The optimal lattice for complex numbers

$$\omega = e^{2\pi i / 3}, \qquad \omega^2 + \omega + 1 = 0$$

The square lattice gives 4 neighbors per point; the hexagonal one — 6. The proper arithmetic object is the Eisenstein ring $\mathbb{Z}[\omega]$.

Full paragraph (abstract translation)

Gauss's 1831 treatise, in addition to the famous complaint, also introduced the Gaussian integers $\mathbb{Z}[i]$ — the square lattice encoded by that very 90° convention. But this lattice is not geometrically optimal: the hexagonal lattice is denser, with six equidistant neighbors per point instead of four.

The proper arithmetic object is the ring of Eisenstein integers $\mathbb{Z}[\omega]$, where $\omega = e^{2\pi i / 3}$ satisfies $\omega^2 + \omega + 1 = 0$. It was introduced by Eisenstein (1844) in his proof of the cubic reciprocity law (for details, see Ireland & Rosen, 1990).

Its norm

$$N(a + b\omega) = a^2 - ab + b^2$$

is the natural quadratic form of the 60°-geometry; and, like $\mathbb{Z}[i]$, it is a Euclidean ring.

Who first noticed that packing is hexagonal

Thue (1910) — the hexagonal packing of circles in the plane is denser than any other. This is the 2D analog of the same intuition that will later lead to the hierarchy of sphere packings in 3D, 8D, and 24D — we'll touch on it in the final part.

References

• Eisenstein, G. (1844) Beweis des Reciprocitätssatzes für die cubischen Reste… Journal für die reine und angewandte Mathematik, 27, 289–310. degruyterbrill.com
• Ireland, K. & Rosen, M. (1990) A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer. link.springer.com