A long time ago, we have looked at Soddy’s Hexlet, where a chain of six spheres is interlinked with a chain of three spheres.
There are variations of this. For instance, you can have two interlinked chains of four spheres each.
The alert visitor will have noticed that I am only displaying halves of spheres. This is because it is easier to add the other halves on one’s mind instead of thinking them away in order to see what’s behind.
There is more. If you take a suitable chain of five spheres, you can fit 10 around and through, but you will need to make three turns until the chain closes. This means that the spheres will touch their immediate successors, but intersect the ones after one and two turns, respectively.
There still is more, of course, which we leave to the reader to explore. Finding these chains is not difficult, provided you do this in the 3-dimensional sphere, and place the spheres inside complementary tori with suitable radii.
The Inner Frame 