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 lower edge of the square up to to create the line . Having accomplished this, the lines and trisect the angle .
Let us see why this is true. Consider the diagram below. We have drawn in , which is the location of the segment after it is folded, , the fourth side of the isosceles trapezoid , and , the second diagonal of . We must show that , where and .
Because and are parallel, , and because is the altitude of the isosceles triangle , . Thus . Now, is an isosceles trapezoid and is an isosceles triangle, so and are congruent isosceles triangles. Thus . It follows that .
The geometric properties of origami constructions are quite interesting. Every point that is constructible using a compass and straightedge is constructible using origami. But more is constructible. As we’ve seen, it is possible to trisect any angle using origami (I’ll leave the obtuse angles as an exercise). It is possible to double a cube. It is possible to construct regular heptagons and nonagons. In fact, where the constructability of -gons is related to Fermat primes, the origami-constructibility of -gons is related to Pierpont primes. While the field of constructible numbers is the smallest subfield of that is closed under square roots, the field of origami-constructible numbers is the smallest subfield that is closed under square roots and cube roots. In fact, it is possible to solve any linear, quadratic, cubic, or quartic equation using origami!
There are quite a few places to read about geometric constructions using origami, but a good starting point is this online article (pdf) by Robert Lang.