Abstract
The paper describes a method of redistributing the points of the collinear sets in a Desarguesian plane so as to produce a (hybrid) projective plane which is non-Desarguesian. The method is applied to the construction: (i) of a plane over a prescribed subfield of the real field, and (ii) of a plane (over a Galois field) which is proved to be identical with the Hughes plane. On the basis of this construction algebraic relations in the field can be interpreted as incidence relations in the hybrid plane. In order to verify that the planes of type (i) are not isomorphic with Desarguesian planes, some conditions are established which show that all planes of this type (as well as of type (ii)) contain Fano subplanes.
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1. Subplanes of the Hughes plane of order 25;Archiv der Mathematik;1987-08
2. On 3-Blocking Sets in Projective Planes;Combinatorial Design Theory;1987
3. The combinatorial structure of the Hughes plane;Mathematical Proceedings of the Cambridge Philosophical Society;1970-09