I made a little tool exploring higher-order Lissajous patterns. Well, I'm not entirely sure they're actually called that, but this name sounds very mathematical :-). Some outputs look like human tongues or crests, some like butterflies, some like atomic mushrooms.

I'm sure everyone knows the principle of the basic Lissajous patterns. You iterate the following through a reasonable range of T and plot the points as you go:

X = sin(T / Rx) Y = cos(T / Ry) |

Rx == Ry gets you a circle, Rx == 2 and Ry == 1 is a paraboloid, and Rx == 1 and Ry == 2 looks like an hourglass. The images generated by my cart extend this a little bit. I sum a few "higher harmonics" before plotting the result:

X = 0 Y = 0 for I = 1 to speficied_order: X += sin(T * I / Rx) * R Y += cos(T * I / Ry) * R R *= 0.75 |

These are somewhat phase-shifted 1:2 of order 1, 2, 3, and 4:

I am pretty excited by this little discovery. (Or a lack of a proper literature research?) Regardless, it's been fun :-)

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