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 ofinner spheres Maximum radius of inner spheresBest packing of m>1 equal spheres in a sphere setting a new density record PackingdensityThe best known packings of equal spheres in a sphere (complete up to N = 900), Packomania 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. Sphenocorona 11 0.3445... 0.45003... Proven optimal. Diminished icosahedron 12 0.3445... 0.49095... Proven optimal. Icosahedron
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