Nested Spheres

16/Nov/2018

This is a new shape that I have made based on sphere inversions. It is a nested set of spheres, such that each level of spheres is a Ford-Farey sphere packing over the Riemann sphere.

The structure is a tree, it has no self contact, but is also probably the densest packing that such a tree can be.

Each level is a graphical depiction of the Eistenstein rationals. So the total shape depicts an infinite cascade of Eisenstein rationals, rather like each nested sphere depicts the rationals over the infinitesimal gap around its parent's location on the complex plane. 

As such, one might describe it as a cascade-like depiction of the complex numbers. 

Here is another picture:

and a close up:

The structure is a cluster-sponge-tree, which means that if the sphere radii were reduced it would be a cluster (disparate spheres) and if they were increased it would be a normal sponge-tree. A sponge-tree is a tree structure (branches) which also has holes through it (like a sponge). See https://sites.google.com/site/simplextable/.

When the height is 1 m, then its volume is 0.21 m^3, its surface area is 0.74 m^2.56, and the number of spheres is 0.54 m^2.56.  These approximate numbers refer to the Minkowski content and dimension of the shape. And the sphere count refers to the Minkowski content and dimension of the set of sphere centre points.

Update July 2020: We can parameterise this shape by the packing ratio, here we see it with the ratio changing evenly from 0.4 up to 1.2:

The shape self-intersects when this ratio is greater than 1, taking the shape from a tree structure to a sponge structure.

The Fragmentarium code to render it is SphereTree.frag:
  https://github.com/TGlad/FragmentariumExamples

There is one more option that I have looked at. You can remove the 30 degree rotation in the formula and then you need to scale it differently. If gives a more typical Kleinian fractal look, but is still very nice, a cluster-sponge (when packing ratio is 1), made of lots of circles:

The spheres shrink to nothing for larger iterations, but here is a method that avoids those spurious spheres:

Despite looking like an empty (void) sponge, it is in fact made of spheres (or rather, balls), so it solid with mass.

The nested spheres structure is now discussed in detail in my book Exploring Scale Symmetry, in Chapter 4.