# Kaleidoscopic pseudolines

Leah Berman has just published a paper, Symmetric Simplicial Pseudoline Arrangements, in the Electronic Journal of Combinatorics. It's related to my own paper in EJC on simplicial arrangements, of course, but much more closely to a blog post in which I suggested a physical model for simplicial line arrangements (and some pseudoline arrangements) involving shooting a laser into a two-mirror kaleidoscope. If you use an actual laser, that shoots light obeying the law of reflection that the angle of incidence equals the angle of reflection, you get a line arrangement, in which there lines are formed by the laser beams and their reflections, the mirrors and their reflections, and sometimes also the line at infinity of the projective plane. With some care in choosing the angles of the laser and the mirror this arrangement can be made simplicial. If you generalize the situation to allow other reflection angles, you get a pseudoline arrangement, and in that post I used this generalized model to describe a (previously known) infinite family of simplicial pseudoline arrangements formed in this way by two lasers.

Leah has taken that idea and run with it. She goes through a careful case analysis of all simplicial arrangements formed in this way from a pair of mirrors and two laser beams (forming arrangements with three or four symmetry classes of pseudolines), and describes two infinite classes of simplicial arrangements with three laser beams; these include all known infinite families of simplicial pseudoline arrangements. She also finds some sporadic examples with even more beams, and conjectures that there are infinite families with more than three beams.

Relating to a later post of mine on how to define pseudolines, the definition she gives of a single pseudoline is the classical one, "a simple closed curve that is topologically equivalent to a line" (in the projective plane). Here "topologically equivalent" is misleadingly terse: it would be easy to misread it as meaning homoemorphic, but I believe that the intended meaning is for the curves to be homotopic — one can continuously transform an ordinary line into a given pseudoline. But for most of the paper the curves she uses are in a more restricted class, polygonal chains topologically equivalent to lines: these are formed by replacing a finite portion of a straight line by a polygonal chain. The requirement that a pseudoline start and end along the same straight line imposes some constraints on the angles of the laser beams and the number of reflections they can each make, which simplifies the task of classifying arrangements of this type.

As usual with this sort of subject, there are lots of pretty pictures.