Geometric Axioms for the Hall Plane

It is proved that the geometric incidence axiom H below provides necessary and sufficient conditions for a plane Π of the following specification to be a Hall plane, namely, (i) Π is a translation plane with axis l ∞ , (ii) Π has a square‐root subplane Δ meeting l ∞ , in a subline, (iii) Δ is a plane over a field (on which axiom H imposes the restriction that it shall contain an irreducible quadratic form). Axiom H. X , Y ∈ l ∞ ∩ Δ, Z ∈ Δ\ l ∞ , A ∈ Δ\{ l ∞ ∪ XZ }, C ∈ Δ\{ ZX ∪ ZY }, are arbitrarily selected points in their respective sets. { S i } = XZ \Δ, T i = AS i ∩ l ∞ , R i = CS t ∩ YZ , B t = R i T i ∩ Δ. (Axiom) (i) { B i } is a single point, B say, (ii) if C ∉ l ∞ then AX , BY , CZ are concurrent.