Non-Desarguesian plane

Non-Desarguesian plane

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In mathematics, a non-Desarguesian plane, named after Gérard Desargues
Gérard Desargues
Girard Desargues was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are named in his honour.Born in Lyon, Desargues came from a family devoted to service to the French...

, is a projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

 that does not satisfy Desargues's theorem, or in other words a plane that is not a Desarguesian plane
Desarguesian plane
In projective geometry a Desarguesian plane, named after Gérard Desargues, is a plane in which Desargues' theorem holds. The ordinary real projective plane is a Desarguesian plane. More generally any projective plane over a division ring is Desarguesian, and conversely Hilbert showed that any...

. The theorem of Desargues is valid in all projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

s of dimension not 2, that is all the classical projective geometries
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

 over a field (or division ring), but Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

 found that some projective planes do not satisfy it. Understanding of these examples is not complete, in the current state of knowledge.

Examples


Known examples of non-Desarguesian planes include:
  • The Moulton plane
    Moulton plane
    In incidence geometry, the Moulton plane is an example of an affine plane in which Desargues' theorem does not hold. It is named after the American astronomer Forest Ray Moulton...

    .
  • Every projective plane of order at most 8 is Desarguesian, but there are three non-Desarguesian examples of order 9, each with 91 points and 91 lines.
  • Hughes plane
    Hughes plane
    In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by .There are examples of order p2n for every odd prime p and every positive integer n....

    s.
  • Moufang plane
    Moufang plane
    In mathematics, a Moufang plane, named for Ruth Moufang, is a type of projective plane, characterised by the property that the group of automorphisms fixing all points of any given line acts transitively on the points not on the line. In other words, symmetries fixing a line allow all the other...

    s over alternative division rings that are not associative, such as the projective plane over the octonion
    Octonion
    In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...

    s.
  • André plane
    André plane
    In mathematics, André planes are non-Desarguesian planes with transitive automorphism groups found by ....

    s.