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. (There’s a very neat reason for this.) 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

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