Cell decompositions of the projective plane with Petrie polygons of constant length (Q1356075)

The author defines a Petrie polygon of an abstract finite face-to-face decomposition of a combinatorial 2-manifold to be a (possibly infinite) abstract polygon such that every two consecutive abstract edges, but not three, belong to one cell. A dual pair of abstract face-to-face cell decompositions \((\mathcal D_{P^2},\mathcal D^*_{P^2})\) of the real projective plane is called a Petrie decomposition when its induced pair of cell decomposition \((\mathcal D_{S^2},\mathcal D^*_{S^2})\) on the 2-sphere forms a dual pair of combinatorical types of convex polyhedra and when it shares the following two properties with dual pairs of Platonic solids: (1) Every Petrie polygon is a finite simple closed polygon of length \(2(n-1)\) for some fixed \(n\). (2) Every pair of Petrie polygons of \(\mathcal D_{S^2}\) or of \(\mathcal D^2_{S^2}\) has precisely two common edges. It is shown that Petrie decomposition \((\mathcal D_{P^2},\mathcal D^*_{P^2})\) with Petrie polygon length \(2(n-1)\) are in one-to-one correspondence with reorientation classes of uniform oriented matroids in rank 3 with \(n\) elements. In particular those Petrie decompositions in which one part, say \(\mathcal D_{P^2},\) has only 3-valent vertices are studied. A \(p_3\)-maximal pseudoline arrangement with \(n\) elements in turn determines a neighborly 2-manifold with Euler characteristic \(\chi=n(7-n)/6,\) and vice versa, this neighborly 2-manifold uniquely determines its generating \(p_3\)-maximal pseudoline arrangement. An algorithm for generating \(p_3\)-maximal pseudoline arrangements which leads in particular to uniqueness assertions for small number of elements \(n \) is described. A complete list of those \(n, n\leq40,\) for which there exists at least one \(p_3\)-maximal pseudoline arrangement with \(n\) element is provided.