Is a nested polygon (please see attached picture) a fractal?
Cheers
Hey muchof.
What kind of iteration scheme are you using? How do you construct the object using your iteration scheme?
Fractals are characterized by this property of iterative-ness which contributes to the quantitative characteristic known as the Hausdorff dimension which should be fractional in the case of a fractal, and a major part of this is related to the self-similarity since this self-similarity creates weird dependencies in the actual information content of the object itself.
This is how you can show something is a fractal.
Hey chiro
Thank you for your response.
The image was taken from the internet (i searched polygon fractal), and have no idea of the iteration used.
Please forgive my ignorance , i thought the self similarity was the repeated triangle making a evolutionary type diagram of a fractal?
I don't think this is a fractal since it has a finite number of breaks in it and doesn't have the infinite iteration property that fractals need to have.
Fractal objects have this property where you have continuous resolution at all parts and by that I mean that the behaviour of "zooming in" means that the object is continuously being defined with more detail everywhere at every resolution (think of it in terms of zooming in).
Your object doesn't have that: the resolution of the object reaches a minimum (or maximum) very quickly and no extra detail or sense of self-similarity at any scale beyond a specific resolution is present and this is not a fractal object.
The other thing is that dimension wise, all of these objects are one-dimensional and don't have a fractal dimension. You can define all your polygons by a one-dimensional parameterization by knowing only the number of sides and fractals have the property that the Hausdorff dimension is fractional which is an additional (yet related) property to the self-similarity.
The fractal objects have infinite-resolution so unlike your polygon example, what this means is that you can zoom in at any scale and you will never find a situation where you get "grainy" details.
For example if you zoom in on say a computer bitmap file (BMP) you will see the pixels soon enough and everything will look blocky: this never happens in an infinite-resolution scenario at all no matter how far you go and zoom either in or out, but if you zoom in on the polygon you will soon enough get this blocky behaviour where the resolution shows a break-down.
Yes i think you are right.
I understand that fractals are infinite, i was looking at the picture as very basic representation of a 10 step progression of a fractal (with lines removed)
I have recently learned of fractals, they fascinate me. The image was going to be used as a tattoo design. If however the image cant be classed as a fractal or even quasi fractal i guess i better rethink the image.
interestingly enough, what you would get if you carried out the iteration scheme to its "limit" is not a circle, as one might suppose, but a vertical line! (the other sides of the n-gons are "trying to reach the common side", but they don't "have enough pieces" so they aren't "flexible enough").
This picture is not a fractal but represents a moment of the fractal
The fractal will be:
Beggining with the simplest polygon, add a face in each iteration
To chiro:
1.- You don't see the infinite iteration property? What about the addition of a face in each iteration?
2.- A fractal could have a negative movement in the fractal dimension so, instead of be a fraction of the last iteration you could have a reunion of some fragments from the last iteration. This is negative movement from one or more fractal dimensions!
3.- Do you can't see a tendency from a part? If you don't you never will see a fractal because you never ever will see a complete factal giving its nature (infinite)
4.- don't you see the factal dimension of this polygons? Give you a clue: the number of faces of each iteration...
5.- Fractal objects are never ever finished because they are infinites!!!!!!!! This is why you can't see the whole domain of this particular factal, dude!
Conclusion:
Your picture is the representation in time of a fractal (in the range of it's fractality)
You will see the same in yourselves: you are a fractal (in a lot of senses) moment. And you have the full potential of infinity but you need time (in the sense of iteration) to develop
Please, don't be naif to think that fractality is only a matter of geometry. Fractals are mainly developed in this category because Benoit likes geometry not because fractal are only a mater of shape. They are a lot more than that
Saying its the moment of a fractal is not really the same as characterizing a fractal.
This is not an infinite object with continuous symmetry breaking that has a fractional Hausdorff dimension with the appropriate topology.
I've already mentioned some characteristics that fractals need to have and this is not just a matter of a visual description: Hausdorff dimension is an algebraic definition that is based on topology with set theory and is an accurate way of categorizing a fractal.
Don't start re-defining your definition of a fractal and saying its equivalent with what others have said: your definition is completely different.
Only geometric fractals have Hausdorff dimension
Again you are making a rookie error considering fractal only when it has geometrical shape (respect, Benoit Mandelbrot does the same error)
Fractal is a methodology not a shape nor a geometric stuff
Everything you could iterate in a way or another is make by a fractal
You said I try to re-define a fractal but perhaps it's you who choose an incomplete partial definition so I need to re-define it, don't you think so?
It's not an actual fractal though.
Each so called iteration of the actual object is 1-dimensional. It doesn't have the iterative characteristic that a proper fractal has.
In the limit, you just get a circle and this object is 1-dimensional as well.
Again, don't re-define stuff.
If you want to give the iterative procedure and show me that this object indeed does not have an explicit definition and some proper Hausdorff dimension let alone all the other properties then be my guest.
The fractal does not need time to develop: it already exists independent of time.
Time is just a constraint we add to make things easier to analyze the different levels of detail of something, but the fractal already exists in its infinite form regardless of how we classify it even if that means resorting to iterative definitions that use an iteration index to define more or further detail.
Time is not required for a fractal to emerge: it already has that form. We just aren't that bright to be able to analyze the thing in its entirety because we lack the capacity to analyze it in its full form.
Again, don't re-define things.
Chiro, man, a fractal without iterations has full potential but it is only a function and this is why fractals doesn't become full developed until Benoit put them in a computer
Muchof, thanks, did you know Nassim Haramein? He talks about mixing science with spirituality (we, as fractals, have infinite potential that we develop by searching to inside in the quest of creativity. Creation capacity (via fractality) is what they mean when they said: We are made in the image and likeness of God. Nothing esotheric, here, only creators!)
The next iteration in science is to mix both and study the cases in a holistic way in the sense that biology only explain a part, you need to mix biology, phisics, etc... in a fractal way (as biology as one part, phisics as another an so on)
In that sense, today science is linear and needs to become fractal (like in a tree) in a formal way
But there is people like Chiro who refuses the main scientific precept (everything we think we know could be erroneus) because is easier. It is easier too, to repeat what other have tell you, isn't it Chiro? (or you are the main researcher of fractals in the world?, no, right? you're only repeating what others tell you as a parrot)
Mandelbrot has the same problem with mathematicians in his time and we have seen how they have to eat their own words before (because the nature of science itself)
I know Nassim Haramein and am aware of his work especially with his addition (in a joint paper with colleagues) where he adds rotational attributes to objects (physics and space-time) and I think that the stuff he is doing is ground-breaking: you should really think before you go blabbing.
I actually think the way science is as it stands now is not right: the approach is rather mechanical, and the way things are analyzed are done in such a way where everything analyzed is disjoint rather than seen as a connected whole.
I don't think that many scientists have any idea how to think in these terms, but as you have pointed out, some are thinking beyond the boundaries that their forefathers were constrained by.
But you are actually constrained by the same sort of things in some respects as they are: you are choosing your iteration as a way of creating a disjoint classification between iterations.
Time is just a constraint: it is a way that many people require in order to analyze something in a particular way but it is not a required constraint at all.
What you have done with your iterations is similar to the activities of people that force things in a way where they project their representation to something where it is structured time-wise or is analyzed with respect to specific local relationships and is then looked at in a disjoint way: the exact same thing as the parrots you make out despise.
The fractal, like anything else is a whole thing and you can decompose it in many ways (one of which is as you have done), but it does change under any decomposition: this is something you have introduced to try and attempt to make sense of it in the same way that the scientists you are so eager to criticize also do.