What is the golden ratio?
A ratio compares two lengths by division. Cut a line into two parts, a (big) and b (small). The cut is golden when
(a + b) ÷ a = a ÷ b
The whole compares to the big part in the same way the big part compares to the small part. This shared value is called φ (the Greek letter phi). φ ≈ 1.618.
A proportion is when two ratios are equal. So the golden cut is a special proportion. It also keeps a kind of balance (harmony) that is not perfect symmetry: the two parts are different, but related in a neat way.
Finding the exact value of φ
Let b = 1 and a = x. The rule becomes (x + 1) ÷ x = x ÷ 1.
Multiply both sides by x: x + 1 = x², so x² − x − 1 = 0.
The quadratic formula gives x = (1 + √5) ÷ 2 (we keep the positive answer, because a length is positive).
φ = (1 + √5) ÷ 2 = 1.6180339… It never ends and never repeats, so φ is an irrational number.
Useful facts: φ² = φ + 1 = 2.618…, and 1/φ = φ − 1 = 0.618…
Golden rectangle and golden spiral
A golden rectangle has length ÷ width = φ. Cut off a square whose side is the width. The rectangle left over has the same shape as the first one: it is golden too. Do it again and again and the squares get smaller and smaller.
Draw a quarter circle inside each square, from corner to corner. The joined curve is the golden spiral. It grows by the same factor (φ) every quarter turn, so it looks the same at every size. Shells, horns and storms often show spirals of this kind (not always exactly golden).
Fibonacci numbers lead to φ
The Fibonacci sequence starts 1, 1, and each next number is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
Divide each number by the one before it: 2/1 = 2, 3/2 = 1.5, 5/3 = 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615… The answers swing above and below and settle on 1.618 = φ.
Where people see it: leaves and seeds arranged at about 137.5° (the golden angle) so each gets light; old temples and paintings measured by artists; music where a climax is placed about 0.618 of the way through a piece. Be careful: many claims are exaggerated. Always measure before you believe. Musical scales also use simple ratios (2:1 octave, 3:2 fifth) found by the Pythagoreans; those are different from φ.
Try it at home
Measure your height and the height of your navel from the floor. Divide: height ÷ navel height. Many people get between 1.55 and 1.7. Then measure 5 rectangles at home (book, phone, TV, card, door) and work out length ÷ width. Which is closest to 1.618?
Key formulas and definitions
- (a + b) ÷ a = a ÷ b = φ
- φ = (1 + √5) ÷ 2 ≈ 1.618
- φ² = φ + 1 ≈ 2.618
- 1/φ = φ − 1 ≈ 0.618
- Golden rectangle: length = φ × width
- Fibonacci: F(n+1) ÷ F(n) → φ
Worked examples
1. A golden rectangle is 10 cm wide. How long is it?
Length = φ × width = 1.618 × 10 = 16.18 cm.
2. A 100 cm stick is cut in the golden ratio. Find the two parts.
Big part a = 100 ÷ φ = 100 × 0.618 = 61.8 cm. Small part b = 100 − 61.8 = 38.2 cm. Check: 61.8 ÷ 38.2 ≈ 1.618.
3. Show that φ = (1 + √5)/2 satisfies x² = x + 1.
√5 ≈ 2.236, so φ ≈ 1.618. φ² ≈ 2.618 and φ + 1 ≈ 2.618. Exactly: φ² = (1 + 2√5 + 5)/4 = (6 + 2√5)/4 = (3 + √5)/2 = φ + 1.
4. Find 21 ÷ 13 and 34 ÷ 21. Which is closer to φ?
21 ÷ 13 = 1.6154; 34 ÷ 21 = 1.6190. Differences from 1.6180: 0.0026 and 0.0010. So 34 ÷ 21 is closer.
5. A poster is 70 cm long. What width makes it golden?
Width = length ÷ φ = 70 ÷ 1.618 ≈ 43.3 cm.
6. From a golden rectangle 16.18 × 10, cut a 10 × 10 square. Is the rest golden?
The rest is 10 × 6.18. 10 ÷ 6.18 ≈ 1.618. Yes, it is golden again.
Common mistakes
- Thinking φ = 1.6 exactly. It is (1 + √5)/2 = 1.6180339…, an irrational number.
- Dividing small ÷ big and calling 0.618 the golden ratio. The golden ratio is big ÷ small = 1.618; 0.618 is 1/φ.
- Believing every famous building or face is exactly golden. Many such claims are not supported by real measurements.
- Thinking Fibonacci ratios equal φ. They only get closer and closer; no Fibonacci ratio is exactly φ.