There are exactly 3 associative real normed division algebras as the Frobenius theorem from 1879 tells. Each of them produce natural Lie groups . Each of them produces natural dense sphere packings . Each of them produces natural rings , the ring of integers, the Eisenstein ring and the ring of quaternion integers. Each of them produces natural Goldbach conjectures: positive even integers larger than 2 are sums of primes, every Eisenstein integer with entries is a sum of positive Eisenstein primes, every Eisenstein quaternion with entries larger than 1 is a sum of two Hurwitz quaternions with positive entries. Each of them produces a natural sphere packing which are known to be optimal. The rational integers, the Hex lattice and D4 are known to be the densest lattice sphere packings (it is only conjectured that D4 is also the densest packing overall). The integers and the Hex lattice are fixed points of the soft Barycentric renormalization map. I wondered whether the D4 lattice has such a property too but this requires to get some soft Barycentric refinement for delta sets. The D4 is not a simplicial complex as already the unit sphere S(v) which is the 24-cell is not a simplicial complex. Unlike the Z or the hex lattice, the D4 lattice is not a simplicial complex. The fundamental space parts are 16-cells. Its tessellation is called the 16-cell honeycomb. Since there is a dual honeycomb, there should be a dual operation similarly as the soft Barycentric refinement for simplicial complexes and the lattice should be a fixed point.
Here are some pointers about the Eisenstein-Quaternion story (something I worked on in the summer 2016): Project page https://people.math.harvard.edu/~knill/primes with the ArXiv paper. The more detailed experiments paper.