The Most Irrational Number

φ and sunflowers

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $\varphi = [1; 1, 1, \dots]$, with Fibonacci-ratio convergents.
• All-small terms make φ the hardest number to approximate: the “most irrational”.
• The golden angle (≈137.5°) packs sunflower seeds evenly.

Steps

1. All ones
2. Self-similar
3. Packing seeds
4. Spokes
5. Hurwitz’s theorem
6. Ten ones