Great camera work filming itWhere have I seen this? Oh, in my back yard.
Great camera work filming itWhere have I seen this? Oh, in my back yard.
Perhaps you're looking at natural fractal patterns from the wrong perspective.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.
Yes, it took 40 years from the time Sierpinski planted the triangle to fully form the tree....Great camera work filming it
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.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.
(I.e. the only place where there are fractals, so no need to specify.)In mathematics,
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.a fractal is a set of exact iterations generated by a specific mathematical code,
Right, and thus there are no fractals in nature.but nature is a dynamic environment which may interfere with the mathematical precision in which a fractal pattern code can express itself.
Sure, but these fractal patterns aren't fractals, which is exactly what I was pointing out in the quote.OTOH, a dynamic environment may create fractal patterns, which might not become expressed in a very stable environment.
I've never heard of that term; what is a growth formula?In nature, the fractality lies in the growth formula of each species.
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.If the growth pattern is random as perhaps with shrubbery then it is not fractal.
You mean "gives them fractal patterns."But most trees have a specific growth pattern that makes them fractal.
(A formula that tells a tree how to grow? Yeah, I'm going to ignore the "maths are real" sentiment this time.)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.
Right, but they aren't exactly, thus it is not a fractal.Tree leafs clearly show similar fractal patterns, each leaf is almost identical to all the other leafs of that tree.
I've never claimed that, and it's obviously not a requirement for fractals.Ferns clearly have a fractal growth pattern, which does not mean all leaves need to be exactly the same size,
"At all sizes" being a very limited range from, say, centimeter to decimeter. No infinite iteration there, so no fractal.but display a fundamental similarity at all sizes from the largest leaves to the smallest. Ferns are a very old species of foliated plants.
Sounds plausible.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.
Welcome to the world of misnomers. Fractal antenna's aren't actually fractals, they are fractal-like.The antenna in your cellphone is fractal,
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.htmlbecause fractals shapes are able to receive and process a wide range of EM wave-lengths,
I'm quite sure I've seen non-fractal-like solar panels. But at best solar panels are not fractals, they are fractal-like.Solar panels are fractal, for that very reason.
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"?)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,
Yes, and you've pointed out one instance where this sloppy language happens (antennae).which expands the scope of fractal expressions and applications to include not only pure abstract mathematics, but also real forms in nature.
Fractal
A fractal is a natural phenomenon or a mathematical set
https://en.wikipedia.org/wiki/Fractalthat 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[
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.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.
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...
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
There is no such point.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?
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.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."
If you can reconcile these apparently conflicting statements, please help me out in understanding
Yes, I think that's the point - approximation.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?
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?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.
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.I don't need to write Wiki (in fact, I agree with Wikipedia here), you need to learn definitions and proper usages of words.
Exactly, some fractals are self-limiting.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.
Because even he underestimated the power of the potential of fractal mathematics in the arts and sciences.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...
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.Where's the conflict?
And similar pages could no doubt be found showing examples of "natural" parabolas and circles and catenaries and so forth.I just showed you pages and pages where we use the word phrase "examples of natural fractals."
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.Exactly, some fractals are self-limiting.
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.Rather than saying our mathematics are attempts to model reality, I propose that reality is trying to model these ideal forms,
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?Nothing about a set of exact iterations, or a mathematical code.
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:
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.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.
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
There is no "natural fractal instruction".I agree, but that is not the problem with the natural fractal instruction
So nothing like a "self-limited fractal" is involved.the problem lies in the limitations imposed by the physical environment in which it is expressed.
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.hard as nature tries to assume this state of perfection
Then what is DNA, if not a form of coding system?There is no "natural fractal instruction".
Then what are the "growth" and "stop growth" switches in DNA, if not a form of coding system?So nothing like a "self-limited fractal" is involved.
"Symmetry", "State of lowest energy"? Are these not natural "tendencies" of trying to find "equilibrium"?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.
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.That is an argument from authority.
and has been tested and proven correct.The basic formula for the Mandelbrot set is: Z = Z 2 + C . The Mandelbrot set is determined by iterating with this equation.
Yes, even scientific magazines can use sloppy language.I just showed you pages and pages where we use the word phrase "examples of natural fractals." in scientific magazines
I'm only arguing certain usages of the word "fractal" is wrong, nothing more.and the most lifelike rendering of natural landscapes and alien creatures using fractal art. But that's all wrong ?
Doesn't allow for what specifically?Just cannot be done. The mathematical theory of fractals just doesn't allow for that?
Please stop pushing fringe theories. I'm specifically talking about mainstream and you know it. This is dangerously close to being intellectually dishonest.The thing is that fractal art allows for all of it, because it is a fundamental property of spacetime itself .
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.Well then we must be in agreement somewhere. I have been using quotes from Wiki throughout, yet I am told it's all wrong.
You might be joking, but that's actually closer to the truth than you think.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.
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.Exactly, some fractals are self-limiting.
So now you know better than the inventor?Because even he underestimated the power of the potential of fractal mathematics in the arts and sciences.
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.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.
This however I can agree with, in general.Just like the abstract mathematically ideal circles, spirals and triangles , the universe seems to be filled with their approximations.
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.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.
We impose the coding. DNA is just behaving according to its properties. The coding model is our doing.This I truly don't understand statement. Is DNA not a coding mechanism?
Yes, we (humans) have been able to translate that into mathematics. We model their behaviors with mathematics.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/