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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)

#31

Post by eternalban »

franciscop wrote: post #26
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"

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

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

#32

Post by fatneckbeard »

eternalban wrote: post #30
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

amazing

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

#33

Post by RosanaAnaDana »

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.

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.

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

#34

Post by mfragin »

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…

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"

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

#35

Post by idiotsecant »

ouid wrote: post #21
ajuc wrote: post #15

Earlier quoted context omitted.

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.

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.

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

#36

Post by robotresearcher »

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

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

#37

Post by robotresearcher »

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…

> 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.

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

#38

Post by _a_a_a_ »

ouid wrote: post #21
ajuc wrote: post #15

Earlier quoted context omitted.

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.

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

did you even look?

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

#39

Post by dekhn »

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…

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.

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

#40

Post by dekhn »

mfragin wrote: post #34
ajuc wrote: post #15

Earlier quoted context omitted.

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…

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 i…

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.
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