[HN Gopher] Fern leaves and cauliflower curds are not fractals (...
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Fern leaves and cauliflower curds are not fractals (2012)
Author : minicaionut
Score : 75 points
Date : 2023-05-11 11:24 UTC (11 hours ago)
(HTM) web link (www.ncbi.nlm.nih.gov)
(TXT) w3m dump (www.ncbi.nlm.nih.gov)
| d--b wrote:
| 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.
| wrs wrote:
| 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...
| kergonath wrote:
| > 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.
| wrs wrote:
| 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.
| anigbrowl wrote:
| 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.
| kqr wrote:
| You can estimate Hausdorff dimensions of non-infinitessimal
| things, like coastlines.
| Y_Y wrote:
| Indeed, and there's a nice list of them here: https://en.wiki
| pedia.org/wiki/List_of_fractals_by_Hausdorff_...
|
| 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.
| anonymouskimmer wrote:
| People who are complaining about the article:
|
| 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.
| red_trumpet wrote:
| 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.
| dako2117 wrote:
| it really amazes me that someone took the time to write this
| article.
| anigbrowl wrote:
| 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.
| throwbadubadu wrote:
| > 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..
| sfvisser wrote:
| 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.
| ksaj wrote:
| 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.
| RosanaAnaDana wrote:
| 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.
| Clamchop wrote:
| 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.
| robotresearcher wrote:
| > 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.
| ksaj wrote:
| 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.
| robotresearcher wrote:
| 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.
| EamonnMR wrote:
| The thick branches, of course, start out as thin twigs.
| bannedbybros wrote:
| [dead]
| CoolGuySteve wrote:
| 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.
| robotresearcher wrote:
| > 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.
| JadeNB wrote:
| > 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.
| CoolGuySteve wrote:
| There were fractal body plans in the precambrian era but
| they went extinct pretty quickly.
|
| https://en.wikipedia.org/wiki/Rangeomorph
| robotresearcher wrote:
| Crystals perhaps? Bottom up rather than top down, but
| locally recursive.
| JadeNB wrote:
| > Crystals perhaps? Bottom up rather than top down, but
| locally recursive.
|
| Great example!
| [deleted]
| LesZedCB wrote:
| imagine publishing a whole paper to win an "Um, actually"
| argument
| ajuc wrote:
| 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.
| mfragin wrote:
| 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"
| dekhn wrote:
| 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.
| ska wrote:
| > 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.
| ouid wrote:
| As far as i can tell, fractal doesn't have a definition. The
| closest it ever came was the one you roughly gave.
| _a_a_a_ wrote:
| https://html.duckduckgo.com/html?q=fractal%20definition
|
| did you even look?
| idiotsecant wrote:
| 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.
| ouid wrote:
| 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."
| franciscop wrote:
| 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"
| eternalban wrote:
| 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
| anigbrowl wrote:
| 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
| ska wrote:
| > "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.
| coremoff wrote:
| 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
| cies wrote:
| 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.
| frou_dh wrote:
| Rosetta Code has a page on how to draw the 'fractal' fern leaf in
| dozens of different programming languages, probably including
| your favourite:
|
| https://rosettacode.org/wiki/Barnsley_fern
| freedude wrote:
| "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.
| jonathankoren wrote:
| 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.
| dekhn wrote:
| 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).
|
| Some interesting reading on development and math modelling:
| https://en.wikipedia.org/wiki/Reaction%E2%80%93diffusion_sys...
| https://en.wikipedia.org/wiki/Multi-state_modeling_of_biomol...
| https://en.wikipedia.org/wiki/Pattern_formation
|
| 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.
| ajuc wrote:
| You can create the same fractal in many ways. For example you
| get Sierpinski triangle if you repeatedly draw smaller
| triangles, or you can just do xor.
| CrazyStat wrote:
| 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.
| ska wrote:
| 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.
|
| Math has many beautiful corners.
| adzm wrote:
| Long story short, fractal has a specific mathematical meaning,
| and a common more vague meaning, and these meanings are not the
| same.
| mkaic wrote:
| 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.
| rsiqueira wrote:
| ...and "3.14159265358979" is also not PI! Because PI is infinite.
| [deleted]
| [deleted]
| dekhn wrote:
| Here's a much more interesting (popsci) article:
| https://www.nytimes.com/2021/07/08/science/cauliflower-fract...
| and the paper related to it:
| https://www.science.org/doi/full/10.1126/science.abg5999 Note the
| comment at the bottom of the article (who is the author of the
| posted article)
|
| After reading a fair amount of this I'd conclude that the posted
| article doesn't really move science farther, it just acts as a
| pedantry gate.
| jamal-kumar wrote:
| > 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
|
| [1]
| https://en.wikipedia.org/wiki/The_Algorithmic_Beauty_of_Plan...
| [1]
| eternalban wrote:
| 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
| fatneckbeard wrote:
| amazing
| intrasight wrote:
| There have been some HN posts on this recently. Some quick
| findings:
|
| https://www.nytimes.com/2021/07/08/science/cauliflower-fract...
|
| https://www.science.org/doi/10.1126/science.abg5999
|
| https://ournarratives.net/cauliflowers-fractal-pattern-a-key...
|
| Even if not a "true" fractal, for the layperson, it's close
| enough and does a good job of showing what is a fractal.
| michaericalribo wrote:
| > A realistic set of mathematical equations to describe fern leaf
| or cauliflower curd development is needed
|
| I wish the author had derived these equations -- something like
| the book The Geometry of Pasta
|
| https://www.amazon.com/Geometry-Pasta-Caz-Hildebrand/dp/1594...
| bannedbybros wrote:
| [dead]
| [deleted]
| chinaman425 wrote:
| [dead]
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