Parametric curves are magical. From gorgeous patterns to gnarly fractals, all these images are generated from the same two expressions. The variations depend solely upon the values of a and b.

Just like coordinate pairs, the left and right elements determine the respective x and y components of the points along each curve.

Intuitively, the curves are made in a manner similar to a sort of “drawing device.” Imagine an arm (or vector) rotating about a circle at a constant rate, like a clock. Attached at the tip of the arm is another smaller circle, whose arm rotates at a faster rate than the previous one.

At the tip of the arm on the last circle is an “abstract pen”–either in purple or black. Hence there are two such devices in these drawings, as given by their respective colors in the expressions above.

In this incredible video, 3blue1brown demos such “devices,” calling them “circle drawings.” He uses them to draw an arbitrary pi symbol. To be clear, the mathematics behind the above images is not quite the same as these circle drawings (which are made with Fourier series), but their construction is quite similar. 3b1b succinctly describes the underlying phenomenon as “everything is rotations.” That is, using nothing but a combination of rotating things, you can draw anything you like.

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Mathematics is beautiful. <3