# Sphere packing in a sphere

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
unit spheres
Maximum radius of inner spheres[1] Optimality Diagram
Exact form Approximate
1 ${\displaystyle 1}$ 1.0000 Trivially optimal.
2 ${\displaystyle {\dfrac {1}{2}}}$ 0.5000 Trivially optimal.
3 ${\displaystyle 2{\sqrt {3}}-3}$ 0.4641... Trivially optimal.
4 ${\displaystyle {\sqrt {6}}-2}$ 0.4494... Proven optimal.
5 ${\displaystyle {\sqrt {2}}-1}$ 0.4142... Proven optimal.
6 ${\displaystyle {\sqrt {2}}-1}$ 0.4142... Proven optimal.
7 0.3859... Proven optimal.
8 0.3780... Proven optimal.
9 0.3660... Proven optimal.
10 0.3530... Proven optimal.
11 ${\displaystyle {\dfrac {{\sqrt {5}}-3}{2}}+{\sqrt {5-2{\sqrt {5}}}}}$ 0.3445... Proven optimal.
12 ${\displaystyle {\dfrac {{\sqrt {5}}-3}{2}}+{\sqrt {5-2{\sqrt {5}}}}}$ 0.3445... Proven optimal.

## References

1. ^ Pfoertner, Hugo (2008-02-02). "Densest Packings of n Equal Spheres in a Sphere of Radius 1. Largest Possible Radii". Archived from the original on 2012-03-30. Retrieved 2013-11-02.
• Huang, WenQi; Yu, Liang (2012). "Serial Symmetrical Relocation Algorithm for the Equal Sphere Packing Problem". arXiv:.
• Gensane, T. (2003). "Dense packings of equal spheres in a larger sphere". Les Cahiers du LMPA J. Liouville. 188.