Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It is the three-dimensional equivalent of the circle packing in a circle problem in two dimensions.
| Number of inner spheres | Maximum radius of inner spheres[1] | Packing density | Optimality | Arrangement | Diagram | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Exact form | Approximate | ||||||||||||||||||
| 1 | 1 | 1.0000 | 1 | Trivially optimal. | Point | ||||||||||||||
| 2 | \dfrac{1}{2} | 0.5000 | 0.25 | Trivially optimal. | Line segment | ||||||||||||||
| 3 | 2\sqrt{3}-3 | 0.4641... | 0.29988... | Trivially optimal. | Triangle | ||||||||||||||
| 4 | \sqrt{6}-2 | 0.4494... | 0.36326... | Proven optimal. | Tetrahedron | ||||||||||||||
| 5 | \sqrt{2}-1 | 0.4142... | 0.35533... | Proven optimal. | Trigonal bipyramid | ||||||||||||||
| 6 | \sqrt{2}-1 | 0.4142... | 0.42640... | Proven optimal. | Octahedron | ||||||||||||||
| 7 |
{\sqrt{2+2\sqrt{3}\cos\left(
\right)}}+1} | 0.3859... | 0.40231... | Proven optimal. | Capped octahedron | ||||||||||||||
| 8 |
| 0.3780... | 0.43217... | Proven optimal. | Square antiprism | ||||||||||||||
| 9 |
1}{2} | 0.3660... | 0.44134... | Proven optimal. | Tricapped trigonal prism | ||||||||||||||
| 10 | 0.3530... | 0.44005... | Proven optimal. | ||||||||||||||||
| 11 | \dfrac{\sqrt{5}-3}{2}+\sqrt{5-2\sqrt{5}} | 0.3445... | 0.45003... | Proven optimal. | Diminished icosahedron | ||||||||||||||
| 12 | \dfrac{\sqrt{5}-3}{2}+\sqrt{5-2\sqrt{5}} | 0.3445... | 0.49095... | Proven optimal. | Icosahedron | ||||||||||||||