About the image to the right

An arrangement of pseudolines is a decomposition of the real projective plane by closed curves that pairwise intersect exactly once. The image to the right is an arrangement of 25 pseudolines such no pseudoline in the arrangement is incident to more than 10 vertices of the arragement (parallel rays intersect at infinity, and the line at infinity is included in the arrangement). This is one of a family of arrangements of n=18j+5 (for each j > 0) pseudolines, such that no pseudoline is incident to more than 8j+2 vertices of the arrangement. Purdy, Smith, and I observed that this arrangement shows that the strong Dirac conjecture does not hold in its natural generalization to pseudoline arrangements.

This is of current interest to researchers in discrete geometry, since most of the older tools in discrete geometry don't differentiate between straight lines and pseudolines. On the other hand, newer algebraic methods apply to algebraic objects (like lines), but not to purely topological objects (like pseudolines). For example, the proof by Tao and Green of the Dirac-Motzkin conjecture, unlike earlier results on the problem, relies on the algebraic structure of an arrangement of lines; hence, the pseudoline generalization of the Dirac-Motzkin problem remains open.

The arrangement displayed had previously been investigated by L Berman.