The 'hexagonal tiling honeycomb' is a beautiful structure in 3-dimensional hyperbolic space. I'm trying to figure out something about it.
It contains infinitely many sheets of hexagons, tiling planes in the usual way hexagons do. These are flat Euclidean planes in 3d hyperbolic space, called 'horospheres'. I want to know the coordinates of the vertices of these hexagons. I have some clues.
The hexagonal tiling honeycomb has Schläfli symbol {6,3,3} . The Schläfli symbol is defined in a recursive way. The symbol for the hexagon is {6}. The symbol for the hexagonal tiling of the plane is {6,3} because 3 hexagons meet at each vertex. Finally, the hexagonal tiling honeycomb has symbol {6,3,3} because 3 hexagonal tilings meet at each edge!
The symmetry group of the hexagonal tiling honeycomb is the Coxeter group {6,3,3}. This is a discrete subgroup of the Lorentz group O(3,1), which acts on 3d hyperbolic space because that space is the set of points (t,x,y,z) in Minkowski spacetime with
t² − x² − y² − z² = 1 and t > 0
The Coxeter group {6,3,3} is generated by reflections, but its 'even part', generated by pairs of reflections, is a discrete subgroup of PSL(2,ℂ), because this is the identity component of the Lorentz group. In fact, this Coxeter group is almost PSL(2,𝔼), where 𝔼 is the ring of 'Eisenstein integers'. These are complex numbers of the form
a + bω
where a,b are integers and ω is a nontrivial cube root of 1. So there should be a nice description of the hexagonal tiling honeycomb using Eisenstein integers! And this is what I'm trying to find... quickly, before May 1st because I'm have a column due then. 😧
