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.

  Last updated: 11 December 2008

9 Comments

• 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:
  a=3.1
  b=20 (max)
  c=30
  A=-0.8

  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!

Older: Guilloché Patterns Newer: Fast Mandelbrot Renderer