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.