SuperFormula 3D

Images from Johan Gielis’s paper on 2d Superformula

SuperFormula is a generic geometric transformation equation that encompasses a wide range forms found in nature, the transformation applied to a circle, for example, produces the archetypal shapes of starfish, shells and flowers. Johan Gielis’s paper on the Superformula, available here, is essential reading for the computational biologist and botanist. It’s cited literature reference list, alone, reads like a who’s who of the history of mathematical and computational modelling of morphogenesis.

The history of this kind of study goes back quite a long way, according to D’Arcy Thompson the relation between shapes of flowers and trigonometric functions was first postulated by the monk Grandus in the 17th Century. Along the way others, including Thompson in his classic work ‘On Growth and Form’, have pondered on the idea of a universal line of code to describe natural forms. A quote from ‘On Growth and Form’:

‘For the harmony of the world is made manifest in Form and Number, and the heart and soul and all poetry of Natural Philosophy are embodied in the concept of mathematical beauty’.

L-systems are a beautiful example of this kind of extreme economy of information to describe and convey very complex branching systems such as those found in plants and nerve pathways.

Recently Gielis’s ideas on the Superformula have been taken up by Paul Bourke and extruded into three dimensions; the natural world exists like this after all. Paul has written an OpenGL based software (shareware) for Mac OS-x and Linux enabling explorations into the world of the Supershape. Vincent Berthoux has contributed a Windows version, which uses a text config file to describe the Supershape – its free to download and also available at Paul’s site.

While I don’t always go with the philosophy of trying to reduce the (in my eyes) irreducible complexities of the ‘nature’ into a line of code I certainly do appreciate this kind of work. Outside of science, at the very least, it arms computational artisans with a vocabulary of algorithms for mimicking the some of nature’s greatest work. It can also allow us to apply these formulas to more practical disciplines such as architecture – which judging by a few hundred years of Euclidean obsession can only be a good thing!

It should comes as no surprise then that the boys at VVVV have been putting their beloved toolkit for video synthesis through its paces with vertex shader implementations of the Superformula, check out Sanch’s elegant series: 1 2 3 4 5 6 7

I’ve been spending some time with VVVV recently, it’s a fantastic patch based environment ideal for fast prototyping of graphical/video based art/installations. Some more on 4V soon.

UPDATE: 020606 (pm)

Sanch (david dessens), who posted grabs of those wonderful Supershapes, kindly mailed me today with some extra info, he says:

‘…..The contributor of the superformula vertex shader is from tonfilm (tebjan halm) who has implemented the superformula equation, Gregsn (Sebastian Gregor) who has implement the calculi of the normal for shading, and me for debugging…

I will post the shader today in the wiki shader gallery, you will love it, possibilities are crazy……’

12 Responses to “SuperFormula 3D”

  1. Kristian Twombly writes:

    This is excellent! The first thing that I thought of when I saw this post was for installation. Now I’ll start thinking about translating this into sound…..

  2. paul writes:

    heh kristian… well the good news is VVVV can deal with sound and midi in a variety of ways if you intend to go that way?

    regarding input it can analyse sound for volume (RMS) or frequency analysis via FFT, it can talk to MIDI (input and output) aswell as communicate with software such as such as PureData or Ableton via a variety of protocols (TCP,UDP,OSC)

    Ofcourse you could try implimenting the superformula in an entirely different environment…. or perhaps you are hinting at using the pure data of the superfomula as a basis for generating sound or music itself?

    Soooo many options :)

  3. DJ Fadereu writes:


  4. tonfilm writes:

    nice to find something here about this. btw, sanch made some incredible new screenshots:

  5. Jack Round writes:

    I’m a home builder who has been looking for software to “generate” homes from a recursion / morphogenetic perspective (see if you haven’t run across it yet…wonderful site). I believe this type of paradigm yields much more beautiful homes than the traditional model. Any guidance or ideas you have would be very much appreciated.

  6. paul writes:

    heh tonfilm…. the stuff you and the crew are doing with VVVV is fantastic, look forward to more posts dedicated to this software!

  7. paul writes:

    Heh jack you may find computational/generative architecture sites like these interesting:

    Both explore the possibility of growing structures/buildings/surfaces utilising algorithms that describe certain forms of morphogenesis such as L_systems.

  8. rand writes:

    The meso guys are really pushing things forward at the moment – the super formula implementation is a fantastic addition to the vvvv armoury. I have uploaded some pics of my experiments with it at
    Those intrigued by the possibilites of this software should also check tonfilms superlative quaternion julia set fractal shader and implemetation of peter de jongs attractor at

  9. johan gielis writes:

    very cool stuff.

    You may want to check out and for more examples.

    Albert Kiefer’s book can be downloaded from

  10. johan gielis writes:

    an addition to my previous mail


    you can download and xcel file in which you can play around, also with ‘supertrigonometric’ waves

    johan gielis

  11. paul writes:

    thank for dropping by johan and thanks for the extra leads! :)

  12. 3d scanning writes:

    I work with 3d scanning equipment, l really enjoyed your post

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