The theorem of Meinrenken that I am referring to is theorem 1.3 in this paper. (I know links are no fun to click on, but the paper has some good pictures.)

Informally the theorem says that one may deform the Coxeter complex up to a certain point where things collapse...

In my (default) simulation the collapse happens when t=1. Three of the triangles collapse to line segments, and one of them collapses to a point.

In general, you can pick your favorite 2x2 matrix, there will be some time, t, where such a collapse first happens.

My paper concentrates on the case when this matrix is the identity, and I show that there is (pardon the pun) "beauty in the break down".