"Bihexagonal" lattice?

[The Shadow]

If you take the set of points in 4D described by hexagonal lattices in the wx and yz planes, does that make a well-known lattice or is it something less interesting? If so, which one? (To further explain what I mean - obviously if you take square lattices in the wx and yz planes, the result is the 4D hypercubic lattice.)

[Klitzing]

First of all the hexagonal tiling itself is no 2D lattice at all.
Only the dual trigonal tiling would be a lattice.
For lattices you might cf. eg. to https://bendwavy.org/klitzing/explain/lattice.htm.

But if you are after a space-filling hyper-honeycomb structure, then yes, there is a so-called comb-product,
which allows to "multiply" euclidean tesselations of whatever dimension.
For the various types of products (prism-, tegum-, pyramid-, and comb-product)
cf. eg. to https://bendwavy.org/klitzing/explain/product.htm.

So eg. the tetracomb x6o3o x6o3o would be a 4D space-filling by x6o x6o (hexagonal duoprisms).
(Its incidence structure is provided here: https://bendwavy.org/klitzing/incmats/hibbit.htm.)

--- rk

[The Shadow]

By the "hexagonal lattice" I meant A2. If you take the product of A2 with itself, is that the link you gave? I'm having trouble understanding your notation.

[Klitzing]

A2 lattice as a tiling then is trat = x3o6o (triangle tiling, https://bendwavy.org/klitzing/incmats/trat.htm), the dual of hexat = o3o6x (hexagonal tiling, https://bendwavy.org/klitzing/incmats/hexat.htm). In fact, trat is the Delone complex of A2, while hexat is the Voronoi complex of A2.

If you then take the comb product of those complexes, which corresponding to the lattice A2xA2, you'll get as according Delone complex tribbit = x3o6o x3o6o (triangle duoprism tetracomb, https://bendwavy.org/klitzing/incmats/tribbit.htm) resp. as its according Voronoi complex hibbit = o3o6x o3o6x (hexagonal duoprism tetracomb, https://bendwavy.org/klitzing/incmats/hibbit.htm).

--- rk

[Challenger007]

I roughly understand what this is about, but I can't quite understand the point: will this system help in solving a complex equation?