aking connections between mathematical and artistic ideas is a major source of joy for me, and consequently I’m always on the lookout for interesting experiments that I can conduct to get more familiar with different areas of mathematics in ways that feel free and creative. A few weeks ago, I watched a YouTube video by physics and mathematics educator Toby Hendy in which she explains how to draw a parabola using a metaphor inspired by Bob Ross, the late host of The Joy of Painting.
In Hendy’s metaphor, we paint a landscape that contains a horizon, a mountain, and a mushroom situated near the top of the mountain.
Hendy explains that it’s possible for us to define the contour of the mountain while only knowing the position of the mushroom and the horizon. She proceeds to draw a set of lines projected downward from the horizon, and another set of lines projected from the mushroom at various angles. A parabola then appears at the points where those two sets of lines meet.
When I saw the parabola appear, I couldn’t quite understand what was happening exactly, so I decided to recreate this experiment in code using p5.js. Here is what I obtained: You can find the code for this animation in the p5.js web editor here.
In this animation, we see that it’s possible to construct an isosceles triangle whose first vertex is situated on the horizon, whose second vertex is situated on the mushroom, This point that Hendy makes us imagine as a mushroom is called the focus of the parabola. Wikipedia describes a parabola as “the locus of points in [the] plane that are equidistant from both the directrix and the focus.” The directrix is the horizon in our metaphor. We can thus see that the present exercise makes us replicate this definition very precisely. and whose third vertex traces a parabola. I also cut the isosceles triangle into two right triangles—I feel like this cut clarifies the symmetry that is present in the isosceles triangle and which is so important here.
To build this ABC triangle, we must first project a vertical line c downward from the vertex A and another line a between the vertices A and B. We must then calculate the angle A and project a line b from the vertex B to form an angle equal to A (so that ABC is isosceles). Edges b and c of the triangle must then meet at vertex C. When this process is repeated starting from several different points on the horizon, the vertex C of each triangle always falls on the path of the parabola.
Onward to the realm of sounds
One of the first things that came to my mind after I drew a single parabola was that I could now create endless copies of it and turn them into a signal, a parabolic wave.
I got really curious about how this would sound. I first imagined that it would sound something halfway between a sine wave and a triangle wave, because it visually seems to have characteristics of both: a smooth curve and a sharp corner. The parabolic wave also looks very similar to the absolute value of a sine wave, but it is a little bit different, as shown here. This intuition turned out to be wrong.
Below you can listen to a sound file containing three notes: an A at 220 Hz played with a sine wave, then a parabolic wave, and then a triangle wave. You’ll hear that the parabolic wave in the middle doesn’t sound at all like it is halfway between the sine wave and the triangle wave.