Fractal geometry is a correct tool to describe natural shapes (for example, trees, bushes, etc.) Lindenmaier used a kind of fractal set (L-System) to describe biological shapes. Johan Gielis in 2003 wrote a "superformula"as a way to mathematically generate a wide variety of forms found in nature. The form generated by the equation can be controlled by setting the values of a few key parameters.
http://en.wikipedia.org/wiki/Superformula
As far as I understood the "superformula" (why "super" ? ) describes smooth regular curves with simple scaling properties, whereas fractal geometry describes non smooth objects with scale invariance. I guess it is possible to approximate arbitrary close a figure generated by the "super" formula by a fractal set (e.g. using an iterated function system and Barsney's collage theorem). I don't think the converse is true.
Now, concerning nature, if you ask which description is the "best" my opinion is that fractal geometry and super formula constitute two tools that can be used depending on the type of question you ask.
Bruno
I was writing a long answer, but somehow the page refreshed and I lost it. So a short summary:
1. super: generalization of superellipses and supercircles (Lamé curves)
They are now known as Gielis transformations (a name coined by geometer's)
2. On http://paulbourke.net/geometry/supershape/ you will find many examples, among others a superfractal (build up from supershapes). Of course we can also morph these shapes along the way.
3. The equation also defines regular polygons, so that Koch curve generation or the like can be defined into one equation, rather than an algorithm. And while the algorithm might be simpler, getting it into one equation has its own merits
4. On www.genicap.com under publications, you will find a number of publication in geometry, applied maths, mathematical physics etc. The company works on some technological advances, among others in communication technology, nanotech and image recognition.
5. It is a simple generalization of the Pythagorean theorem, so relates to deep geometry and other things nice. But most of this work is underway. Some papers are downloadable from the Genicap spaces page under publications.
If you have any further questions or remark, please let me know.
three small additions:
1. Yohan_Fougerolle has some of his pajavascript:pers on research gate, towards image recognition
2. As an implicit equation, it defines not only the boundary, but anything inside or outside as well. It also opens the door to a broader concept of manifold, in particular differentiable manifolds (new ways of defining tangent spaces, curvature....).
3. When thinking about complexity, a huge variety of shape now differs only in a few numbers from circle and sphere.
if you need more information, please let me know. I have good collaborations with Italian mathematicians.
well, one more addition for tonight connected to the ubiquitous and famous power laws: from a circle to Lamé supercircles results from increasing (or decreasing) exponents, so (x)^n + (y)^n = R^n (parentheses here mean absolute values). So it is a generalization of conic sections, since (x/A)^n + (y/B)^n = 1 for the superellipse. But the same generalization can be done for parabola's and hyperbolas. In the case of superparabola's (Neil's parabola is a special case), we have nothing else but the power laws, in which the fractal dimensions y = x^(a/b) are simply integer dimensions y^b=x^a. for a/b rational number.
A group of students, under the supervision of Alexandra Silva, are combining supershapes through R-functions. One of the goals is to make fractal like Koch flakes in one equation :-), will keep you all informed.
A fractal only simulates growth, quite well at times, perhaps cells can not function with too many adjacent singularities. I have come to understand engineering and biology though similar, solve connections quite differently. Perhaps 3D Shapes govern or influence growth by creating gentle gradients that influence cells to multiply in one location and lie dormant elsewhere. Every engineered structure fails at such sharp junctions unless suitably smoothing out the transition zone. Cells seem to use geodesic paths to maintain connectivity and often complex shapes provide ample provision for geodesic paths at the microscopic level. I dare not claim to be an expert just a mundane scientist by training. The Super Formula does not answer all questions but seems useful at the moment. These astonishing Super Formula shapes are presumed to be the product of cellular growth only vaguely seen in pottery or glass blowing.
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