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 postNow 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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The Best Photos of 2012
A man walks inside of the crumbling oval skeleton of the House of the Bulgarian Communist Party on mount Buzludzha in central Bulgaria on March 14, 2012.
[Image: Dimitar Dilkoff/AFP/Getty Images]
We’re not the only ones who think this was pulled straight out of Close Encounters of the Third Kind, right?](http://24.media.tumblr.com/e505c7a9b6c8158cdad1f60410bf605b/tumblr_meojy6ekWn1qcokc4o1_400.jpg)



