Mandalaxies

One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers (Heinrich Hertz)

I love spending my time doing mathematics: transforming formulas into drawings, experimenting with paradoxes, learning new techniques … and R is a perfect tool for doing it. Maths are for me a the best way of escape and evasion from reality. At least, doing maths is a stylish way of wasting my time.

When I read something interesting, many times I feel the desire to try it by myself. That’s what happened to me when I discovered this fabolous book by Julien C. Sprott. I cannot stop doing images with the formulas that contains. Today I present you a mix of mandalas and galaxies that I called Mandalaxies:

This time, the equation that drives these drawings is this one:

$x_{n+1}= 10a_1+(x_n+a_2sin(a_3y_n+a_4))cos(\alpha)+y_nsin(\alpha)\\ y_{n+1}= 10a_5-(x_n+a_2sin(a_3y_n+a_4))sin(\alpha)+y_nsin(\alpha)$
where $\alpha=2\pi/(13+10a_6)$

The equation depends on six parameters (from a1 to a6). Searching randomly for values between -1.2 and 1.3 to each of them, you can generate an infinite number of beautiful images:

Here you can find the code to do your own images. Once again, Rcpp is key to generate the set of points to plot quickly since each of the previous plots contains 4 million points.