Fibonacci and the golden ratio
Build the Fibonacci numbers 1, 1, 2, 3, 5, 8, … as a spiral of squares, and divide each by the one before. Watch the ratio swing above and below and settle on 1.618, the golden ratio found in sunflowers, pine cones and shells.
Core concept
In the Fibonacci sequence each number is the sum of the two before it (Fₙ₊₁ = Fₙ + Fₙ₋₁); the ratio of neighbours Fₙ₊₁ ÷ Fₙ gets closer and closer to the golden ratio φ = (1 + √5) ÷ 2 ≈ 1.618
Topic: Maths in everyday life
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, … Draw squares with these sides next to each other and they make a growing rectangle; a curve through them makes a spiral like a snail's shell. Divide each number by the one before it: 2 ÷ 1 = 2, 3 ÷ 2 = 1.5, 5 ÷ 3 = 1.667, 8 ÷ 5 = 1.6, … The ratio: A way to compare two amounts, like 2 cups of rice to 3 cups of water (2 : 3). All key words jump above and below a number and close in on it: the golden ratio φ = (1 + √5) ÷ 2 = 1.618… Sunflower seeds, pine cones and pineapples often have spirals in Fibonacci numbers, such as 34 and 55.
What you will learn
- Continue the Fibonacci sequence by adding the two numbers before.
- Work out the ratio of neighbouring Fibonacci numbers.
- Describe how the ratio approaches the golden ratio, 1.618.
Before you start
- Know how to divide and round to decimal places.
Materials
- Squared paper to draw the Fibonacci squares
- A sunflower head or pine cone, if you can find one
Equipment
- None
Procedure
- Set n to 2 and record F₂, F₃ and their ratio.
- Record n = 6.
- Record n = 10 and n = 15: how close is the ratio to 1.618?
How this simulation is simplified
- F₁ = F₂ = 1. Fibonacci numbers are found with Binet's formula, Fₙ = φⁿ ÷ √5 rounded to the nearest whole number.