"Solution" to exercise for axiom (O5)

I won't give the whole game away by drawing a picture, but the repeated use of axiom (O5) in this exercise will result in the appearance of a parabola on the paper. Really!
To see why, imagine making one of the folds in this exercise. Before you unfold the flap, take a heavy black pen and draw a line from the point p1 to the folded edge, making it perpendicular to the "folded-up" segment of l1 (as in the left picture above). If our pen is heavy enough, this line will bleed through the paper and mark the underneath side as well, so when we unfold this flap we'll see two lines (as in the right picture above). Note that these two lines have the same length and one is perpendicular to the original line l1. This shows that exactly one point on the crease line we just made is equidistant to the point p1 and the line l1. In other words, the crease line is tangent to the parabola with focus p1 and directrix l1.

This should seem amazing - origami actually allows us to do simple calculus! Just one fold computes a tangent line of a parabola.

Exercise: Prove the above observation rigorously. (Hint: show that the envelope of all the crease lines is a parabola.)

But there's a more important thing to observe here. Parabolas are given by second degree equations. Thus axiom (O5) finds a point for us on some second degree equation. In other words, axiom (O5) solves second degree equations for us! It may seem strange to think of an origami fold as solving an equation, but mathematically this is exactly what is going on.

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