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 inner spheres	Maximum radius of inner spheres		Packing density	Optimality	Diagram
	Exact form	Approximate
1	[math]\displaystyle{ 1 }[/math]	1.0000	1	Trivially optimal.
2	[math]\displaystyle{ \dfrac {1} {2} }[/math]	0.5000	0.25	Trivially optimal.
3	[math]\displaystyle{ 2 \sqrt {3} - 3 }[/math]	0.4641...	0.29988...	Trivially optimal.
4	[math]\displaystyle{ \sqrt {6} - 2 }[/math]	0.4494...	0.36326...	Proven optimal.
5	[math]\displaystyle{ \sqrt {2} - 1 }[/math]	0.4142...	0.35533...	Proven optimal.
6	[math]\displaystyle{ \sqrt {2} - 1 }[/math]	0.4142...	0.42640...	Proven optimal.
7		0.3859...	0.40231...	Proven optimal.
8		0.3780...	0.43217...	Proven optimal.
9		0.3660...	0.44134...	Proven optimal.
10		0.3530...	0.44005...	Proven optimal.
11	[math]\displaystyle{ \dfrac {\sqrt{5} - 3} {2} + \sqrt{5 - 2 \sqrt{5} } }[/math]	0.3445...	0.45003...	Proven optimal.
12	[math]\displaystyle{ \dfrac {\sqrt{5} - 3} {2} + \sqrt{5 - 2 \sqrt{5} } }[/math]	0.3445...	0.49095...	Proven optimal.