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 N\'eron-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 $\mathrm{tr}(A) \gt 0$ and $\det(A) \gt 0$ 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!