View source for 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067250/n0672501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067250/n0672502.png" /> are given, other than the point of intersection of the lines; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067250/n0672503.png" /> is parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067250/n0672504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067250/n0672505.png" /> is parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067250/n0672506.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067250/n0672507.png" /> is parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067250/n0672508.png" />. 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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,   "Grundlagen der Geometrie" , Springer  (1913)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Bieberbach,   "Einleitung in die höhere Geometrie" , Teubner  (1933)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Skornyakov,   "Projective planes"  ''Uspekhi Mat. Nauk'' , '''6''' :  6  (1951)  pp. 112–154  (In Russian)</TD></TR></table>



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The phrase  "(non-) Pascalean geometry"  is obsolete: it has been replaced by  "(non-) Pappian geometryPappian geometry" .

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Twelve geometric esays" , Univ. Illinois Press  (1968)  pp. Chapt. 1</TD></TR></table>