Mandalas

Mathematics is a place where you can do things which you can’t do in the real world (Marcus Du Sautoy, mathematician)

From time to time I have a look to some of my previous posts: it’s like seeing them through another’s eyes. One of my first posts was this one, where I draw fractals using the Multiple Reduction Copy Machine (MRCM) algorithm. That time I was not clever enough to write an efficient code able generate deep fractals. Now I am pretty sure I could do it using ggplot and I started to do it when I come across with the idea of mixing this kind of fractal patterns with Voronoi tessellations, that I have explored in some of my previous posts, like this one. Mixing both techniques, the mandalas appeared.

I will not explain in depth the mathematics behind this patterns. I will just give a brief explanation:

  • I start obtaining n equidistant points in a unit circle centered in (0,0)
  • I repeat the process with all these points, obtaining again n points around each of them; the radius is scaled by a factor
  • I discard the previous (parent) n points

I repeat these steps iteratively. If I start with n points and iterate k times, at the end I obtain nk points. After that, I calculate the Voronoi tesselation of them, which I represent with ggplot.

This is an example:

Some others:

You can find the code here. Enjoy it.

6 thoughts on “Mandalas

  1. Lovely. Suggestion to having unlimited different mandalas each time you run the code:
    iter <- sample(2:5, 1) # Number of iterations (depth)
    points <- sample(4:12, 1) # Number of points
    radius <- 1+(sample.int(401,size=1,replace=TRUE)-1)/100 # Factor of expansion/compression

  2. Borders on unreasonably entertaining to play with! Love it— thanks so much for sharing the code. I love seeing geometry come alive.

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