Golden Ratio – Mathematics' Most Famous Proportion
The Golden Ratio, written φ (the Greek letter phi) and equal to approximately 1.618, is a special number that appears when a quantity is divided into two parts so that the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part. That simple self-referencing definition turns out to connect geometry, algebra, art, and nature in ways that have fascinated mathematicians for over two thousand years.
The ratio was first defined rigorously by Euclid around 300 BCE in his Elements, where he called it the “extreme and mean ratio” and used it to construct a regular pentagon. The name “Golden Ratio” itself is far more recent: it was popularised in the 19th century, and the Greek letter φ was chosen in the early 20th century by the American mathematician Mark Barr, reportedly in honour of the ancient Greek sculptor Phidias, who was believed (though not proven) to have used the ratio in designing the Parthenon. Some widely repeated claims about the Golden Ratio in ancient art and architecture are exaggerated or unverifiable, but its appearance in Renaissance work is well documented: the Italian mathematician Luca Pacioli devoted an entire 1509 book, De Divina Proportione (“On the Divine Proportion”), to it, illustrated by his friend Leonardo da Vinci.
Defining the Golden Ratio
If a line is split into a longer part a and a shorter part b so that (a + b)/a = a/b, that common ratio is the Golden Ratio:
φ = (1 + √5) / 2 ≈ 1.6180339887…
Algebraic Properties
φ is the positive solution of the quadratic equation x² − x − 1 = 0, which gives it two curious properties shared by no other number:
φ² = φ + 1 (squaring it just adds 1)
1/φ = φ − 1 (its reciprocal is just 1 less)
φ² = 1.618² = 2.617924.
φ + 1 = 1.618 + 1 = 2.618.
These match (to 3 decimal places): φ² ≈ φ + 1. ✓
The Golden Rectangle
A golden rectangle has its longer side divided by its shorter side equal to φ. If you remove a square from a golden rectangle, the smaller rectangle left over is also a golden rectangle – you can repeat this forever, and connecting the corners of each square with a quarter-circle arc traces out a spiral, sometimes called the golden spiral.
Longer side = shorter side × φ = 10 × 1.618 = 16.18 cm.
The Golden Ratio and the Fibonacci Sequence
The Golden Ratio is inseparable from the Fibonacci sequence: as you go further along 1, 1, 2, 3, 5, 8, 13, 21…, the ratio of each term to the one before it gets closer and closer to φ. For example, 21/13 ≈ 1.6154 and 34/21 ≈ 1.6190 – both already close to 1.618.
The Golden Ratio in Art and Nature
| Where | How φ appears |
|---|---|
| Nautilus shells and sunflower spirals | Growth spirals whose proportions approximate the golden spiral |
| Da Vinci's Vitruvian Man | Body proportions often cited as approximating φ (though precision varies) |
| Photography and design | The “rule of thirds” is a simplified approximation of golden-ratio composition |
| Credit cards | Standard size (85.60 × 53.98 mm) is very close to a golden rectangle |
Key Takeaways
- φ = (1 + √5) / 2 ≈ 1.618, defined so that (a+b)/a = a/b.
- φ is the only number where φ² = φ + 1 and 1/φ = φ − 1.
- Consecutive Fibonacci ratios converge to φ.
- A golden rectangle can be split into a square and a smaller golden rectangle, repeated indefinitely.