# Non-Pascalean geometry

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A geometry with a non-commutative multiplication. As a consequence of the fact that in affine geometry the property of commutativity is equivalent to the Pascal theorem, the name non-Pascalean geometry is usually attached to a geometry in which the following theorem fails to hold: Suppose that on each of two intersecting straight lines three points and are given, other than the point of intersection of the lines; if is parallel to and is parallel to , then is parallel to . This is sometimes called Pappus' theorem; it is a special case of the theorem of Pascal in the theory of conic sections (namely, when the conic degenerates to a pair of straight lines).

The possibility of constructing a non-Pascalean geometry follows from the fact that Pascal's theorem is not a consequence of the axioms of incidence, order and parallelism when the metric axioms are excluded from Hilbert's system (cf. Hilbert system of axioms). On the other hand, the existence of a non-Pascalean geometry is also connected with the possibility of constructing a geometry over a non-commutative skew-field, that is, a non-Pascalean geometry is at the same time a non-Archimedean geometry.

The significance of non-Pascalean geometry stems from the role of Pascal's theorem in research connected with establishing the independence of axiom systems and logical connections between propositions.

#### References

 [1] D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) [2] L. Bieberbach, "Einleitung in die höhere Geometrie" , Teubner (1933) [3] L.A. Skornyakov, "Projective planes" Uspekhi Mat. Nauk , 6 : 6 (1951) pp. 112–154 (In Russian)