The familiar trigonometric functions can be geometrically derived from a circle.
But what if, instead of the circle, we used a regular polygon?
In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.
We still want to keep using the angle from the x-axis as the function’s input, instead of the distance along the polygon’s boundary. (These are only the same value for the circle!) This is why the square does not trace a straight diagonal line, as you may expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore.
Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.
More on this subject and derivations of the functions can be found in this other post
Now you can also listen to what these waves sound like.
This technique is general for any polar curve. Here’s a heart’s sine function, for instance
OMFG this reminds me of my neuro classes. it blew my mind when one of my profs showed this to us.
