Sphere packing in a sphere

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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
OptimalityArrangementDiagram
Exact formApproximate
11.00001Trivially optimal. Point Spheres in sphere 01.png
20.50000.25Trivially optimal. Line segment Spheres in sphere 02.png
30.4641...0.29988...Trivially optimal. Triangle Spheres in sphere 03.png
40.4494...0.36326...Proven optimal. Tetrahedron Spheres in sphere 04.png
50.4142...0.35533...Proven optimal. Trigonal bipyramid Spheres in sphere 05.png
60.4142...0.42640...Proven optimal. Octahedron Spheres in sphere 06.png
70.3859...0.40231...Proven optimal. Capped octahedron Spheres in sphere 07.png
80.3780...0.43217...Proven optimal. Square antiprism Spheres in sphere 08.png
90.3660...0.44134...Proven optimal. Tricapped trigonal prism Spheres in sphere 09.png
100.3530...0.44005...Proven optimal. Spheres in sphere 10.png
110.3445...0.45003...Proven optimal. Diminished icosahedron Spheres in sphere 11.png
120.3445...0.49095...Proven optimal. Icosahedron Spheres in sphere 12.png

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