Packing Spheres

The greengrocer’s problem

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

• Hexagonal packing of circles fills π/(2√3) ≈ 90.7% of the plane, the best possible.
• The greengrocer’s stacking of spheres fills π/(3√2) ≈ 74%.
• Kepler’s conjecture that nothing beats it was proved by Hales with computer help.

Steps

1. Stacking oranges
2. A square grid
3. Shift the rows
4. The greengrocer’s density
5. Kepler’s conjecture
6. Kissing spheres