Conference rusure::math

I am enjoying working on this problem, but I have no solution yet.

When you can divide a quadrilateral into triangles that are "relatively 
rational"  (that is, their ratios are rational), then you can divide that 
rectangle into a finite number of equal triangles.

I have been able to prove that certain 4-gons can't be divided up into certain 
kinds of triangular configurations, but I have no general proofs.  One 
triangle that I have been unable to subdivide so far has sides of 
sqrt(5),2,sqrt(3), and sqrt(2), going clockwise.  The side of sqrt(2) is 
parallel to the side of 2, which makes some of the proofs and calculations 
easier.  

My guess is that the answer to the problem is no, and that the 4-gon I've 
described is a counter-example, but if someone can find a division of it into 
equal triangles, that may give a clue to a more general construction 
procedure.

-Jim Grant, Littleton

It depends on what you mean by 'dissect'. If that implies a Euclidean 
ruler-and-compass construction, the answer is surely 'no', for exactly the
same reason that a general angle cannot be trisected. If you are asking if
a dissection EXISTS, that's another issue. If you are asking for a CONSTRUCTIVE
existence proof, that is yet another issue. - Lynn

I've not been assuming the Euclidean compass and rule limitations.

When I used the word "constructive"  I meant in the general logical sense that 
a proof of the divisibility of 4-gons into equal triangles might tell us how to
figure out what, where, and how many triangles we need for a given rectangle.

If there were such a "constructive" procedure, that we could prove the validity
of, that would be a proof.  However, it is not a necessity, and by the way, I 
think we need subtler tools than construction procedures to prove this one, if 
it's true. (And I don't know what these subtler tools are.)

So, given that we have "infinitely accurate" rulers, protractors, and 
calculators, can we divide a given 4-gon into equal area triangles?  In 
particular, can someone find a way to divvy-up a 4-gon whose sides are 
sqrt(2), sqrt(3), 2, and sqrt(5)?  This 4-gon has two parallel sides that are 
1 unit apart, and has a total area of (sqrt(2) + 2)/2.     (sqrt is a way of 
saying "square root of".)

-Jim Grant, Littleton

I placed no restrictions on the dissection, nor did I ask
for a constructive proof.

By the way, for people having trouble drawing Jim's quadrilateral,
the vertices are at the following coordinates:

	(0,0)	(2,0)	(2+sqrt(2),1)	(2,1)

OK, I give up.

Any clues, answers, guesses about whether any quadrilateral can be divided 
into equal area triangles?  

      -Jim

The answer is no.  I found this result in the June 1985 issue
of the Abstracts of the AMS, page 263, abstract 85T-51-111,
which I reprint below:

Preliminary report by Elaine K. Rooney and Sherman K. Stein,
University of California, Davis, California 95616.

	Cutting a Polygon into Triangles of Equal Area

J. Thomas (Math Mag. 41(1968)187-190) and P. Monsky (Amer. Math
Monthly 77(1970)161-164) proved that a square cannot be cut into
an odd number of triangles of equal area.  D. G. Mead (Proc. Amer.
Math. Soc. 76(1979)302-304) generalized this result by showing that
the n-cube can be cut into m simplices of equal volumes if and only
if n!|m.  The proofs use valuation theory and an extension of
Sperner's Lemma.  With these techniques and a generous exploitation of
linear maps we show:

1. If the regular n-gon (n=5,6,8) is cut into m triangles of equal area,
   then n|m.

2. There are quadrilaterals that cannot be cut into triangles of equal area.

3. For each integer m>1, there is a quadrilateral which can be cut
   into n triangles if and only if m|n.  Moreover, there are polygons
   for which the method seems to yield no information.