I know some of you must be thinking, as I have, that the spherical radius ratio theory is flawed. First of all, why the radius, why not surface area or volume, as they sound more intuitive.
Here’s why: Taxicab Geometry
The only metric in Taxicab geometry is the radius, the rest is meaningless, as it is in the quantum world. There are no real spheres as these are just abstractions and constructions. Stuck here in MST we see spheres in space all the time. The planet is a sphere, so is the sun. Even the event horizon of a black hole is spherical, and yet its entropy is proportional to its surface area by virtue of holographic projection from a non 3D + time underlying quantum reality. See Leonard Susskind on The World As Hologram.
In our translation from the TPPT we only really claimed a relationship of ratio with the idea of octahedral close packing based on radius, not that the things had to related to spheres specifically. Like I said, spheres and surface area are constructions that require real numbers and we are a long way from these abstractions and illusions.
Thinking about octahedral close packing, 3D fractals and limits in the context of taxi cab geometry is not easy, but we will get there. Maybe there is some room for us to consider this mess in the context of yet another projection. There may be a role for Geometric (Clifford) Algebra here.