Icosahedral Sets in PG(5, 2)

Starting out from the 15 pairs of opposite edges and the 20 faces of a coloured icosahedron, a simple new construction is given of a ‘double-five’ of planes inPG(5, 2). This last is a recently discovered configuration consisting of a setψof (15+20=)35 points inPG(5, 2) which admits two distinct decompositionsψ=α1∪α2∪α3∪α4∪α5=β1∪β2∪β3∪β4∪β5into a set of five mutually skew planes. Moreover,Ir=αr∩βris a line, for eachr,whilenrs=αr∩βsis a point, forr≠s.The new construction illuminates why the symmetry group ofψis isomorphic toA5×Z2. The setψis a set of hyperbolic type, and it has a cubic equation. stead of the 15 pairs of opposite edges and the 12 vertices of the icosahedron yields a setφ⊂PG(5, 2) of (15+12=)27 points. The setφis of elliptic type, but, likeψ,has cubic equation and icosahedral symmetry. The setsψandφare of Tonchev type 3b; see [15, Table I].