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See here for a survey: http://en.wikipedia.org/wiki/Duality_(projective_geometry)

A true duality principle might be that a theorem for some projective plane induces the dual theorem for its dual. Mostly one is only concerned with projective geometries over fields, which are always self-dual.

One way to see that the Hall planes are not self-dual is that the defining quasifields are not semifields (they do not satisfy both distributivity laws), as being self-dual would imply the quasifield being a semifield.

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See here for a survey: http://en.wikipedia.org/wiki/Duality_(projective_geometry)

A true duality principle might be that a theorem for some projective plane induces the dual theorem for its dual. Mostly one is only concerned with projective geometries over fields, which are always self-dual.

One way to see that the Hall planes are not self-dual is that the defining quasifields are not semifields (they do not satisfy both distributivity laws), as being self-dual would imply the quasifield being a semifield.