A new geometrical construction for the near hexagon with parameters \((s,t,T_2)=(2,5,\{1,2\})\) (Q1430537)

A near polygon \({\mathcal S}=(P,L,I)\) is a partial linear space with the property that for every pioint \(x\) and every line \(l\), there exists a unique point on \(l\) nearest to \(x\) (with respect to the distance \(d(\cdot,\cdot)\) in the collinearity graph \(\Gamma\) of \(\mathcal S\)). If \(d\) is the diameter of \(\Gamma\), then the near polygon is called a near \(2d\)-gon. \(\mathcal S\) has the order \((s,t)\) if every line incident with \(s+1\) points and every point incident with \(t+1\) lines. Let \(T_2=\{|\Gamma(x)\cap \Gamma(y)|-1\;|\, d(x,y)=2\}\). For every set \(X\) of order \(2n\), \(n\geq 3\), a near \(2(n-1)\)-gon \(H_{n-1}\) has as points the partitions of \(X\) in \(n\) sets of order 2, as lines the partitions of \(X\) in \(n-2\) sets of order 2 and one set of order 4 and a point \(x\) is incident to a line \(l\) if and only if \(x\) is a refinement of \(l\). Theorem 1. Let \(\mathcal S\) be a near hexagon with parameters \((2,5,\{1,2\})\). Then \(\mathcal S\) and \(H_3\) are isomorphic.