Plans on projective incidence planes.
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Detailed Description
Plans on projective incidence planes.
 Todo:
 Projective planes with polarity

First task is to sort out the cases of degeneration.

We have "empty projective incidence planes", "nearempty projective
incidence planes", "pencils" and "nearpencils".

Especially we need to sort out the cases of degeneration for the polar cases.

Here the notions of "dominating vertices" are central.

See the "friendship theorem" in CourseCombinatorics_LintWilson/Chapter21.hpp.

Then we should find all types of polar projective planes of order 2.

One type is already given by fano_gl.

The question is whether this is everything? (That is, if we rearrange the rows and columns of fano_m to obtain a symmetric matrix, are all symmetric matrices obtained in this way, as square matrices, isomorphic to fano_m?)

See "Duality and polarity" in ComputerAlgebra/CombinatorialMatrices/Lisp/Basics.mac.

We need to find out what is known about projective planes with polarities.

Are the possible orders known?

Instead of searching for arbitrary projective planes of a given order, searching for projective planes with polarity is easier, since we just have to search for design graphs with loops  is this restriction interesting or "harmful" ?

See "Exactly one common neighbour" in ComputerAlgebra/Graphs/Lisp/StrongRegularity/plans/general.hpp for the notion of a "(weak) design graph with loops".

See the "friendship theorem" in CourseCombinatorics_LintWilson/Chapter21.hpp for considerations centred around the statement that polarities of nondegenerated projective incidence planes must have absolute points.
Definition in file ProjectivePlanes.hpp.