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Fern leaves and cauliflower curds are not fractals (2012)

ncbi.nlm.nih.gov

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Re: Fern leaves and cauliflower curds are not fractals (2012)

#21

Post by ouid »

ajuc wrote: post #15
throwbadubadu wrote: post #5

> Actual fern leaves and cauliflower curds have a very small number of anatomically variable and non-iterating bifurcations, which superficially look self-similar, but do not allow for scaling down of their structure as real fractals do. Sorry, can't help it, but really? You cannot zoom into real-life fractals infinitely like in those math animations, only a few times? What comes next? Even the coast line or mountain…

Fractals don't even need to be self-similar. They just need to have fractal dimension (if you double the size of every feature - the exponent near the scaling factor must be non-integer). Self-similarity is the easiest way to make a fractal, but not the only one. In fact the idea of fractal was invented for real-life non-self-similar objects. The simple self-similar ones are just examples that are easiest to understa…

As far as i can tell, fractal doesn't have a definition. The closest it ever came was the one you roughly gave.

Re: Fern leaves and cauliflower curds are not fractals (2012)

#24

Post by EamonnMR »

sfvisser wrote: post #19
throwbadubadu wrote: post #5

> Actual fern leaves and cauliflower curds have a very small number of anatomically variable and non-iterating bifurcations, which superficially look self-similar, but do not allow for scaling down of their structure as real fractals do. Sorry, can't help it, but really? You cannot zoom into real-life fractals infinitely like in those math animations, only a few times? What comes next? Even the coast line or mountain…

Agreed. I’m not a biologist, but I always assumed the self similarity of plants and trees is definitely some recursive process with some boundary conditions or external constraints as an implicit parametrization. Which is very similar to how you can describe (some) fractals. It feels very unlikely to me that the tree dna for thick branches is completely different from that for thin twigs.

The thick branches, of course, start out as thin twigs.

Re: Fern leaves and cauliflower curds are not fractals (2012)

#25

Post by ajuc »

freedude wrote: post #6

"Like fern leaves or any other plant branching system at the organ level, the cauliflower curd develops from the inside out through a process totally different from fractal drawing." Soooo, what the author is stating is they are actually even more amazing and wonderfully made than we originally thought. Even though they look like the mathematical model of a built fractal they grow entirely different and with an inter…

You can create the same fractal in many ways. For example you get Sierpiński triangle if you repeatedly draw smaller triangles, or you can just do xor.

Re: Fern leaves and cauliflower curds are not fractals (2012)

#26

Post by franciscop »

throwbadubadu wrote: post #5

> Actual fern leaves and cauliflower curds have a very small number of anatomically variable and non-iterating bifurcations, which superficially look self-similar, but do not allow for scaling down of their structure as real fractals do. Sorry, can't help it, but really? You cannot zoom into real-life fractals infinitely like in those math animations, only a few times? What comes next? Even the coast line or mountain…

Next comes this:

> "The fern leaf thus develops from the inside out and not by randomly dispersed dots that gradually fill the leaf area, as is done with chaos computer programs."

Yeah, real life is not a computer simulation (AFAIK), and thus is not made "of randomly dispersed dots"

Re: Fern leaves and cauliflower curds are not fractals (2012)

#29

Post by CrazyStat »

ajuc wrote: post #25
freedude wrote: post #6

"Like fern leaves or any other plant branching system at the organ level, the cauliflower curd develops from the inside out through a process totally different from fractal drawing." Soooo, what the author is stating is they are actually even more amazing and wonderfully made than we originally thought. Even though they look like the mathematical model of a built fractal they grow entirely different and with an inter…

You can create the same fractal in many ways. For example you get Sierpiński triangle if you repeatedly draw smaller triangles, or you can just do xor.

My favorite way to draw the Sierpinski triangle is Monte Carlo:

1. Pick a point which is in the triangle (e.g. one of the corners of the triangle). Draw that point.

2. Choose one corner of the triangle at random.

3. Move to the point half way between your current point and the chosen corner. Draw that point.

4. Repeat steps 2-3 as long as desired.

Obviously this only ever reaches a countable subset of the triangle based on where you start, but that subset is everywhere dense in the triangle so it doesn't matter.

You can also start at an arbitrary point that's not actually in the triangle. If you discard the first k (say 10) iterations you'll still get something visually indistinguishable from a Sierpinski triangle.

You can also do this with other self-similar fractals, you just have to find the right set of transformations to use. It's quite fun watching the random points coalesce into the shape of the fractal.

Re: Fern leaves and cauliflower curds are not fractals (2012)

#30

Post by eternalban »

jamal-kumar wrote: post #8

> A realistic set of mathematical equations to describe fern leaf or cauliflower curd development is needed Well if we're talking about Lindenmayer's work on L-Systems being limited to abstract representations of plants, without getting into all the other structures we're seeing that AREN'T self-similar within the plant itself, yet still branching and perhaps representable by a totally different L-system representati…

I remember reading about L-Systems in Santa Fe Institutes publications in '90s. Just found this book below when searching for that - this looks comprehensive at a glance:

The Algorithmic Beauty of Plants, Prusinkiewicz & Lindenmayer, 2004

http://algorithmicbotany.org/papers/abop/abop.pdf

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