887. Three properties matter for art and composition:

Phi is irrational — its decimal expansion never terminates or repeats.
1/φ = φ − 1 ≈ 0.618 — the reciprocal of phi is its own decimal part.
φ² = φ + 1 — phi has the property that adding successive powers produces a Fibonacci-like sequence.

For composition, what matters is that the golden ratio divides a unit length into segments of 0.382 and 0.618. These are the percentages used in the Phi grid for off-centre subject placement.

Origin

The earliest documented treatment is in Euclid's Elements (c. 300 BC), Book VI, Proposition 30. Euclid's interest was purely mathematical — there is no claim in the Elements that this ratio is beautiful or that artists should use it. The aesthetic claim came much later, with Luca Pacioli's De Divina Proportione (1509), illustrated by Leonardo da Vinci, which named the ratio "the divine proportion" and argued for it as a universal organising principle of beauty.

Use in composition

The golden ratio shows up in three composition overlays:

Phi Grid — divides the frame at 38.2% and 61.8% on both axes, giving four intersection points for off-centre subject placement. The direct alternative to the Rule of Thirds.
Golden Spiral — a logarithmic spiral with phi growth factor, constructed inside a Phi rectangle. For radial subjects.
Golden Triangle — diagonal-led composition with three triangles whose areas approximate golden proportions.

In Grid Maker Pro

The golden ratio is implemented as the Golden Ratio overlay (Phi grid), the Golden Spiral overlay, and the Golden Triangle overlay. A dedicated Golden Ratio tool landing page hosts the live tool. For a long-form treatment, see the Golden Ratio in Art & Photography pillar.

Related terms

Rule of thirds — the simpler 33.3 / 66.6 alternative.
Golden spiral — Fibonacci spiral derived from phi.

Citations

Euclid. Elements, Book VI, Proposition 30. c. 300 BC.
Pacioli, Luca. De Divina Proportione. Venice, 1509. Illustrated by Leonardo da Vinci.
Livio, Mario. The Golden Ratio: The Story of Phi. Broadway Books, 2002.
Wolfram MathWorld. Golden Ratio.

Definition

Golden ratio (φ ≈ 1.618033988…) — the positive real solution to the equation x² = x + 1, or equivalently, the ratio at which a line is divided so that the whole to the larger part equals the larger part to the smaller. Geometrically: a/b = (a+b)/a = φ. The conjugate property is unique among irrational numbers: 1/φ = φ − 1 = 0.618.

Applied to a rectangle, the golden ratio produces the phi rectangle (1:1.618). Applied to a line, it divides at 38.2% / 61.8%. Applied to a sequence: the limiting ratio of consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… → 1.618).

Etymology and origin

The proportion is first documented in Euclid's Elements (c. 300 BCE), Book VI Definition 3 and Proposition 30, under the name extreme and mean ratio. Euclid gave the canonical compass-and-straightedge construction.

The Renaissance renamed it divine proportion via Luca Pacioli's 1509 treatise De Divina Proportione, illustrated by Leonardo da Vinci. The German psychologist Adolf Zeising introduced the modern "golden" terminology in 1854, simultaneously inflating its claimed presence in nature and art beyond what the data supported.