@johncarlosbaez
oh i just accidentally ran into these the number of rooted 2 connected planar graphs with n edges (allowing multiple edges between vertices) runs as G_2=1,2,6,22,91 whereas the schroder numbers run 1,2,6,22,91.
roughly you can build 2-connected planar graphs out of 3-connected planar graphs and the schroder numbers count graphs which are "trivial" in this sense
@johncarlosbaez
(oops schoder numbers run 1,2,6,22,90; typo)
@johncarlosbaez
the first nontrivial graph is the tetrahedron which explains the 90-91 difference
@johncarlosbaez
roughly the idea is to treat vertical lines as vertices and boxes as edges. thing is, this shape you had before gives the tetrahedron with this rule, but there are 2 such shapes.
so to do this in more generality we'd need to think of these as directed graphs and add some more criteria?
@johncarlosbaez
actually I guess for the purposes of the "divide square into similar rectangles" problem we could just allow the side length of the interior rectangle to be negative
@alexthecamel - Cool thought. But the center is square so the outside ones must be too. This forces the outside ones to have side length 1/2 that of the whole square. This forces the central one to have side length 0.
@johncarlosbaez
in this particular case the side length must be 0
but for more general configurations which you can't get from guillotine cuts it seems like a useful idea
@johncarlosbaez
when all the rectangles are oriented the same way you could also think of these as networks of unit ohm resistors?
@alexthecamel - Yes!!!
I was just thinking about this stuff last night, and I want to post about it.
Note that when all the rectangles are oriented the same way you can rescale one axis to make them all squares. Then this resistor network is called a 'Smith network':
@johncarlosbaez
(I think "some side length is zero" is the only way things can break if we take x= a +ve real root of the polynomial we get, side lengths being negative should still get you a well behaved dissection)
@alexthecamel - interestingly, side lengths being negative still gives a positive area. Can you give me an example of a dissection that involves negative edge lengths? Do the areas add up to the area of the square or rectangle being dissected?
@johncarlosbaez
roughly the pinwheel above but with the width and height of the central square negative gives you the same pinwheel but reflected
@johncarlosbaez
so im thinking something like "pick a random rooted planar 2-connected graph, (vertices horizontal lines, faces vertical lines), pick a direction for each edge, knowing the aspect ratio for each rectangle gives us a bunch of equations to solve and if we get a negative number for some edge that means we should have picked the other direction for it.
@alexthecamel - ah, the tetrahedron! Nice.