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  2. Projective plane - Wikipedia

    en.wikipedia.org/wiki/Projective_plane

    The order of a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes). The affine planes which arise from the projective planes PG(2, q ) are denoted by AG(2, q ).

  3. Real projective plane - Wikipedia

    en.wikipedia.org/wiki/Real_projective_plane

    The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map. It follows that the fundamental group of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2.

  4. Incidence geometry - Wikipedia

    en.wikipedia.org/wiki/Incidence_geometry

    The order of a finite projective plane is n = k – 1, that is, one less than the number of points on a line. All known projective planes have orders that are prime powers. A projective plane of order n is an ((n 2 + n + 1) n + 1) configuration. The smallest projective plane has order two and is known as the Fano plane.

  5. Non-Desarguesian plane - Wikipedia

    en.wikipedia.org/wiki/Non-Desarguesian_plane

    Hanfried Lenz gave a classification scheme for projective planes in 1954, [6] which was refined by Adriano Barlotti in 1957. [7] This classification scheme is based on the types of point–line transitivity permitted by the collineation group of the plane and is known as the Lenz–Barlotti classification of projective planes.

  6. Fano plane - Wikipedia

    en.wikipedia.org/wiki/Fano_plane

    Since it is a projective space, algebraic techniques can also be effective tools in its study. In a separate usage, a Fano plane is a projective plane that never satisfies Fano's axiom; in other words, the diagonal points of a complete quadrangle are always collinear. [1] "The" Fano plane of 7 points and lines is "a" Fano plane.

  7. Finite geometry - Wikipedia

    en.wikipedia.org/wiki/Finite_geometry

    If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. The Fano plane is called the projective plane of order 2 because it is unique (up to isomorphism). In general, the projective plane of order n has n 2 + n + 1 points and the same number of lines; each line ...

  8. Hall plane - Wikipedia

    en.wikipedia.org/wiki/Hall_plane

    The Hall plane of order 9 is the unique projective plane, finite or infinite, which has Lenz–Barlotti class IVa.3. [10] Its automorphism group acts on its (necessarily unique) translation line imprimitively, having 5 pairs of points that the group preserves set-wise; the automorphism group acts as S 5 on these 5 pairs. [11]

  9. Hughes plane - Wikipedia

    en.wikipedia.org/wiki/Hughes_plane

    A Hughes plane H: [1] is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1, has a Desarguesian Baer subplane H 0, is a self-dual plane in which every orthogonal polarity of H 0 can be extended to a polarity of H, every central collineation of H 0 extends to a central collineation of H, and