Can you fold a square sheet of paper into three equal rectangles, without using anything but folding?

(That is, you are not allowed to use anything but your hands, and you cannot do anything but folding... everything else is forbidden!)

Note: It is hard to describe folding without a diagram.  I'll do my best.  Letter the square corners in order ABCD.

First a method I came up with (4 folds):

Fold and unfold diagonals AC and BD.  Call this intersection E.
Fold and unfold D up to E.  Call the new intersection F.
Create a fold containing C and F.  Call the point where this crease intersects AD by X.
It can be shown by similar triangles that XD is one third of AD.  Having this trisection point of a side allows you to fold your rectangles.

Second a method I happened to see recently in one of my origami books (a complete coincidence, really) (2 folds):

Fold and onfold A to B.  Call the crease on AB by E.
Fold C to E.  Call the point where CD' intersects AD by F.
AF is two-thirds of AD.

Third a method by iteration, similar to those already given (∞ folds):

Make a guess at point X, such that AX is one third AB and fold a crease at X.
Fold B to X.  Call the crease Y.
Fold A to Y.  Call the new crease X'.
Fold B to X'.  Call the new crease Y'.
Repeat indefinitely.  The X's and Y's reach the one-third points in the limit.

Posted by Jer on 2007-02-19 12:15:52