Golden Ratio and the Fibonacci Sequence

Key points

• Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …: each term is the sum of the two before it.
• Ratios of consecutive terms (8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615 …) approach the golden ratio φ = (1 + √5)/2 ≈ 1.618.
• Squares with Fibonacci sides tile a near-golden rectangle; quarter arcs through them draw the golden spiral.
• Sunflower seeds grow about 137.5° (the golden angle) apart, so they pack without gaps; seed spirals often come in consecutive Fibonacci counts such as 34 and 55.

The ratio settles

φ = (1 + √5) / 2 ≈ 1.618

In nature (often, not always)

• Sunflower heads: 34 and 55 (or 55 and 89) spirals
• Pinecones: 8 and 13 spirals
• Many flowers: 3, 5 or 8 petals (lily 3, buttercup 5, cosmos 8)
• Not every spiral is golden: the nautilus shell's spiral, for example, is not.

Why plants do it

• Each new seed or leaf grows about 137.5° round from the last (360°/φ²).
• This angle never lines seeds up in straight rows, so they fill the head without gaps.
• A simple fraction of a turn, like 135° (3/8), lines seeds into spokes with gaps.

Try it yourself

Goal: Change the turn between seeds and find the angle that packs best, then test whether any start leads to φ.

• Move the angle slider and watch the seeds; try 135°, 137°, 137.5° and 138°.
• Compare the closest two seeds and the biggest hole for each angle.
• At 137.5°, colour the spirals with 21, 34 and 55.
• Enter your own two starting numbers and watch the ratio.

Open the simulation full screen

Quick quiz

3 questions. Choose an answer for each, then submit to see your score and the explanations.