Finite projective geometry

In mathematics, a finite projective space "S" is a set "P" (the set of points), together with a set of subsets of "P" (the set of lines), all of which have at least three elements, satisfying these axioms :
* Each two distinct points "p" and "q" are in exactly one line.
* Veblen's axiom : when "L" contains a point of the line through "p" and "q" (different from "p" and "q"), and of the line through "q" and "r" (different from "q" and "r"), it also contains a point on the line through "p" and "r".
* There is a point "p" and a line "L" that are disjoint.

The last axiom is there to prevent degenerations.

A subspace of the projective space is a subset "X", such that any line containing two points of "X" is a subset of "X". The full space and the empty space are also considered subspaces.

The geometric dimension of the space is said to be "n" if that is the largest number for which there is a strictly ascending chain of subspaces of this form:

: $varnothing\; =\; X\_\{-1\}subset\; X\_\{0\}subset\; cdots\; X\_\{n\}=P.$

As a consequence of the third axiom: "n" ≥ 2.

Classification

Oswald Veblen and John Wesley Young (1879–1932) proved that, if "n" ≥ 3, every finite projective space is isomorphic with a "PG"("n", "K"), the "n"-dimensional projective space over some division ring "K". This is now known as the Veblen-Young theorem, and appeared in the first volume, from 1910, of their two-volume book "Projective Geometry".

There are: 1, 1, 1, 1, 0, 1, 1, 4, 0, … OEIS|id=A001231projective planes of order 2, 3, 4, ….

The two-dimensional case

One can check that the definition for "n" = 2 is completely equivalent with that of an projective plane (identifying lines with the set of points incident with it). However it turns out that these are much harder to classify, as not all of them are isomorphic with a "PG"("d", "K"). Jacques Tits generalized this phenomenon with his generalized "n"-gons : the generalized 3-gons with at least three points on every line are precisely the axiomatic projective planes.

Minimal projective plane

The smallest projective plane is usually said to be the Fano plane, PG [2,2] .

However PG [2,2] does not meet Coxeter's (1974) revised set of axioms, according to which the smallest projective plane is PG [2,5] comprising 31 points and 31 lines.

External links

*
* http://eom.springer.de/P/p075350.htm
* http://planetmath.org/encyclopedia/ProjectiveSpace.html
* [http://www.uwyo.edu/moorhouse/pub/planes/ Projective Planes of Small Order]

References

* Beutelspacher A./Rosenbaum U.; "Projective Geometry. From Foundations to Applications", Cambridge University Press (1998)
* Coxeter, H. S. M.; "Projective Geometry", 1st ed. University of Toronto Press (1974), 2nd ed. Springer Verlag (2003).
* Dembowski,P.; "Finite Geometries", Springer (1968)
* Greenberg, M.J.; "Euclidean and non-Euclidean geometries", 2nd ed. Freeman (1980).
* Hilbert, D. and Cohn-Vossen, S.; "Geometry and the imagination", 2nd ed. Chelsea (1999).