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’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.
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
A psychologist walked around a room while teaching stress management to an audience. As she raised a glass of water, everyone expected they’d be asked the “half empty or half full” question. Instead, with a smile on her face, she inquired: “How heavy is this glass of water?”
Answers called out ranged from 8 oz. to 20 oz.
She replied, “The absolute weight doesn’t matter. It depends on how long I hold it. If I hold it for a minute, it’s not a problem. If I hold it for an hour, I’ll have an ache in my arm. If I hold it for a day, my arm will feel numb and paralyzed. In each case, the weight of the glass doesn’t change, but the longer I hold it, the heavier it becomes.” She continued, “The stresses and worries in life are like that glass of water. Think about them for a while and nothing happens. Think about them a bit longer and they begin to hurt. And if you think about them all day long, you will feel paralyzed – incapable of doing anything.”
It’s important to remember to let go of your stresses. As early in the evening as you can, put all your burdens down. Don’t carry them through the evening and into the night. Remember to put the glass down!
I think this just changed my life.
Did you know these are the same dancers from the Chimney scene in Mary Poppins!?!
Still defining epic, after all these years.
Harlem Shake - ’90s Cartoon Edition [x]