Dual linear spaces generated by a non-Desarguesian configuration

Key Points

A method can create non-Desarguesian projective planes through dual linear spaces, expanding geometry options.
Analysis identifies 105 unique dual linear spaces, enabling targeted extensions in geometry research.
This observational analysis builds upon the principles of Desargues' Law to explore nonlinear configurations.
Insights may enable the development of new geometric structures beyond traditional configurations.

Abstract

We describe a method to try to construct non-Desarguesian projective planes of a given finite order using a computer program. If a projective plane of order n exists, then it can be constructed from a dual linear space by a sequence of one-line extensions. This motivates a careful analysis of the failures of Desargues' Law in a dual linear space. Up to isomorphism, there are 105 dual linear spaces generated by a non-Desarguesian configuration with which to start the extension process. A further reduction allows us to limit the number of starting configurations to 15.

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