I find fractals extremely fascinating and way, way, way out of my depth. One example being the Barnsley Fern, which I find so cool. It feels a bit like someone cracked a little piece of the Matrix code.
But is there anything really significant about the fact that it looks like a fern from a botanical/mathematical perspective? Do the two connect in any real way? Can we somehow find the math genetically or learn something about the mathematical properties of other leaves, for example? How “real” is it?
If I could make an oak leaf from fractals, would it advance mathematics and/or botany or would it be equivalent to creating a cartoon using Geogebra (nice to look at, but basically meaningless)?
The Barnsley Fern was constructed specifically to resemble the species of fern that it does. There are versions of it that have been modified to resemble other ferns. The fractal isn’t some secret mathematical code for why ferns look like they do, it’s more like a drawing of a fern. If someone made a fractal to look like another leaf, it would be just that, not an advancement into the secrets of botany.
The short answer: no. The two do not connect beyond the fact that ferns have a design reminiscent of a fractal, which is likely what inspired the fractal’s creation.
How “real” is it? It is a real set of functions, but if I design a set of functions to look like William Dafoe, it doesn’t mean I’ve cracked the matrix code into his genetics.
Well, you can define the structure of a L-system pretty simply. There’s probably a shared interest between ferns in having a simple set of instructions at a genetic level and mathematicians in working with mathematical structures that have descriptions simple enough for us to reason about.
Here’s some good reading on L-systems, written by the guy they’re named after:
