I try to understand the Fano plane as an easy example of a projective plane.
Wouldn't it be sufficient, if the circle consisted only of two $\frac{1}{3}$ arcs to satisfy the axioms?
2025-01-12 23:58:55.1736726335
Does the circle in the Fano plane have to be closed?
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A projective plane is defined just by the incidence relations, that is, by which points are on which lines. If we use an arc that is not the whole circle but that still connects all three points on the circular arc, it defines the same incidence relations as the usual diagram, and therefore defines the same projective plane.