2024/11/23 by John C. Baez · 2 voices
#math.HO #math.AG #math.MG
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 𝔼 denotes the Eisenstein integers, the Néron-Severi group of the abelian surface ℂ2/𝔼2 is isomorphic to the lattice \mathfrakh2(𝔼) consisting of 2 × 2 hermitian matrices with Eisenstein integer entries. The points A ∈ \mathfrakh2(𝔼) with tr(A) \gt 0 and det(A) \gt 0 come from ample line bundles on ℂ2/𝔼2, 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!