The Hexagonal Tiling Honeycomb

TL;DR

This paper explores the geometric and algebraic structure of the hexagonal tiling honeycomb in hyperbolic space and its connection to algebraic geometry via Eisenstein integers and abelian surfaces.

Contribution

It establishes a novel link between the hexagonal honeycomb structure and the Néron-Severi group of certain abelian surfaces using Eisenstein integers.

Findings

Centers of hexagons correspond to points with determinant 1 in a hermitian matrix lattice.

The honeycomb structure relates to ample line bundles on abelian surfaces.

Connections between hyperbolic tilings and algebraic geometry are elucidated.

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!