The Projective Plane of Order 4

Abstract

The projective plane of order 4 is one of the most remarkable finite geometries. Here are some facts that make it important for us. The projective plane of order 4 is the only projective plane apart from the Fano plane that can be one-point extended to a 3-design. This one-point extension can be further extended, first to a 4 — (23, 7, 1) design and finally to the famous 5 — (24, 8, 1) design. See [24, p. 22] for a concise geometrical description of this extension. The geometry of secant lines of a fixed hyperoval turns out to be the generalized quadrangle of order (2, 2). We use this fact to rebuild PG(2, 4) around a two- and a three-dimensional model of this generalized quadrangle. This yields two highly symmetric models of PG(2, 4). All this is something like a pictorial version of the first part of [24, Chapter 6]. The projective plane of order 4 is the smallest projective plane of square order and therefore the smallest projective plane containing unitals. All unitals in PG(2, 4) are isomorphic, and as point/line geometries they are isomorphic to the affine plane of order 3. We use this fact to rebuild PG(2, 4) around a nice model of this affine plane. The projective plane PG(2, 4) can be partitioned into three Baer subplanes. We describe such partitions in the two- and the three- dimensional models derived from the generalized quadrangle.