Apollonian Sphere Packing or Soddy Spheres

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Apollonion Sphere Packing is a powerful algorithm that efficiently fills a hollow ball with spheres of varying diameters without any intersections, ensuring that they touch each other perfectly. At infinity, there's no empty space left inside the ball. The algorithm begins by placing four balls at the vertices of a tetrahedron. This package includes five stages of the Apollonion Sphere Packing process, featuring a total number of spheres including a circumscribing sphere that is not included in the STL-file. The files are labeled with numbers representing the radius of the smallest spheres occurring within them. For printing purposes, most STL-files have been halved to make it easier to print the packed spheres. To assemble the printed parts, simply glue both sides together while ensuring the correct orientation is maintained. However, be cautious as there may be a small overlap between the spheres in the STL-file. Note that despite the Soddy Spheres being non-intersecting, there is a tiny overlap in the STL-files, which can be useful when printing and connecting the spheres afterwards. Print Settings Rafts: No Supports: No Design Process Creating Soddy Spheres To grasp the mathematics behind Soddy Spheres, refer to http://mathworld.wolfram.com/TangentSpheres.html or http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.127.4067&rep=rep1&type=pdf or http://paulbourke.net/fractals/apollony/. I developed a C++ program to efficiently find all spheres up to about 145,000 for the most dense packing. The output was then exported to openscad and converted to STL. Although the most dense packing exceeded openscad's capacity. The algorithm begins with four spheres of radius one at the vertices of a tetrahedron: [1, 0, -sqrt(1/2)], [-1, 0, -sqrt(1/2)], [0, 1, sqrt(1/2)], and [0, -1, sqrt(1/2)]. These spheres are circumscribed by a centered sphere with radius one / (1 + sqrt(6) / 2). Then, through an iterative procedure, quadruplets of spheres are selected, the two touching spheres are computed, and added to the collection.