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Comment by throwbadubadu | Hacker News Reader

> 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 mountains fractal analogies are wrong in the mathematical sense, you won't even the same shape once if you try hard??

> The above cases demonstrate a general problem of using mathematical tools to investigate or illustrate biological phenomena in an irrelevant manner.

No, I think we have a different problem here..

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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 understand.

When I was teaching math and comp sci, I used broccoli in an example to explain self-similarity: imagine you're playing with your Barbies or GI Joes and you want to make their dinner plates look like they have real food on them. You can break off a much smaller piece of broccoli and it will look "to scale" on the plate. Try that with a banana!

That was to explain the concept of self-similarity, something we CAN see in fractals.

Note: for anyone wanting an easy way to experiment with L-systems, there's a built-in feature in Inkscape that is pretty fun to use. It's under "Extensions/Render/L-System"

I don't perceive broccoli as self-similar at any scale (maybe I'm missing something), but romanesco, for sure. I see at least 3 levels of self-similarity.

> maybe I'm missing something

I think you are. At the very least, you can clearly break a piece off a floret to resemble a scaled version of the whole head.

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."

https://html.duckduckgo.com/html?q=fractal%20definition

did you even look?

Thanks, this and the post to which you replied taught me something that I did not know. TIL...

> 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 bifurcations', rather than a recursive process that bottoms out at a size boundary. Now I know.

> 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.

There were fractal body plans in the precambrian era but they went extinct pretty quickly.

https://en.wikipedia.org/wiki/Rangeomorph

> There were fractal body plans in the precambrian era but they went extinct pretty quickly.

Cool! I didn't know about that.

Crystals perhaps? Bottom up rather than top down, but locally recursive.

> Crystals perhaps? Bottom up rather than top down, but locally recursive.

Great example!

That may be true but there were lifeforms with more fractal structures like the precambrian Charnia (https://en.wikipedia.org/wiki/Charnia ) and other Rangeomorphs ( https://en.wikipedia.org/wiki/Rangeomorph ).

The main difference the author is getting at (I suspect, I'm not a biologist) is that these earlier life forms display a multilevel fractal self-similarity within their fronds, basically fronds of fronds of fronds of fronds, instead of modern plants where the fronds have structural differences like leaves/buds that keep the pattern from replicating past a certain point.

The weird thing is that as far as body plans go, a fractal shape would maximize surface area while minimizing body volume which would seem advantageous for a filter feeding organism. It's not clear why such a simple design went extinct so quickly.

imagine publishing a whole paper to win an "Um, actually" argument

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's largely due to the exogenous origin of branching.

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.

> 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.

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.

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

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"

> "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."

That one is kind of silly, because while the chaos game is one way to realize an IFS like the Barnsley fern, there are others that are more geometric. For example specifically in that case, you can iterate any closed set in the plane under the contractive mapping that defines it, and you will end up converging to the set. Nothing chaotic about that. The algorithm they are referring to (chaos game) does it pointwise which is easier in a computer, but relies on the fact that the resultant sequence of points will, at least after a while, stay distributed over a probability measure supported by the set.

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

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