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.
Then I got curious: exactly how does the re-parametrization redistributes the points along this curve? In the original parametrization, the points are bunched up in the middle of the spiral, and more spaced on the outside. The arc-length parametrization makes them equally spaced along the whole path. So how do they compare?
First, I tried this with black points, but it was too confusing. Same thing for a few dots highlighted. So I decided to color them all based on the angle in the original parametrization. This is the result.
It is really interesting how the colors are bent around. It seems that the distribution is quite non-uniform, even though the spiral is rather uniform in growth.
I originally rendered this with four times as many frames, but due to the amount of colors and dimensions of the GIF, Tumblr wouldn’t accept it. It was too large. Below is the animation with twice as many frames.
Hint: try squinting! It blurs the colors and it looks really trippy!
Another animation of the fractal Harriss spiral. This exploits the plastic number (1.3247…) a number whose cube is itself plus one. This number is considered by some to be the forgotten cousin of the golden ratio (whose square is itself plus one).