A theorem usually attributed to Soddy (but I guess actually disovered
originally by some unknown Japanese) says that given any two tangent
spheres, internally tangent to a third, a chain of exactly six tangent
spheres can be nested about the first two and inside the larger.
http://mathworld.wolfram.com/Hexlet.html
This is a picture from the simplest case, when the two original spheres
are the same size, and the union of their collinear diameters forms a
diameter of the larger sphere. Then the six spheres in the chain are
all identical and with diameters 1/3 of the largest sphere. Again,
rendered using the "reflective spheres and spotlights" trick.
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