I am looking for a classification (or attempt at enumeration) of affine simplicial line arrangements.
A line arrangment is a family of straight lines in $\Bbb R^2$. It is simplicial if all regions are triangles. Grünbaum and Cuntz did a lot of work in enumerating the finite simplicial arrangements (which are surprisingly rare) but there is still no full proof of completeness. I am looking for similar attempts for arrangements of infinitely many lines that are still discrete in the sense that every disc intersects only finitely many lines. I call them affine arrangements because of their relation to the affine reflection groups (see below). I have several questions:
Questions:
- Have there been made similar attempts at enumerating the affine simplicial arrangments? Do they seem to be similarly rare like the finite arrangements?
- I do not know whether every affine simplicial arrangement must be periodic or must have a maximum number of lines intersecting in a point. Is anything know about that? Is there an enumeration if we restrict to, say, the periodic arrangements?
The family of reflection lines of an affine reflection group in $\Bbb R^2$ gives an affine simplicial arrangement. There are three arrangements that can be obtained in this way:
Others can be obtained as restrictions of reflection arrangements from higher dimension. For example, the following arrangement is the intersetion of the hyperplanes of the affine $\tilde E_8$ reflection group with a 2-dimensional flat.
In the finite case there exist arrangements that are not known to be related to any reflecion group, whether 2-dimensional or higher. I wonder this can happen for affine arrangements as well.
