# What qualifies as science?

Where have I seen this? Oh, in my back yard.
Great camera work filming it

Terms like "fractal features" and "fractals aspects" are often used to denote things that share some features with fractals, but aren't actually fractals. Shrubbery is tree-like: woody stems, leaves. But they aren't actually trees.
Perhaps you're looking at natural fractal patterns from the wrong perspective.

First, a fractal is not necessarily infinitely reducible or expandable. It depends on the type of fractal. There are self-limiting fractals, especially in nature, where size and shape are important for survival.

In mathematics, a fractal is a set of exact iterations generated by a specific mathematical code, but nature is a dynamic environment which may interfere with the mathematical precision in which a fractal pattern code can express itself.

OTOH, a dynamic environment may create fractal patterns, which might not become expressed in a very stable environment.

In nature, the fractality lies in the growth formula of each species. If the growth pattern is random as perhaps with shrubbery then it is not fractal. But most trees have a specific growth pattern that makes them fractal. Sometimes the pattern lies in the way the branches bifurcate and if this is repeated from the largest branches to the smallest twigs, it's due to the fractal growth instruction, the formula which tells the tree how to space and grow the branching nodes. Tree leafs clearly show similar fractal patterns, each leaf is almost identical to all the other leafs of that tree.

Ferns clearly have a fractal growth pattern, which does not mean all leaves need to be exactly the same size, but display a fundamental similarity at all sizes from the largest leaves to the smallest. Ferns are a very old species of foliated plants.

If I recall some of the earliest organisms had fractal growth patterns in their RNA because it is naturally a very efficient way to grow. One simple growth pattern instruction for all offspring is naturally efficient. But it also has natural drawbacks.

The antenna in your cellphone is fractal, because fractals shapes are able to receive and process a wide range of EM wave-lengths,
Solar panels are fractal, for that very reason.

To limit the term fractal only to a specific type of self-similar form or ability is the very reason that Mandelbrot finally settled on the statement I posted earlier, which expands the scope of fractal expressions and applications to include not only pure abstract mathematics, but also real forms in nature.

Great camera work filming it

Yes, it took 40 years from the time Sierpinski planted the triangle to fully form the tree....

Perhaps you're looking at natural fractal patterns from the wrong perspective.

First, a fractal is not necessarily infinitely reducible or expandable. It depends on the type of fractal. There are self-limiting fractals, especially in nature, where size and shape are important for survival.
If you are saying fractals can have finite size (volume), then I agree, and I have never claimed otherwise. If that's not what you are saying, then I don't understand what you are getting at.

In mathematics,
(I.e. the only place where there are fractals, so no need to specify.)

a fractal is a set of exact iterations generated by a specific mathematical code,
No, you are describing the iterative procedure to make fractals. This is not the definition of fractals. Please read the Wikipedia quote I gave earlier. Nothing about a set of exact iterations, or a mathematical code.

but nature is a dynamic environment which may interfere with the mathematical precision in which a fractal pattern code can express itself.
Right, and thus there are no fractals in nature.

OTOH, a dynamic environment may create fractal patterns, which might not become expressed in a very stable environment.
Sure, but these fractal patterns aren't fractals, which is exactly what I was pointing out in the quote.

In nature, the fractality lies in the growth formula of each species.
I've never heard of that term; what is a growth formula?

If the growth pattern is random as perhaps with shrubbery then it is not fractal.
I see my comparison has confused you. I'm not saying that shrubbery or trees are fractals; I was pointing out that two things can be similar, yet not the same.

But most trees have a specific growth pattern that makes them fractal.
You mean "gives them fractal patterns."

Sometimes the pattern lies in the way the branches bifurcate and if this is repeated from the largest branches to the smallest twigs, it's due to the fractal growth instruction, the formula which tells the tree how to space and grow the branching nodes.
(A formula that tells a tree how to grow? Yeah, I'm going to ignore the "maths are real" sentiment this time.)

Tree leafs clearly show similar fractal patterns, each leaf is almost identical to all the other leafs of that tree.
Right, but they aren't exactly, thus it is not a fractal.

Ferns clearly have a fractal growth pattern, which does not mean all leaves need to be exactly the same size,
I've never claimed that, and it's obviously not a requirement for fractals.

but display a fundamental similarity at all sizes from the largest leaves to the smallest. Ferns are a very old species of foliated plants.
"At all sizes" being a very limited range from, say, centimeter to decimeter. No infinite iteration there, so no fractal.

If I recall some of the earliest organisms had fractal growth patterns in their RNA because it is naturally a very efficient way to grow. One simple growth pattern instruction for all offspring is naturally efficient. But it also has natural drawbacks.
Sounds plausible.

The antenna in your cellphone is fractal,
Welcome to the world of misnomers. Fractal antenna's aren't actually fractals, they are fractal-like.

Additionally, most cellphones I know of have a monopole antenna, not a fractal one. (But I might be wrong about that.)

because fractals shapes are able to receive and process a wide range of EM wave-lengths,
Which is not something cellphones actually need: "Cellular phone signals are transmitted on two bands, one between 800 to 900 megahertz (MHz) and the other between 1.8 gigahertz (GHz) to 1.95 GHz." Source: https://sciencing.com/difference-waves-cell-phone-waves-6624355.html
Solar panels are fractal, for that very reason.
I'm quite sure I've seen non-fractal-like solar panels. But at best solar panels are not fractals, they are fractal-like.

To limit the term fractal only to a specific type of self-similar form or ability is the very reason that Mandelbrot finally settled on the statement I posted earlier,
And his "settlement" is not what we use today. Please stop it with the argument from authority. (Or do you also pronounce GIF as "JIF"?)

which expands the scope of fractal expressions and applications to include not only pure abstract mathematics, but also real forms in nature.
Yes, and you've pointed out one instance where this sloppy language happens (antennae).

At best, you could argue there are two definitions of the word. The strict, mathematical one (A), and the looser "fractal-like" one (B). However, the latter (B) is not used in mathematics, and the former (A) cannot exist in reality, and thus you cannot equate the two (fallacy of equivocation). In other words, you cannot switch between the two definitions. If you point at an antenna and call it a fractal (B), you cannot switch to it having fractal (A) properties.

Note that I haven't seen anybody define fractals like (B); I've merely found lots of sloppy language.

Even a quote given by yourself hints to this:
"Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges."

"Approximate fractals" (what I've been calling fractal-likes) are not fractals, hence the word approximate.

You need to write Wiki, that they have it all wrong. Better yet , write Mandelbrot he 's got it wrong. Too bad he has passed.

A fractal is a natural phenomenon or a mathematical set
that exhibits a repeating pattern that displays at every scale. It is also known as expanding symmetry or evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern[
https://en.wikipedia.org/wiki/Fractal

p.s. this might be of interest. The geometry of rough surfaces . https://www.ted.com/talks/benoit_mandelbrot_fractals_the_art_of_roughness

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You need to write Wiki, that they have it all wrong. Better yet , write Mandelbrot he 's got it wrong. Too bad he has passed.
No, you need to learn that what one person says (even if (s)he is the inventor) doesn't mean it's so. That is an argument from authority. We use the word fractal differently than what Mandelbrot wanted. In other words, you are using the word fractal in a manner contrary to its normal usage.

I don't need to write Wiki (in fact, I agree with Wikipedia here), you need to learn definitions and proper usages of words.

At 2:37: "each part is like the whole, but smaller." I've mentioned atomic theory multiple times now; this property cannot hold for any material object. Mandelbrot actually refers to this at 5:40. 7:46: "Mathematicians began to create shapes that didn't exist." I wonder why he explicitly mentions that...

Really? And at what point does the space a 3D cube encloses attain a dimension of less than 2? Would that be when it ceases to be a cube and reverts to a Sierpinski triangle?
There is no such point.

You stated that one could not embed an infinite number of fractals in a 4X4 cube, and I handed you two ways to do exactly that.
And it appears you missed this; Must be an error in that scientifically observed and confirmed statement, somewhere, no?
"Natural phenomena with fractal features
Further information: Patterns in nature
Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges."

They talk about natural shapes being approximately fractal, so do I - I used the same word, "approximate", a couple of times in this thread iirc. They talk about the necessary property of self-similarity, and how it extends over finite scale ranges only, in the natural world; so do I.
I even drew a direct analogy between fractals and other mathematical entities that natural forms sometimes approximate over certain scale ranges - circles, parabolas, etc.

Where's the conflict?

(Sorry about the late edit - don't mean to wrongfoot anyone)

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There is no such point.

You stated that one could not embed an infinite number of fractals in a 4X4 cube, and I handed you two ways to do exactly that.

They talk about natural shapes being approximately fractal, so do I - I used the same word, "approximate", a couple of times in this thread iirc. They talk about the necessary property of self-similarity, and how it extends over finite scale ranges only in the natural world, so do I.
I even drew a direct analogy between fractals and other mathematical entities that natural forms approximate over certain scale ranges - circles, parabolas, etc.
Where's the conflict?
Yes, I think that's the point - approximation.

Fractal mathematics can help model some natural phenomena, just as the circles and parabolae that you mention. It seems to me the key misunderstanding is to confuse the idealised and simplified mathematical models we habitually make in science with physical reality itself. It is very plain from my own discipline that many of the systems we study cannot be modelled exactly using mathematics - sometimes there are even mathematical proofs that it cannot be done!

I occasionally wonder if a kind of arrogance can creep in, with some mathematical physicists, who can model exquisitely some very pared down, simplified and isolated phenomena in particle physics, but then go on to extrapolate from their own silo within the academy to the whole of science, when this does not necessarily follow.

No, you need to learn that what one person says (even if (s)he is the inventor) doesn't mean it's so. That is an argument from authority. We use the word fractal differently than what Mandelbrot wanted. In other words, you are using the word fractal in a manner contrary to its normal usage.
I just showed you pages and pages where we use the word phrase "examples of natural fractals." in scientific magazines and the most lifelike rendering of natural landscapes and alien creatures using fractal art. But that's all wrong ? Just cannot be done. The mathematical theory of fractals just doesn't allow for that?

The thing is that fractal art allows for all of it, because it is a fundamental property of spacetime itself .
I don't need to write Wiki (in fact, I agree with Wikipedia here), you need to learn definitions and proper usages of words.
Well then we must be in agreement somewhere. I have been using quotes from Wiki throughout, yet I am told it's all wrong. Oh, I forgot, Wiki isn't wrong, I am, because I don't understand what Wiki is saying, even if I quote it to illustrate my narrative.
At 2:37: "each part is like the whole, but smaller." I've mentioned atomic theory multiple times now; this property cannot hold for any material object.
Exactly, some fractals are self-limiting.
Mandelbrot actually refers to this at 5:40. 7:46: "Mathematicians began to create shapes that didn't exist." I wonder why he explicitly mentions that...
Because even he underestimated the power of the potential of fractal mathematics in the arts and sciences.

He showed a rudimentary mountain he built using a "new tool" (a computer). Today, at the movies, we can't tell reality from fractal created illusion.

And now that we are trying to model and explore the universe at Planck scales, our old friend the fractal turns up because it is able to create models at that scale.

Just like the abstract mathematically ideal circles, spirals and triangles , the universe seems to be filled with their approximations.

Rather than saying our mathematics are attempts to model reality, I propose that reality is trying to model these ideal forms, but can only approximate them because the universe is in constant change and any pure idealized symmetry can only be achieved fleetingly.

Where's the conflict?
There isn't any. The conflict only lies in the semantics, my use of wrong words and terms, which apparently shows my inability for abstract thinking.

I just showed you pages and pages where we use the word phrase "examples of natural fractals."
And similar pages could no doubt be found showing examples of "natural" parabolas and circles and catenaries and so forth.

All of them will be approximately parabolic, circular, etc. They will not be actual parabolas, circles, etc.

Exactly, some fractals are self-limiting.
I don't see any "self" limiting in those examples. The limits on how closely the thing approximates a fractal appear to be imposed by physical reality.

Rather than saying our mathematics are attempts to model reality, I propose that reality is trying to model these ideal forms,
And that proposal seems to be more of a confusion than an enlightenment. It's presentation of a reality "trying" to do things, especially, seems like a fast track to the high weeds.

Nothing about a set of exact iterations, or a mathematical code.
This I truly don't understand statement. Is DNA not a coding mechanism? And if in nature some things follow a regular pattern, i.e. a natural form of coding system, have we not been able to translate these natural codes into the symbolic language of mathematics?
8.5 Trees
The fractals we have examined in this chapter so far are deterministic, meaning they have no randomness and will always produce the identical outcome each time they are run. They are excellent demonstrations of classic fractals and the programming techniques behind drawing them, but are too precise to feel natural
In this next part of the chapter, I want to examine some techniques behind generating a stochastic (or non-deterministic) fractal. The example we’ll use is a branching tree. Let’s first walk through the steps to create a deterministic version. Here are our production rules:

Again, we have a nice fractal with a recursive definition: A branch is a line with two branches connected to it:

Emulate the Processing code in Example 8.6 and number the branches in the above diagram in the order that Processing would actually draw each one.

http://natureofcode.com/book/chapter-8-fractals/

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I don't see any "self" limiting in those examples. The limits on how closely the thing approximates a fractal appear to be imposed by physical reality.
I agree, but that is not the problem with the natural fractal instruction, the problem lies in the limitations imposed by the physical environment in which it is expressed.

One example of our recognition of these environmental limitations is our use of the baseline "in a vacuum", which assumes an objective environmental stability with no external environmental interferences.

But in physical reality there exists no pure objective stable condition. Only in a laboratory setting can we approach such a state, but even there we are dealing with the earth's gravity, etc.

On earth there is no possibility to create a perfect sphere, but in space, where gravity is minimal, we have been able to get really close in creating a perfectly spherical object. Thus we can say that in theory a circle or a sphere is a real object but alas, hard as nature tries to assume this state of perfection (symmetry) "almost perfect" will have to do, which IMO is a good thing.

I doubt that a state of perfect vacuum and stasis, would be conducive to dynamical processes.

Does arguing about fractals qualify as science?

I really only know about Mandelbrot and patterns which look very nice. I do understand they are a lot lot deeper than just looking nice

But pages and pages on a Humpty Dumpty?

Coffee time

Does arguing about fractals qualify as science?

I really only know about Mandelbrot and patterns which look very nice. I do understand they are a lot lot deeper than just looking nice

But pages and pages on a Humpty Dumpty?

Coffee time

I agree in principle, but from the current state of cosmological science, it would seem fractal functions are not trivial .

p.s. Being an avid coffee drinker myself, I use a Keurig coffee machine. It is able to create a cup of coffee equal to the best barista.

I agree, but that is not the problem with the natural fractal instruction
There is no "natural fractal instruction".
the problem lies in the limitations imposed by the physical environment in which it is expressed.
So nothing like a "self-limited fractal" is involved.
hard as nature tries to assume this state of perfection
Nature is not trying to assume any such state, which would not be one of "perfection" anyway (perfect what?). This is the weed patch mentioned above.

There is no "natural fractal instruction".
Then what is DNA, if not a form of coding system?

So nothing like a "self-limited fractal" is involved.
Then what are the "growth" and "stop growth" switches in DNA, if not a form of coding system?

Nature is not trying to assume any such state, which would not be one of "perfection" anyway (perfect what?). This is the weed patch mentioned above.
"Symmetry", "State of lowest energy"? Are these not natural "tendencies" of trying to find "equilibrium"?

I am not speaking of "intention" but of "natural ordering". Is the earth's ecosphere a "self ordering process", or just chaotic processes that happens to result in a "self correcting" system?

Does an exponential growth function in a limited space result in a eventual natural limiting of growth?

We can argue these are all physical phenomena, but our use of terms such as "universal constants" does suggest that some things "follow" deterministic natural laws, an underlying hierarchy of ordering states.

As an atheist I do not ascribe "intelligent design" to these phenomena. But why was this term even invented by theists? Think about it.

My point is that no matter what causality you want to ascribe to this apparent ordering process, it is clear that some underlying recurring rules of pattern creation do exist. The entire world of physical sciences is founded on the observation and description of these recurring patterns in nature.

If not a god or some other intentional (motivated) causality, what then shall we call these underlying natural imperatives? Laws of Nature?

My view is that a form of mathematical ordering functions are responsible for these phenomena. I may not be correct in this, but to me the term Physics is inadequate in describing these ordering principles into a cohesive totality or wholeness and it's dynamical properties..

Is it possible to form a TOE? If not, then a lot of people are wasting their time and that is not a satisfactory answer either, IMO..

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That is an argument from authority.
Is citing E =Mc^2 an argument from authority? Of course it is. Einstein was the knowledgeable authority and it has been tested and proven correct. The authority cited is from a reliable authority.

As was Mandelbrot, given that he had to wait for the development of computers which could handle large amounts of data to develop his fractal "Set" . It is the standard which encompasses all other forms of fractal sets mathematically and naturally.
And like Einstein his fundamental formula is simple and irreducible and complies with Occam's razor.
The basic formula for the Mandelbrot set is: Z = Z 2 + C . The Mandelbrot set is determined by iterating with this equation.
and has been tested and proven correct.
Like Einstein, Mandelbrot is the reliable authority, which can be quoted in all discussion of fractals.

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I just showed you pages and pages where we use the word phrase "examples of natural fractals." in scientific magazines
Yes, even scientific magazines can use sloppy language.

and the most lifelike rendering of natural landscapes and alien creatures using fractal art. But that's all wrong ?
I'm only arguing certain usages of the word "fractal" is wrong, nothing more.

Just cannot be done. The mathematical theory of fractals just doesn't allow for that?
Doesn't allow for what specifically?

The thing is that fractal art allows for all of it, because it is a fundamental property of spacetime itself .
Please stop pushing fringe theories. I'm specifically talking about mainstream and you know it. This is dangerously close to being intellectually dishonest.

Well then we must be in agreement somewhere. I have been using quotes from Wiki throughout, yet I am told it's all wrong.
The fact that I agree with one sentence on Wikipedia doesn't mean I agree with all of Wikipedia, because the fact that one sentence on Wikipedia is correct doesn't mean that all sentences on Wikipedia are correct.

Oh, I forgot, Wiki isn't wrong, I am, because I don't understand what Wiki is saying, even if I quote it to illustrate my narrative.
You might be joking, but that's actually closer to the truth than you think.

Exactly, some fractals are self-limiting.
That is not what I was referring to. Please stop misrepresenting my position. The quote is about the infinite iterations bit, not the self-limiting bit.

Because even he underestimated the power of the potential of fractal mathematics in the arts and sciences.
So now you know better than the inventor?

He showed a rudimentary mountain he built using a "new tool" (a computer). Today, at the movies, we can't tell reality from fractal created illusion.

And now that we are trying to model and explore the universe at Planck scales, our old friend the fractal turns up because it is able to create models at that scale.
Please stop misrepresenting fringe theories as mainstream. I've already shown you that CDT is not mainstream (at the moment). You are once again dangerously close to being intellectually dishonest.

Just like the abstract mathematically ideal circles, spirals and triangles , the universe seems to be filled with their approximations.
This however I can agree with, in general.

Rather than saying our mathematics are attempts to model reality, I propose that reality is trying to model these ideal forms, but can only approximate them because the universe is in constant change and any pure idealized symmetry can only be achieved fleetingly.
And that is not mainstream, it's possibly not even scientific. If you want to argue this, please do so in the philosophy or fringe sections of these forums.

This I truly don't understand statement. Is DNA not a coding mechanism?
We impose the coding. DNA is just behaving according to its properties. The coding model is our doing.

And if in nature some things follow a regular pattern, i.e. a natural form of coding system, have we not been able to translate these natural codes into the symbolic language of mathematics?

Again, we have a nice fractal with a recursive definition: A branch is a line with two branches connected to it:

Emulate the Processing code in Example 8.6 and number the branches in the above diagram in the order that Processing would actually draw each one.

http://natureofcode.com/book/chapter-8-fractals/
Yes, we (humans) have been able to translate that into mathematics. We model their behaviors with mathematics.