Klein's arrangements of lines and conics (Q6572415)

In 1878, \textit{F. Klein} [Math. Ann. 14, 428--471 (1879; JFM 11.0297.01)] discovered a maximally-symmetric genus-3 complex quartic curve: its symmetry group has 168 elements. This group also determines the \textit{Klein arrangement} \(\mathcal{L}\) of 21 (complex) lines with 21 quadruple and 28 triple intersection points. (As \(\mathcal{L}\) has no \textit{pairwise} intersections, the Sylvester-Gallai theorem rules out any realization by real lines!)\N\NLet H be a \(\{7/2\}\) heptagram with vertices \(\{p_1,\ldots,p_7\}\) and edge intersections \(\{q_1,\ldots,q_7\}\). If we construct \(\{7/3\}\) heptagrams P and Q on these two sets of points, there are seven more points \(\{r_1,\ldots,r_7\}\) each lying on two edges of \(P\) and two edges of \(Q\); in total, each of the 21 points lies on four of the 21 lines. This is the \((21_4)\) Grünbaum-Rigby configuration.\N\NThis paper considers various interesting properties of these two configurations, and other configuration, of lines and of conics related to them. It has numerous beautiful illustrations, warranting attention even from the nonspecialist, and some interesting (and probably challenging) conjectures.