Non-Pascalean geometry

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 $A,B,C$ and $A_1,B_1,C_1$ are given, other than the point of intersection of the lines; if $CB_1$ is parallel to $BC_1$ and $CA_1$ is parallel to $AC_1$, then $BA_1$ is parallel to $AB_1$. 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)

Comments

The phrase "(non-) Pascalean geometry" is obsolete: it has been replaced by "(non-) Pappian geometry" .

References

[a1]	H.S.M. Coxeter, "Twelve geometric esays" , Univ. Illinois Press (1968) pp. Chapt. 1