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Fern leaves and cauliflower curds are not fractals (2012)

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

#41

Post by ouid »

idiotsecant wrote: post #35
ouid wrote: post #21

Earlier quoted context omitted.

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

https://en.wikipedia.org/wiki/Fractal_dimension Not only are fractals very well defined in mathematical terms, but there are further mathematics based on those definitions.

From the article you linked:

>Ultimately, the term fractal dimension became the phrase with which Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word fractal, a term he created. After several iterations over years, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants."

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

#42

Post by wrs »

kergonath wrote: post #18
wrs wrote: post #14

Earlier quoted context omitted.

It isn’t random, and it also isn’t a fractal (as is explained in the very short article). In other news, the golden ratio doesn’t really occur in nature as much as people would like to believe. [0] [0] https://www.maa.org/external_archive/devlin/devlin_05_07.htm...

> as is explained in the very short article It’s a straw man. The fractal is an analogy; nobody seriously thinks you can zoom in or out of a cauliflower infinitely. We know about atoms and stuff. It’s just pointless pedantry.

Analogies are only useful to the extent they have explanatory power. The assertion here is that this one doesn’t: there are several distinct growth mechanisms applied in sequence, not a recursive application of the same mechanism.

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

#43

Post by ksaj »

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.

Yes indeed. Without boundary conditions, they would grow indefinitely. Imagine a 3 story (and growing) cauliflower.

There are fractals that simulate tree growth. Even they have boundaries to stop them from turning into a giant fuzzy ball.

Also, unlike mathematical fractals, nature has limits as to how tiny things can be. At some point, it becomes quantum effects rather than fractal ones. Fractal math doesn't usually bother with that distinction, and will happily drive towards an infinitely small point, which in nature would be meaningless.

Hell, even we can only appreciate mathematical fractals if we zoom in digitally, meaning the numbers are reset to larger quantities than they were before the zoom. That effectively makes the digital zoom a bit of a mirage.

Nature doesn't ever have to do that, because of natural constraints. Once you reach the quantum level, things start to look really similar to each other, which never seems to be considered in mathematical fractals.

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

#44

Post by Clamchop »

RosanaAnaDana wrote: post #33
sfvisser wrote: post #19

Earlier quoted context omitted.

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.

It's really not that complicated. all vascular plants have basically the same node -- axially meristem -- internode -- node pattern to growth. All plants are actually fractals but it's not a long the leaf dimension, it's along the stem axis. It's also why it's so easy to effectively take cuttings and get a an 'fully mature' plant from them (eventually). Every node--etc.. section is the same pattern as every other. It…

Not all plants have the growth patterns you describe. Many monocots and gymnosperms (cycads) either don't branch, or do so irregularly (they'll look "messy", think Joshua trees or branching palms), and cuttings aren't a viable means of propagation.

Whatever the definition of a fractal (seems contentious), these plants aren't clearing the bar.

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

#45

Post by ksaj »

robotresearcher wrote: post #37
RosanaAnaDana wrote: post #33

Earlier quoted context omitted.

It's really not that complicated. all vascular plants have basically the same node -- axially meristem -- internode -- node pattern to growth. All plants are actually fractals but it's not a long the leaf dimension, it's along the stem axis. It's also why it's so easy to effectively take cuttings and get a an 'fully mature' plant from them (eventually). Every node--etc.. section is the same pattern as every other. It…

> All plants are actually fractals but it's not a long the leaf dimension, it's along the stem axis. > Every node--etc.. section is the same pattern as every other. Common sense says this isn't true. Some sections grow leaves and flowers rather than smaller branches, others do not.

It boils down to an if statement. Once you've reached a certain number of iterations, or the initial branching meets certain conditions, another rule set applies. But those rules are still not all that different - follow the leaf veins, and all their branching throughout.

The same applies to the roots.

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

#46

Post by robotresearcher »

ksaj wrote: post #45
robotresearcher wrote: post #37

Earlier quoted context omitted.

> All plants are actually fractals but it's not a long the leaf dimension, it's along the stem axis. > Every node--etc.. section is the same pattern as every other. Common sense says this isn't true. Some sections grow leaves and flowers rather than smaller branches, others do not.

It boils down to an if statement. Once you've reached a certain number of iterations, or the initial branching meets certain conditions, another rule set applies. But those rules are still not all that different - follow the leaf veins, and all their branching throughout. The same applies to the roots.

The article denies that, at least for the species and structures it discusses. It specifically says it's a non iterative process. That's the point the paper is trying to make. I'm not sure if that's the structures you are talking about, but a few comments up declared 'All plants are actually fractals'. The paper says no, at least in the aspects they are talking about, which are structures that appear fractal-like but are not. They also say it's misleading to think of them as fractals, as the processes are importantly different.

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

#47

Post by anigbrowl »

eternalban wrote: post #31
franciscop wrote: post #26

Earlier quoted context omitted.

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"

They were referring to this distinction: "Organic form itself is found, mathematically speaking, to be a function of time.... We might call the form of an organism an event in space-time, and not merely a configuration in space." - D'ary Thompson https://en.wikipedia.org/wiki/D%27Arcy_Wentworth_Thompson https://en.wikipedia.org/wiki/On_Growth_and_Form

This may be true even for inorganic matter, eg cracks in dried mud follow a predictable development that can be modeled quite economically.

http://irep.ntu.ac.uk/id/eprint/28373/1/5915_Goehring.pdf

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

#48

Post by anigbrowl »

wrs wrote: post #42
kergonath wrote: post #18

Earlier quoted context omitted.

> as is explained in the very short article It’s a straw man. The fractal is an analogy; nobody seriously thinks you can zoom in or out of a cauliflower infinitely. We know about atoms and stuff. It’s just pointless pedantry.

Analogies are only useful to the extent they have explanatory power. The assertion here is that this one doesn’t: there are several distinct growth mechanisms applied in sequence, not a recursive application of the same mechanism.

Different mechanisms can use the same template. The Romanesco cauliflower is so obviously self-similar that ignoring this in favor of talking about mechanisms that vary according to scale needs is missing the forest for the trees.

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

#49

Post by anigbrowl »

Somewhat off-topic, but since this thread has attracted people with a prior interest in L-systems: is anyone aware of an algorithm/ research/ anything really into reverse-engineering the - that is, given a degenerate tree or a set of trees, extracting a decent model that produces similar trees?

By degenerate tree, I mean messy real-world examples of trees in the computer programming/graph sense. Imagine applying algorithm to large directory trees or taxonomies; I'm interested in whether there's a mathematical way to approximately model the structural features.

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

#50

Post by JadeNB »

robotresearcher wrote: post #36
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…

> You cannot zoom into real-life fractals infinitely like in those math animations, only a few times? The point of the paper is that these are not 'real life fractals', so your correct declaration about the obviousness of real life fractals being bounded in their depth is not relevant, and does not make this paper pointless. It's not obvious to me that fern branches are 'anatomically variable and non-iterating bifurc…

> It's not obvious to me that fern branches are 'anatomically variable and non-iterating bifurcations', rather than a recursive process that bottoms out at a size boundary.

But hardly any (no?) real-life phenomena are the latter! As a mathematician, I'm as bothered by imprecise use of mathematical terminology as anyone, but, if we're going to call anything in real life a fractal, then it surely means something more like "appears to contain structure at multiple scales" than "the same structure at all scales". As @throwbadubadu points out (https://news.ycombinator.com/item?id=35903049), the classical example of a coast-line will also have different structure at small scales and at large ones, so either we throw out calling that a fractal (OK with me!), or we accept that we're using precise language imprecisely.

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