Fibonacci in a sunflower
Starter
What fraction of a full turn is 180°? 120°? 144°? 135°?
1/2, 1/3, 2/5, 3/8.
How many straight spokes does each of these angles give?
2, 3, 5 and 8: the bottom number of the fraction.
Why does a fraction of a turn always make spokes?
After as many seeds as the bottom number, the turns add up to a whole number of turns, so the next seed lands on the first spoke again.
Core
Work out 8/5, 13/8, 21/13, 34/21 and 55/34 to three decimals. What do you notice?
1.600, 1.625, 1.615, 1.619, 1.618: the ratios close in on the golden ratio φ ≈ 1.618 from both sides.
Work out 360° ÷ φ², with φ = 1.618034.
360 ÷ 2.618034 ≈ 137.5°.
The golden angle is 0.381966 of a turn. Multiply by 13, 21 and 34. How far is each from a whole number?
4.9656, 8.0213, 12.9868: 0.034, 0.021 and 0.013 of a turn away. Seeds 21 or 34 apart nearly line up, which makes the spirals.
Challenge
Show that 360° × (1 − 1/φ) = 360° ÷ φ², using φ² = φ + 1.
From φ² = φ + 1, dividing by φ: φ = 1 + 1/φ, so φ − 1 = 1/φ. Then 1 − 1/φ = (φ − 1)/φ = (1/φ)/φ = 1/φ².
Why do sunflowers show 21 and 34 spirals rather than 20 and 30?
Spirals appear between seeds that nearly line up, and for the golden angle those are a Fibonacci number apart: 13, 21, 34, 55…