"But look, I made you some content
Daddy made you your favorite, open wide
Here comes the content
It's a beautiful day to stay inside"
- Bo Burnham
| complete_undulation | Lissajous curves | heart_and_shell |
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| traced path of the double pendulum | harmonics of revolving arcs | ordered chaos |
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This piece represents the 10th complete graph where each vertex is randomly placed and oscillating in a random direction with random frequency. You could also say it's a collection of pendula swinging in their own planes and we're looking up at them. But that's not very catchy, so let's name it complete_undulation.
This piece helps show how the Lissajous curves are constructed parametrically from the trigonometric functions with different frequencies. Or rather, each curve is drawn by attaching a pen via vertical and horizontal rods to gears rotating at different speeds.
These are the traced out paths the bottom mass of a double pendulum makes for different initial conditions and system parameters.
With each arc subtending a quarter of circle and revolving at a speed proportional to the distance from the center, we see there are moments of order hidden with the chaos.
This one is a little more niche and jargony than the others. But if you're curious about the underlying math, each shape is the overlay of either Bezier or B-spline curves evaluated over the permuted vertices (because order matters in the evaluation of the curves) of the complete graphs with 3, 4, 5 vertices, respectively. The top row is BSplines and the bottom is Bezier.
There's another version in the project folder where each is evaluated on the 3, 4, 5 roots of unity; granted the only difference is with n = 2, the vertices are arranged in a triangle with one in the middle (showing that the graph is planar), versus forming a square.