Dirac's conjectured lower bound is asymptotically the best possible, as the even numbers formula_1 greater than four have a matching upper bound formula_77. The construction, due to Károly Böröczky, that achieves this bound consists of the vertices of a regular formula_21-gon in the real projective plane and another formula_21 points (thus, formula_80) on the line at infinity corresponding to each of the directions determined by pairs of vertices. Although there are formula_81 pairs of these points, they determine only formula_21 distinct directions. This arrangement has only formula_21 ordinary lines, the lines that connect a vertex formula_84 with the point at infinity collinear with the two neighbors of formula_84. As with any finite configuration in the real projective plane, this construction can be perturbed so that all points are finite, without changing the number of ordinary lines.