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All Shapes in One Equation?
asadodetira writes ""One simple equation can generate a vast diversity of natural shapes, a Belgian biologist has discovered. Nature has the story. "The Superformula" sounds impressive, apparently its only for shapes, i thought you could solve lots of PDE's or tensor integrals or something with this, but not, it's only for shapes."
I have a java applet online that allows you to fiddle with the values in the equation and generate the 'super'shapes in realtime:
bodytag.org/supershapes1/
I doubt we can do anything with that formula to express Nature's art. Sure, we can build simpler graphical engines, but that's it.
I don't know how far this "transformed circle formula" is from a circle formula, but as long as it's an integer-dimension thing, we get nothing from it. It doesn't scale.
The concept of locality is too important - the behaviour of a cell is really parametered by its neighbors; the same ADN is in your brain, your liver and your nails.
Fractals are still a relatively simple method of describing 3D structures - sure it's really hard to start with a real object and map it to a fractal (but Nature works the other way around!). A very small disturbance can create as many shapes as you want; the number of different vegetal organisms showing very similar DNAs seem to support this.
Besides the biologist's own homepage, there is also the Genicap homepage, featuring a link to a PDF whitepaper.
It describes the superformula as a "generalized superellipse equation". The 3D version is based on superquadrics.
the equation can be found here
(link found on page with java demo linked to in parent comment - thanks!)