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Butterfly curves

Published on 5 November 2008

Taken from Clifford Pickover's book, Computers and the Imagination, is this experiment that creates butterfly like curves.

The formula is expressed in polar coordinates as:

By changing the A, B, a, b and c parameters you can get some nice results.
Click on the image below to give it a go.

butterfly 1225924540

  Last updated: 11 December 2008


  • Justin commented at 14 December 2008 at 22:35

    These are beautiful, and your site in general is fantastic. Keep up the good work! :)

  • richard mutter commented at 22 December 2008 at 02:43

    wooww... very interesting nice & cool curve...!
    it is possible to put this as a Plugins in AE..?
    i'm a vj & really into making own footages..
    i just find the Droste Effect and i just love it!
    its so helpfull for me to edit my own photo & video!

    thank you..
    richard mutter

  • I, Gallery commented at 20 January 2009 at 08:30

    Try setting:
    b=20 (max)

    It becomes such a cutie! =)

  • Raysun Goergen commented at 28 August 2009 at 20:37

    This is awonderful project! Is there a way to save high quality (maybe vector) images of thebutterflies?

  • Amarja commented at 1 November 2009 at 21:33

    I've tried to recreate this in Autocad but I've not been able to draw it with out tracing it, can I please please get a vector file of the butterfly curve or how to create it in Autocad, would really be appreciated.
    Thank you:)

  • Tom commented at 2 November 2009 at 21:49

    Amarja: unfortunately it doesn't support vector output at the moment and I've not used Autocad so I don't know how you would go about generating the curves within it.

  • Ronald commented at 28 March 2010 at 21:36

    Many points and lines, the curves would be a better option

  • Guillaume commented at 13 November 2011 at 12:27

    This is a really inspiring website. I am thinking of an idea to build a video with guilloché ( do you think illustrator and after effect is a good idea, since i dont know flash) and thats because of this site. If you want i can post a link of your site on my blog.

  • Jon commented at 8 March 2012 at 07:42

    Really elegant approach to developing some pretty images. Now I only need to plug in a variable field, and loop the graphing software in to my video compiler and make those wings flap!