David Richeson: 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…

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