The Hexagonal Tiling Honeycomb

Abstract

The hexagonal tiling honeycomb is a beautiful structure in 3-dimensional hyperbolic space. It is called {6,3,3} because each hexagon has 6 edges, 3 hexagons meet at each vertex in a Euclidean plane tiled by regular hexagons, and 3 such planes meet along each edge of this honeycomb. It also appears naturally in algebraic geometry. If $\mathbb{E}$ denotes the Eisenstein integers, the Néron-Severi group of the abelian surface ${\mathbb{C}}^2/{\mathbb{E}}^2$ is isomorphic to the lattice ${\mathfrak{h}}_2(\mathbb{E})$ consisting of $2\times 2$ hermitian matrices with Eisenstein integer entries. The points $A\in {\mathfrak{h}}_2(\mathbb{E})$ with $\mathrm{tr}(A)>0$ and $\det (A)>0$ come from ample line bundles on ${\mathbb{C}}^2/{\mathbb{E}}^2$, and among these points, those with $\det (A)=1$ correspond to principal polarizations. But these points are precisely the centers of the hexagons in the hexagonal tiling honeycomb!

2 pages, figure by Roice Nelson