Given a finite incidence structure or equivalently a finite collinearity structure (satisfying standard set of axioms), I am interested in sufficient conditions for realizability in Euclidean plane (or real projective plane).
For now my goal is to know how to determine whether a structure (with no more than $n$ points) is realizable i.e. it is isomorphic to a subset of Euclidean plane. What I gathated so far:
- It cannot contain configurations known to be not realizable (such as Fano plane, Möbius–Kantor configuration) as a substructure.
- It must satisfy Desargues and Pappus theorems.
I am absolutely not sure if that is enough (and which configurations to test). If the general conditions are not known, any more "tests" like this would help me, too. I am particularly interested in "complete" test for $n=10$.
Just to clarify, standard set of axioms are planar Hilbert's axioms (without Playfair's parallellism axiom) for incidence structure and the following for collinearity structure (from "From Affine to Euclidean Geometry" by W. Szmielew):
- $(a=b \vee b=c \vee a=c) \implies L(abc)$.
- $(a\neq b \wedge L(abp) \wedge L(abq) \wedge L(abr))\implies L(pqr)$.