Cool Curves with Graphmatica

Graphmatica is a freeware mathematical graphing program that's popular in schools for teaching curve sketching and the properties of mathematical curves. One reason why it's so popular is that it's really easy to play around with different equations to produce all sorts of cool shapes. The Mactutor History of Mathematics Archive has a webpage featuring famous curves: not Marilyn Monroe or Marlene Dietrich, but more abstract classic beauties like the Hyperbola, Ellipse, and rather more obscure exotics like the Lemniscate of Bernoulli and the Quadratrix of Hippias.

So while playing with Graphmatica I found a way to construct the Yin-Yang symbol using only two simple polar equations: it is quite striking to see how the ancients concocted a symbol whose shape is so purely mathematical. Simply graph both r = tanh t and r = -tanh t and the two 'tadpoles' which form the picture will fit into each other. For the little circles, x^2 + (y + 0.5)^2 = 0.0525 and x^2 + (y - 0.5)^2 = 0.0525 seem to fit the dimensions best.

In my opinion, though, the cutest one I've found so far is the 'egg in egg' equation, r = t sin t, which really does look like an egg nestled in another egg! There are also a whole lot of equations to draw heart-shaped curves. The cardioid family of curves is perhaps best known, but they don't have the tapered bottom of the classic valentines heart. Math World has an encyclopaedia entry on the topic.