Here are some pattern emerging from (simple) mathematical properties that I find very beautiful.
Cardioid
Cardioid is based on two moving circles, they form very nice patterns, have a look at the wiki page to find more infos or click here for the R script used.
Networks
Networks represent relationships amongst individuals, fortunately drawing lines between fixed dots create mesmerizing pictures, click here to see the R script used.
Mandalas
Mandalas are ancient drawings bringing inner peace, they can also be generated from regular points spacing and voronoi tessalation, click here to see the R script used, heavily inspired from asincron.
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