Parametric curves are magical. From gorgeous p…

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.

[D35M05 53T 1N 50URC3 L1NK.]

Mathematics is beautiful. <3

Lecturing on polar coordinates, a favorite Pro…

Lecturing on polar coordinates, a favorite Professor of mine once said, “The whole point is the pole. That’s where all the action is.”

Even then, I was astonished to find artful figures (like the above) deeply nestled at the heart of otherwise unremarkable rose curves. The action is at the pole indeed.

Remarkably, the slightest tweak in initial conditions results in new designs. To my endless delight, I am exhausted by the possibilities.

And that’s only for the given expressions. I am really overwhelmed considering the vast expanses opened by altering expressions, initial conditions, etc. 

The source contains the expressions. It’s interactive! Customize and make it your own. Do it! There are too many for me to find.

Just don’t forget to zoom in. 🙂

Mathematics is beautiful. <3

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