Last time I mentioned a fold that was equivalent to the shared tangent of two parabolas. In this article, I’d like to talk more about parabolas and origami - finding a parabola’s tangent is actually quite common in origami.
You might know that a parabola is the shape of a graph of a quadratic function; that is, the graph of something in the form of y=ax^2+bx+c.
You might also have heard of parabolas in the context of parabolic reflectors, where rays of light or sound emanating from one point are reflected parallel to each other or vice versa.
Parabolas in origami occur because of another definition of a parabola: the set of all points equidistant from a point (called the parabola’s “focus”) and a line (called the parabola’s “directrix”).
Okay, okay, maybe this is too obtuse and it seems like it’s not going anywhere, but let me explain. When you fold two points so they lie on top of each other, you get the perpendicular bisector of the segment containing the two points, that is, the set of all points equidistant from the two original points. Now, if you keep one of the original points fixed and move the other point around in a line, it stands to reason that the lines will all have a point that’s on a parabola - the parabola whose focus is the fixed point and directrix is the line on which the other point is moving.
Perhaps it’s time for the algorithm. Draw a point close-ish to one of the sides of a piece of paper. Then, take that side and fold it so that when it lands, it goes through the point. Make many folds of this type. Eventually, you should see a parabola emerge - there’ll be a heavily creased region, and a region without any creases, and the boundary between the two is a parabola.
