ProjectivePlanes.hpp File Reference

Plans on projective incidence planes. More...

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Detailed Description

Plans on projective incidence planes.

Projective planes with polarity
  • First task is to sort out the cases of degeneration.
    1. We have "empty projective incidence planes", "near-empty projective incidence planes", "pencils" and "near-pencils".
    2. Especially we need to sort out the cases of degeneration for the polar cases.
    3. Here the notions of "dominating vertices" are central.
    4. See the "friendship theorem" in CourseCombinatorics_LintWilson/Chapter21.hpp.
  • Then we should find all types of polar projective planes of order 2.
    1. One type is already given by fano_gl.
    2. 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?)
    3. See "Duality and polarity" in ComputerAlgebra/CombinatorialMatrices/Lisp/Basics.mac.
  • We need to find out what is known about projective planes with polarities.
    1. Are the possible orders known?
    2. 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" ?
    3. See "Exactly one common neighbour" in ComputerAlgebra/Graphs/Lisp/StrongRegularity/plans/general.hpp for the notion of a "(weak) design graph with loops".
    4. See the "friendship theorem" in CourseCombinatorics_LintWilson/Chapter21.hpp for considerations centred around the statement that polarities of non-degenerated projective incidence planes must have absolute points.

Definition in file ProjectivePlanes.hpp.