Originally posted on Division by Zero:

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 $latex {\theta}&fg=000000$ formed by the bottom edge of the square of origami paper and a line (a fold, presumably), $latex {l_{1}}&fg=000000$, meeting at the lower left corner of the square. Create an arbitrary horizontal fold to form the line $latex {l_{2}}&fg=000000$, then fold the bottom edge up to $latex {l_{2}}&fg=000000$ to form the line $latex {l_{3}}&fg=000000$. Let $latex {B}&fg=000000$ be the lower left corner of the square and $latex {A}&fg=000000$ be the left endpoint of $latex {l_{2}}&fg=000000$. Fold the square so that $latex {A}&fg=000000$ and $latex {B}&fg=000000$ meet the lines $latex {l_{1}}&fg=000000$ and $latex {l_{3}}&fg=000000$, 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 $latex {l_{3}}&fg=000000$ to create a new fold $latex {l_{4}}&fg=000000$. Open the paper and fold it to extend $latex {l_{4}}&fg=000000$ to a full fold (this fold will extend to the corner of the square, $latex {B}&fg=000000$). Finally, fold the lower edge of the square up to $latex {l_{4}}&fg=000000$ to create the line $latex {l_{5}}&fg=000000$. Having accomplished this, the lines $latex {l_{4}}&fg=000000$ and $latex {l_{5}}&fg=000000$ trisect the angle $latex {\theta}&fg=000000$.

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