Centering small generalized polygons-projective pottery at work

The generalized triangle, quadrangle, and hexagons of order 2 are small point-line geometries that play a role in the theory of generalized polygons and buildings that is comparable to that of the Fano plane in the theory of projective planes. Virtually everybody working in discrete mathematics is familiar with the generalized triangle of order 2 aka the Fano plane, the smallest projective plane, PG(2,2), or the unique symmetric block design with parameters 2-(7, 3, 1), and its elementary representation as the geometrical configuration of an equilateral triangle together with its three medians and inscribed circle. The main purpose of this paper is to derive an elementary representation of the generalized hexagons of order 2 which extends naturally to a special representation of the projective space PG(5,2). Along the way, we also derive a similar representation of the generalized quadrangle of order 2 which turns out to be equivalent to the well-known representation of the quadrangle in terms of synthems and duads and which extends naturally to a representation of PG(3,2).