How to trisect an angle with origami

There are a couple things that can’t be done with a compass and straightedge - squaring a circle, doubling a cube, finding the distance from a point to an ellipse.

Among these is the task of trisecting a given angle. However, despite not being able to be done using classical methods, origami has an algorithm for this construction.

It begins with the angle drawn from the corner (I will, WLOG, refer to the example with the angle in the bottom right corner) of the piece of paper with one ray lying on the paper’s edge and the other ray marked or folded. A fold is made parallel to the edge of the paper with the angle’s ray on it, close to the top. Then, the bottom half of the paper is folded in half.

Next, the point at the intersection of the top fold and the left edge of the paper and the vertex of the angle are folded in such a way that they meet the angle’s ray and the bottom horizontal fold, respectively.

Finally, if you fold from the corner to the point where the diagonal fold intersected the bottom horizontal fold and the point where the angle’s vertex landed, on the bottom horizontal fold, you will get two more lines that perfectly trisect your original angle.

Now, here’s the crucial question: What did using origami add that a straightedge and compass couldn’t? Well, it was that diagonal fold. It added what would amount by straightedge and compass to an infinite amount of information. That fold maps to finding the shared tangent of two parabolas, which requires a cubic equation, which cannot be done using Euclidean methods. Indeed, origami is even more powerful than straightedge and compass.

Origami, Geometry, and Math

Origami, Geometry, and Math have many shared concepts and uses.  Other subjects sometimes taught separately such as Algebra and its advanced topics blend right into Calculus.  Math in its many manifestations are representations of the same underlying concepts; for example, (a+b) squared, can be worked out with algebra, but it can also be represented in squares and rectangles of a square with a side of length (a+b). https://www.mathdoubts.com/a-plus-b-whole-square-geometric-proof/   Using either the geometry or the algebra, the answer turns out to be the same.  Why? it's the same underlying idea, we are looking at and solving the same problem through either an algebra viewing lens, or a geometry viewing lens.

Origami of course is closest to geometry in physical appearance. There are, however, things that can be easily done with origami that are hard to do with conventional tools of geometry (straightedge, compass), such as trisecting an arbitrary angle.  See https://plus.maths.org/content/trisecting-angle-origami for details. 

Starting with basic constructions, it is possible to arrive at either exact 30 and 60 degree angles, or usable approximations of a 72 degree angle (a whole 360 degree circle divided into fifths), and those useful approximations can be used in origami construction that are sometimes clumsy using a protractor and ruler.

Coming back to the original point, Origami, Geometry, Algebra, Calculus and other aspects of Mathematics are simply different viewing lenses, or windows with which we look at a set of concepts and principles.  Sometimes Origami can give us an angle that is useful and pleasing to look at.