Worksheet · Numbers and geometry

Fibonacci in a sunflower

Name: Date:

Before the film: predict

Seeds are placed one by one, each turned by the same angle from the one before. What shape do you get with half a turn (180°)? With a third of a turn (120°)?

Starter

What fraction of a full turn is 180°? 120°? 144°? 135°?
How many straight spokes does each of these angles give?
Why does a fraction of a turn always make spokes?

Core

Work out 8/5, 13/8, 21/13, 34/21 and 55/34 to three decimals. What do you notice?
Work out 360° ÷ φ², with φ = 1.618034.
The golden angle is 0.381966 of a turn. Multiply by 13, 21 and 34. How far is each from a whole number?

Challenge

Show that 360° × (1 − 1/φ) = 360° ÷ φ², using φ² = φ + 1.
Why do sunflowers show 21 and 34 spirals rather than 20 and 30?