E:\paul\Mathematica\tex\test\origami.txt

In the past sessions we have been trying to find all rectangles that can be constructed by folding a 1X1 piece of paper. Trivially all rectangles that can be inscribed in the paper can be achieved by drawing the desired rectangle and then folding the paper according to the lines drawn. The following graph descibes these rectangles (X and Y axes are the length and breadth of the rectangles). We were able to prove that all rectangles in which [Graphics:origami.txtgr1.gif] >=2 are impossible to achive. We were able to prove that all rectangles described in the area under the curve [Graphics:origami.txtgr2.gif] are possible as well as all rectangles within the unit square.

We were able to show that rectangles that lay under the curb [Graphics:origami.txtgr7.gif]

(1-

) can be achieved by folding. This asserts that some uninscribable rectangles can be achieved by folding the unit square. To prove that not all rectangles for which [Graphics:origami.txtgr9.gif]

>=2 holds can be achieved we used the following argument:
The points of any rectangle ABCD of lenght L and breadth B which can be achieved by folding from the unit square can be placed in the unit square. There is a corner R from which the point A is of least distance. The points B&D that are a distance >=L from A have to be of distance >=L from R. Let T be the corner that is on the other side of the diagonal from R, then at least one of the points B&D lies on or outside the 90° triangle whose hypotinus is of lenght B, and is at corner T, as drawn. This limitation proves that many of the described rectangles are imposible to achieve by folding a unit square paper.

We are yet to discover the set of all rectangles that are achievable from a single fold. Nor have we discussed a limit on the number of folds for given rectangles. We should concentrate on the following 2 problems:
1) Can we describe new classes of achievable rectangles?
2) Can we rule out other rectagles?