The quaternion integers (the D4 lattice)
We had looked at the lattice Hex last time, which represent the Eisenstein integers. The lattice D4 represents the quaternion integers. There are lots of similarities. Both are known since a long time to be the densest sphere packings: the hex lattice is the unique densest circle packing in R^2 and D4 is the unique densest LATTICE packing in R^4 [I left out the LATTICE in the talk. It is an open problem whether it is also the densest packing overall.] . Both lattices also carry a natural arithmetic. The Hex lattice a ring of algebraic integers, the lattice D4 is the quaternion ring or integers. The unit sphere in D4 is the 24-cell, which I brag about having nicely 3D printed 2 days ago. Both rings have primes and in both rings carry natural Goldbach conjectures (see https://arxiv.org/abs/1606.05958 from Summer 2016, I had been told by a referee then that the paper has `` too much information" and that it needed to be cut. Compare with • Amadeus - "There are simply too many notes." but who cares. That 2016 paper is beautiful as it is and I do not need publications). The pictures were taken on Tuesday noon during a wobble run through Winchester (I almost guillutined my big toe last week in our trapdoor at home, but it was able only cut through it half the way). But it had been a gorgeous day. The colors are so nice now.