Following the post on Apollonius gasket (Viete way), I made a try on 3D. for me the equivalent is Soddy's kissing sphere.

1. place 4 tangent spheres, themselves tangent to a containing one. (5 spheres)

2. group them by 4 tangent sphere

3. compute the radius (following Descartes theorem for 3D - Descartes' theorem - Wikipedia

4 compute the center of the sphere

5 regroup the sphere to get 5 spheres

6 loop!

it seems that I got the same result as Nicolas Hannachi  Kissing Spheres (pagesperso-orange.fr)

BUT, I feel that I am missing some cavities and also some spheres are computed 2 several times (it is like a "geometrical leak" (no issue with the software - it is more on the algo)... which does not happen in 2D because Apollonius gaskets are closed. I wanted to generate a Voronoi structure but because of that it is impossible.

I did not study variation of initial spheres. in yellow the initial internal sphere curvature quadruplet is (1, 1, 1, 1). I made a try and render with (2, 3, 3, 3)

3dxml below

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