Friday, 2 June 2006
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……’