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These biplanes do not fly. They are simply symmetric designs with λ=2. More precisely, a biplane is a system of points and lines (= blocks) satisfying the two axioms:

B1.

Two distinct points are contained in two distinct lines.

B2.

Two distinct lines intersect in two distinct points.

In other words, a biplane is an incidence structure consisting of a set P of points and a set B of blocks, with an "incidence" relation between P and B, having the properties that any two points are incident with just two blocks, and any two blocks with just two points (see [1]). If a biplane has k points on each block then we say that its order is k-2. As in the case of finite projective planes, the classical Bruck-Ryser-Chowla theorem imposes some restrictions on the existence of biplanes as well. However, unlike the projective planes, it is not known whether there is an infinite family of biplanes. Indeed only a very few of them have been discovered so far. We know that any finite Desarguesian projective plane can be embedded in an abelian group such that if {p₁, p₂,…,p_n} is a line in the projective plane, then p₁+p₂+…+p_n = 0 in the group. Here we construct similar group embeddings for some of the well-known biplanes. This was part of unpublished research done in collaboration with late Professor N.S. Mendelsohn during the 1990's.

1. Cameron, Peter J. Biplanes. Math. Z. 131 (1973), 85–101.