It is well known that it is impossible to trisect an arbitrary
angle using only a compass and straightedge. However, as we will
see in this post, it is possible to trisect an angle using
origami. The technique shown here dates back to the 1970s and is
due to Hisashi Abe. Assume, as in the figure below, that we
begin with an acute angle formed by the bottom edge of the
square of origami paper and a line (a fold, presumably), ,
meeting at the lower left corner of the square. Create an
arbitrary horizontal fold to form the line , then fold the
bottom edge up to to form the line . Let be the lower left
corner of the square and be the left endpoint of . Fold the
square so that and meet the lines and , respectively. (Note:
this is the non-Euclidean move—this fold line cannot, in
general, be drawn using compass and straightedge.) With the
paper still folded, refold along to create a new fold . Open the
paper and fold it to extend to a full fold (this fold will
extend to the corner of the square, ). Finally, fold the