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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[1] Packing
density
Optimality Arrangement Diagram
Exact form Approximate
1 1.0000 1 Trivially optimal. Point
2 0.5000 0.25 Trivially optimal. Line segment
3 0.4641... 0.29988... Trivially optimal. Triangle
4 0.4494... 0.36326... Proven optimal. Tetrahedron
5 0.4142... 0.35533... Proven optimal. Trigonal bipyramid
6 0.4142... 0.42640... Proven optimal. Octahedron
7 0.3859... 0.40231... Proven optimal. Capped octahedron
8 0.3780... 0.43217... Proven optimal. Square antiprism
9 0.3660... 0.44134... Proven optimal. Tricapped trigonal prism
10 0.3530... 0.44005... Proven optimal.
11 0.3445... 0.45003... Proven optimal. Diminished icosahedron
12 0.3445... 0.49095... Proven optimal. Icosahedron

References

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