Leah Berman has an interesting new paper out on EJC, on movable configurations of points and lines in which each point touches four lines and each line touches four points. As Berman describes, it's easy to make configurations in which some parts can move relative to others (even factoring out all the affine transformations), but much less so to do so while preserving some symmetry in the configuration. Anyway, the math gives rise to lots of pretty pictures.

...which led me to contact her, and then her advisor Branko Grünbaum, regarding the configuration of points and planes I reported a while back. Apparently, Grünbaum says, little or nothing is known about \( (n_5) \) point-plane configurations, though there has been some work on \( (n_4) \) and \( (n_6) \). So, despite its simplicity, and despite over a century of research on arrangements, my \( (9_5) \) may be new.

Grünbaum also mentioned a connection to simplicial arrangements: the nine planes of the \( (9_5) \), together with its three symmetry planes and two more planes for the remaining facets of its convex hull, form a simplicial arrangement of \( 14 \) planes, one of very few known. It's hard to see this as more than a coincidence without more examples, but it's an intriguing coincidence...