Sphere packings in 3D
How does one fill a sphere with smaller spheres of various sizes so that every possible void is filled?
There are only five known configurations, all obtained by a sphere inversion transformation, the 3D equivalent of a circle inversion. Initial spheres are positioned at the vertices of a platonic solid. The sphere containing and touching all these spheres is part of the constellation also, but is obviously never drawn in the images.
Inversion spheres are placed so that they are orthogonal to the spheres at the faces of the platonic solid. One inversion sphere is at the center, orthogonal to all initial spheres. For the case of the tetrahedron, cube and dodecahedron, there is only one possible configuration, consisting of touching initial spheres. In the case of the octahedron, there is a constellation with touching spheres, and one other with nontouching spheres. No packing can be obtained starting from the icosahedron.
The colors of the individual spheres in the images correspond to the iteration count.
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Dodeca 2

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