Spheres & Flow · 3D
Sphere Packing
13 spheres of the vector equilibrium
Twelve equal spheres can touch a thirteenth: the densest possible cluster. Their centres form Fuller’s vector equilibrium. Its middle layer is six circles around one, the heart of the Fruit of Life.
- Spheres
- 13
- Kissing number
- 12
Turn it in your hands, strike it to hear the chord of its edges, slice it with a plane or lift it into four dimensions.
Facts
- In 3D, at most 12 equal spheres can touch one: the kissing number (proved 1953).
- Stacking this pattern forever gives the densest packing (Kepler’s conjecture, proved 2014).
- Its shadow down a 3-fold axis is a hexagonal flower.
History and meaning
Newton and Gregory argued in 1694 about whether 13 spheres could touch one. Newton said 12, and was right.
Across the dimensions
- Equator of 24-CellIn 4D, twenty-four spheres can touch one. Their centres are the vertices of the 24-cell.
Learn it step by step
- Journey The Breathing SolidStep 1: Twelve around one
