> 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.
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.
>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."
> 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.
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.
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.
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 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 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.
"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
> 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 representation from say the branching of stems and leaves (Say networks of vasculature within plants such as xylem and pholem, which we see as 'veins' within the leaves for example) - Then yeah, plants really are WAY more complex than that and they deserve a more accurate representation.
I think L-systems are beautiful and I really recommend anyone who's interested check out Lindenmayer's work on it all [1] but I think if this article has any point it's that we need more complex models to really do plants justice. I think that the fractal appearance of stuff like romanesco broccoli sure is cool but I think it's better understood as sort of a holographic projection of fractalline growth into 3d space rather than an actual 3d fractal
Like these models are super excellent for making renders of plants in 3D modelling engines but they're not REALLY plants, even if we can make them look incredibly realistic using just that basic level of modelling, some nice shaders, and some trig functions to make it look like it's blowing in the wind
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
There's this trope about physics and math envy, which I think ultimately boiled down an envy of the funding that physicists received in the cold war (it lead to a lot of equations in Ecology, for instance).
But ignoring the reasons for math envy for a moment, there's also a push among some of the more computationally inclined to try to explain other sciences using their tools. This has been going on for a least a few decades, with "holographic universe" ideas, Wolfram's "New Kind of Science", and the like, and of course actual computational modeling such as weather models. And conflating a prediction engine with intelligence (like that guy at Google did).
It's good to show that these reductions, at our current level of computational technology, are not that accurate. "Computational hacking" is really no different from "P-hacking", in that it is easily abused by the lay audience and scientists who conflate possibly fictitious correlative modeling with reality.
The model is not the thing. The P-score is not the thing. Mechanisms are the thing. Mechanisms are the core of science. All else is preliminary, at best.
I don't think I get your point. Especially regarding weather models. I'm pretty sure that to program good weather predictions, you need to know a lot about the mechanisms. And a prediction model can be used to test theories by comparing predictions against reality. That makes the theories falsifiable.
And what is it about comparing computational weather models to p-hacking? Good weather predictions provide value to lots of people, while p-hacking is looking for results where there are none, only benefitting the hacker.
TLDR: Math and models can aid with understanding, but fundamentally they're about divination, not explanation. And explanation is the core of science. Over application of models is as bad as p-hacking, in that false explanations for phenomena are promulgated. At least with p-hacking, one other paper can say that a correlation was not shown. Whereas with model-hacking, there is often a tendency to "tweak" the model to fit, ala the earth-centric model of the solar system with its epicycles.
> And what is it about comparing computational weather models to p-hacking?
P-hacking is just an extension of assuming p-values past a certain point are very good evidence of real-world correlation. The Princeton Engineering Anomalies Research has evidence showing human-operator specific effects on computerized random processes at the 7 sigma statistical level: https://pearlab.icrl.org/pdfs/1997-correlations-random-binar...
Do you believe that this evidence of possible psychic phenomena is real without validated mechanistic means?
> Good weather predictions provide value to lots of people, while p-hacking is looking for results where there are none, only benefitting the hacker.
Except when the p-hacking is, serendipitously, correct. P-hacked results are not necessarily wrong, they're just fraught.
> I'm pretty sure that to program good weather predictions, you need to know a lot about the mechanisms.
I'm not saying that all use of models and mathematics in science is bad. I'm saying that premature generalizations based on looks is bad.
Do you at least get my point regarding reducing the universe to a hologram or to cellular automata? Weather modeling is much better than these because of its predictive power, but most importantly because the weather modelers keep trying to refine their models based on actual mechanisms.
Knowledge of the mechanisms help guide the parameters used in the models. But the models are not the thing. The equations are not the mechanisms. And the approximations certainly aren't the thing.
And I'm sure that holograms and cellular automata are useful at times in predicting or thinking about reality, just as fractals are useful in some understanding of plants (despite not being accurate models). Just as the Earth-centric model of the solar system with its epicycles was pretty decent at predicting future positions of the planets.
Mathematical models are not the end of the scientific investigation. Mathematical models are the intersection of science (understanding of the universe) and divination (trying to predict the future). Statistics are most useful for trying to identify linkages which probably share mechanisms, or for testing a hypothesized mechanism's effect on particular parameters. In physics, equations for quantum stochastic processes does not mean we've finished with our quantum understanding of the universe.
Math is not science, it's a separate thing that helps with science. Generalized models are generalized models, not fully accurate representations.
And trying to prematurely universalize models is just as bad as p-hacking. Even worse is trying to universalize a single model idea (such as holograms, cellular automata, or fractals) to an entire field. Even if you are right, your model provides a divination understanding of the phenomena, not a scientific understanding of the phenomena.
Crap, I have some good points here, but it's probably TLDR at this point.
Of course it's not an actual fractal, because you know, infinity...
But take the Romanesco Broccoli, it really looks like a fractal. It can't be just random. There definitely is some kind of mechanism that's fractal-ish somewhere.
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.
Biology has the concept of convergent evolution. Would you say a Thylacine is a Wolf, and that it's appropriate to treat one as the other?
Fractals are a singular example of repeating patterns, but there are additional repeating pattern types that aren't fractals. Just like Thylacines and Wolves were/are examples of apex predators, not versions of each other.
It's also worth noting that there are other ways of "measuring" fractal dimension that do not always agree so it can be informative to also look at e.g. box counting dimension too.
3blue1brown has some good fractal videos; one in particular being relevant to this discussion: "Fractals are typically not self-similar" - https://www.youtube.com/watch?v=gB9n2gHsHN4
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.
There are no true mathematical fractals in nature, only approximations of them. But to say that ferns are not even approximating fractals at all is clearly incorrect to me.
Sure, the leaves don't randomly spawn in space, filling the leaf region at various points - they're leaves, they grow like leaves. It's still an iterated function system, the same type of fractal as a Sierpinski triangle. The only difference is it's a limited number of iterations.
A less prescriptivist title would be "I disagree with how people use the word fractal", as words are defined by how they're used and in what contexts, not by dictionaries or institutions. Considering that many, many people call fern leaves and cauliflower curds fractals, I see no issue with them continuing to do so. English as she is spoke.
"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 internal program that cannot be reproduced by our existing knowledge-base.
I read that same line, but came to the opposite conclusion than you. I didn’t find it amazing at all, instead I found it banal.
Essentially the author is saying that fern leaf doesn’t grow as single line, and then subdivides. Well, no shit. Literally no one ever thought that. It betrays a lack of imagination where the author is conflating a single algorithm for the construction of a fractal to an actual fractal shape. It’s the same as complaining that something similar to a Sierpinski’s Triangle can’t be a fractal because it wasn’t made by placing atoms at the midpoints between it and a randomly selected exterior vertex.
Are fern leaves technically fractals? No. (And no, I am not talking about the sophomoric objection then lacking infinite regression.) Are they similar to fractals? Yes. Is it useful as an illustration of both a feactal, and how complex body plans can be encoded in DNA without actually specifying every point like a literal blueprint? Yes.
None of this is different than the discussing the similarities fractals and coastlines. No one thinks Britain is a literal fractal.
Honestly, this idea of fractals is very similar to the idea that the irrational number phi shows up it nature. When you actually look deeply at the purported examples, you’ll find out, it’s not actually there.
Generative processes in biology are complex and fascinating. A lot of modelling builds simpler models which can be analyzed but don't explicitly simulate the full process of growth and development (which would require extreme memory and CPU). There is an open question in the field about how closely models need to recapitulate the underlying biology to be useful (in terms of generalized predictive ability).
When I was growing up, and until not too much longer ago, I assumed it would be practical to build full molecular dynamic simulations with atomic or quantum details, simulating large systems like groups of cells. Now I appreciate that this would be a lot of work that could be better handled by a well-trained deep neural net whose model does not recapitulate the underlying mechanics.
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.
These sets of transforms are known as iterated function systems (IFS), and the algorithm dubbed "the chaos game" by Barnsley. If I recall correctly the algorithm actually distributes uniformly over a probability measure supported by the set, so for picture-making purposes it is typically done with a probability associated with each transforms chosen to even out the visitation (otherwise some details will take forever to be seen).
The whole area is a consequence of Banach's fixed point principle (a very fundamental result), with Hutchinson I think extending it to unions of contractive maps.
I have seen some nature museum as a kid, and the older the life forms were in there, the more they looked "fractalish" to me. Ferns, fossils, the older the repeating'er.
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..