Desargues $\implies$ associativity: Projective planes over non-associative structures? I've been reading about constructing projective planes over division rings (skewfields). There's this very nice fact that if Pappus's theorem holds in a  division ring, this ring is actually commutative.
I've found a lot of vague references to the fact that if Desargues holds in a projective plane (over something), then that something is associative. But I've only ever found references to projective planes over division rings (which are already associative.)
So I guess my questions are: To what kind of structures does the above statement refer to? And more generally, what can you build a projective plane over? Can you build a projective plane over any division algebra?
Any references to projective planes over non-associative structures are welcome.
 A: The topic is quite old and somewhat settled so there are good encyclopedia references and those will point you to books if you like.  Here is a short summary.

On the classic side are Ruth Moufang's planes which are based on Octonions.  You can find Moufang planes on wikipedia.
Once you have had a look at that you can swing to the complete characterization which is Marshal Hall's so-called ternary rings.  Ternary here refers to a valence 3-operation instead of you usual binary $+$ and $\times$.  Think $y=mx+b$, it takes 3 things in, a slope $m$, an run $x$, and a y-intersept $b$.  So it makes some algebraic sense that everything that will have "lines" in some geometric sense might have a coordinate system based on line-like equations.  So if you follow the hunch out far enough you will invent a product $\langle m,x,b\rangle$ of triples and put on the necessary axioms, e.g. $\langle 1,x,0\rangle=x$ and so on, until you find enough that you can prove you get a projective plane.  Find out more again on wikipedia
If you are looking for something field like but without the associativity of multiplication (you still keep associative addition though) you can look at semifields  (see here).  The term was coined by Donald Knuth who used it in his thesis, but the idea is much older, work of A.A. Albert and others in the 1930's.
Take for example a usual finite field $\mathbb{F}_{q}$ over some prime power $q=p^e$ with $e\geq 3$.  Let $x\mapsto x^{\theta}$ be a field automorphism, and $\lambda$ a constant.  Then define a new product
$a*b=ab+\lambda a^{\theta} b^{\theta^2}$.
These give you Albert semifileds $A$ as they are known today, and they are usually non-associaitve.  If you look at the $A$-points ($A$-spaces) and $A$-lines (2 generated $A$-spaces) you get projective planes.
For a somewhat up-to-date list as of 2005 see Kantor  The link is an incomplete PDF but the full paper is:
Finite semifields, pp. 103--114 in: Finite Geometries, Groups, and Computation (Proc. of Conf. at Pingree Park, CO Sept. 2005), de Gruyter, Berlin-New York 2006.
