help me analyze this chaotic attractor/fractal i designed

reply to post by tgidkp


could your "resources be exhausted" due to trying to compute an infinite calculation?

I find this interesting as I will be doing a discrete mathematics course next semester. I am a cybersecurity major, so i know that computers have a hard time with the whole infinite idea. For example, to go from 1 to 2, you have to cross an infinite number of decimal points, but computer architecture treats math differently, as in not smooth but in a discrete finite sense.

I am just beginning to delve into this stuff myself so i may not be much help, but that MAY be a reason why you cannot calculate an infinite function. You may need better hardware!! Goodluck with your research, I will be watching this thread to see what else comes up!

also, you may like this thread that i created, inspiration goes to Ray Kurzweil and the book singularity is near.
Why the universe may be a computer afterall: cellular automaton, rule 110, and the holographic universe

reply to post by tgidkp


You say you are writing a paper or a thesis on this? Seriously drop the graphing calculator language. Python is very easy to learn. Download it (2.7.2) and numpy and matplotlib on to your laptop.
www.python.org...
scipy.org...
matplotlib.sourceforge.net...

In fact I tried to implement your function in python, but there is something I don't understand. If you are using the iterative method to generate a fractal you need at least one and preferrably two or more contractive mappings from R^2 -> R^2, and then you would typically apply them iteratively from random start points

Your function will set yout=xout=p(x,y) wherein p is a 4th order poly. of x and y. I thought perhaps your first poly sets x and your second y, but that doesn't generate the complex fractals you have shown. In other words I was expecting contractive s) (xout, yout) = func(xin, yin) , wherein xin and yin are chosen randomly.

Tell me what you are really doing and I will be happy to supply the python code to get you started.

reply to post by SevenThunders


I think that I understand your hangup. I am sure its my fault, not yours.

after you have written the function, you must run operate it as z,y) NOT x,y)

this is a "lost in the translation" between my calculator and the rendering program.

likewise, if you are attempting to iterate it directly in the rendering program, replace all variables 'x' with 'z'.

so then, 'ultra fractal', the program I am using, uses a loop/endloop format...just like I have shown.

literally, if you paste the functions as I have shown (replacing z for x) between loop and endloop tags you should get the same results.


I get the feeling this doesn't help much.


I will check out the python. I may have some time once my semester has ended.

thank you very much for your help.

reply to post by SevenThunders


here are screencaps....so that you can see exactly what i am looking at.

this one using a single polynomial


this one using the (truncated) iterative function

reply to post by VonDoomen


i fell in love with stephen wolfram's "a new kind of science" about a decade ago. much of the application of cellular automata that he makes toward the various existing science disciplines was above my head at the time. now that i have taken courses in these various subjects, i should probably take another look at it. thanks for the reminder! (btw, if you have never looked at this book, prepare to be impressed.....you think rule 110 is interesting?....you aint seen nothin yet!)

but really, i am such a terrible newb at all of what i am presenting in this thread i am afraid that i am beginning to make a fool of myself in my explanations. i have spent countless hours preparing all of this. i probably could have saved myself a lot of trouble if i already had linear algebra under my belt already. so, again, thanks for the reminder!


but ultimately the reason i have taken to the helm on much of this information is that, the "education" that i receive at school is typically so underwhelming. also, i am impatient and a control freak.


thanks for your input!.....i think i may have the "resources exhausted" problem solved.....

i spent several hours this morning rendering a composite of the first five iterated polynomials. i have not yet had a chance to explore it (time for school).

this is going to be really neat!

Originally posted by tgidkp
this one using the (truncated) iterative function

Help me understand some of this syntax of yours... I'm plugging it into _javascript.

I've converted the decimals to fractions (I find it's easier to see connections this way).

y = 0?
z = #pixel?

a = -(1/2) * (z * (y - 1) + 3 * (y - (5/3)))
y = a
b = 1/2 * z * (y - 1) + (3/2) * y - (5/2)

As you can see I'm a bit confused on y, and in particular z. Also, what do you mean when you place a decimal after a number (like 3.) but without any digits after the decimal?


Love the pictures.

Your best source will be to pack up your equations and trot off to the math department and ask them. They can tell you what kind of fractal it is (what class it belongs to) and what class of things use this kind of mathematics. Generally chaotic systems don't work well for biological systems, since there's feedback loops and everything's not completely random (even the flight of birds.)

And isn't this backwards from the way biology works? I believe you're supposed to observe and then use computational methods to explain -- not compute and then hunt for matches.

And yes, as you keep running through loops and storing the info, you are running out of computer memory (actually, you're using in inefficiently.) 64 bit machines can only calculate up to 2^64 -- anything higher than that returns errors (they can do it symbolically but if you're asking them to write it all out, they go bonkers after 2^64 (or 2^128, or whatever bit length your machine has)

reply to post by tgidkp


so your saying a specific function/format of chaos creates organization?

reply to post by circlemaker

WRT y=0 and z=pixel, these are program defaults. these are the inital values for z and y. the fractal will be drawn from there. the #pixel is a drawing instruction which i do not know anything about.

the only part i have input, which informs the fractal itself, is the "Loop" section.

there you will find 7 lines of code. the first six (without the 7th) will draw the fractal using the "iteration method", which i have found to give unreliable computer-crashing results.

the 7th (without the first six) will draw the fractal as the solutions of the polynomial according to the 'value of z'.

i have iterated the initial function (first six lines) individually, which yeilds a maximum of 5 polynomials before system crash. so, these five polynomials are the best (if not only) representation i will likely achieve.


so, what is the 'value of z'?

z yeilds the coordinates of the solutions along the imaginary axis. to iterate a fractal, you must feed z back in on itself. z is simply the updated value of the variable from the previous iteration of the polynomail. run the function for solution #1. input the solution #1 as the value of z for the second iteration of the loop. it is a recursive function....round and round until 'bailout' is reached.

the 'y' variable yields the coordinate of the current value of 'z' along the real axis.


the interesting part of it all is that the set of numbers which are the solutions for the fractal (and are printed to the screen) are the solutions of the iterated polynomial which do not yield a rational product before the 'bailout' is reached. in other words, the fractal solutions are only those which regress into infinity beyond the 'bailout'.

therefore, i believe (but am not certain) that truncating the decimal points into fractions might not produce any fractal at all. haha...or it might produce a more accurate one. i will check.



it would be really awesome to get a second display of these fractal sets using a different program. however i will understand if that is a bit more of a time investment than you planned on. i will upload a file containing the 5 polynomials which i have composited into the image above (when i get home later).



!!BUT!! the point of interest here is in the nature of the chaotic attractor (and the corresponding number set) which form the basis of these images. the visualizations are useful, but for this thread the images are just candy. i was really hoping to find someone here that could talk me through some of the intermediate (math only) stages of producing these fractal images. this is why i have uploaded zip files containing all of my (sloppily organized) work leading up to this.


if you are willing to hang, we should talk about the attractor.

reply to post by Indellkoffer

....trot off to the math department....

haha. i have little faith in the faculty at this school. i have found that, usually, i am better off learning non textbook stuff on my own.

Generally chaotic systems don't work well for biological systems, since there's feedback loops and everything's not completely random (even the flight of birds.)

it is a common misconception that CHAOS=NOISE. this is not the case.

the very important distinction is that noise is random (and would be a terrible model for biological systems), while chaos has only the appearence of randomness. in fact, a chaotic signal can be anlyzed using statistics and found to have very distinct regularities.


since starting this project, i keep seeing chaos everywhere i go. for example, last week i had my feet in a foot-massaging device while drinking a cup of tea. the vibrations were being transmitted through my body into the cup and into the liquid. the surface of the water was observed to have standing waves which appeared to be perfectly still.

so the point is that, rather than sloshing all about, the cup and the liquid were forming a 'system'. within this system exist certain parameters (such as the amount of liquid, the shape of the cup, etc.) which cause the output of the system to feedback as an input, which forms the regularity. but if you were to observe the motion of the water molecules at the molecular level, it would probably be indistinguishable from noise.

so a chaotic system can only be observed from a "higher level" than the components of the system. this feedback is a very real-world application of a fractal system, and the emergent nature of the chaos is very much a popular theme in theoretical biology.

And isn't this backwards from the way biology works?

are you sharpshooting me? nice try.

in fact, there is a ton of emerging data showing that many (if not most (if not all)) biological processes are maintained by a chaotic oscillation of some sort. the reason that they have been unrecognized up to this point is that, as you have, most researchers mistake a chaotic signal as merely "noise".

one of the real problems, though, is that there is no analytical foundation by which we might predict the characteristics that a biological chaotic system might have. so they are very tough to spot, and even harder to figure out the parameters which form the 'system' for the chaos.

my attempt here in this project is, as a subset of the larger goal of my thesis, to familiarize myself with procedures in the generation and representation of a chaotic signal. in this way, if during my observations i see a potential chaotic system, i will have the tools handy to enable further analysis.



thanks for your comments.

reply to post by ImaFungi


it would be better to think of chaos as being inherently organized.

"organized noise".

reply to post by tgidkp


but I always thought the definition of chaos is a lack of order or organization?

your main idea is that there is the alpha function (big bang) of the universe,,, and it is as chaotic as infinite potential energy can be,,,, but this potential energy is destined to do something under its parameters of chaos,,,, and so the universe is what it has done and is doing?....

I've said this before but I think it may apply here...

Chaos is just order we don't understand yet. And since we can't understand everything, there will always be both order and chaos.

reply to post by circlemaker


so there is no such thing as out of control,,,,, because the universe has its physical laws ( physics) and all energy and systems will abide by them according to their forces and scale and make up,,,, stuff that is out of control will at some point be acquired in an ordered system?


isnt this the opposite of what scientists believe according to ideas of entropy decaying order in the universe?

reply to post by ImaFungi


What I'm saying is that it's a matter of perspective. Another way to say it is that I have difficulty understanding chaos and that chaos itself may be a lack of understanding. Same thing with the idea of randomness. We often think something is random, but we don't have all the variables so it's hard to say for sure.

As far as entropy, I don't know. I'm more into math than physics at the moment.

reply to post by tgidkp


I am one of your non math heads, somewhat. The only non linear math I do would be blackjack related since that is my job and area of EXPERTISE.

Maybe finding a way incorporate the shape of a circle in place of infinity.

Like I said non math head hear.

Maybe 2 starting points instead of a start and finish.

Like I said non math head hear.

Maybe a start function the your non linear equation then start function and your non linear equation again the have it just repeat the process.

Like I said non math head hear.

Why don't I see chaos in the pictures. I suppose chaos can be structured also?

reply to post by circlemaker


if i take a handful of sand, and throw it as hard as i can, is the path the sand takes through the air random/chaotic?

or planned/controlled/organized/ordered?

im not trying to argue with you, just try to figure out what is being talked about as well......

are you saying randomness or chaos can never exist because everything always obeys some law.........

is the deffiniton of the opposite of randomness and chaos..... a big part of it...the will/intent/purpose...... to create order? like we can make skyscrapers and we are intelligent taking random pieces and creating order....... but the wind if it blows fractal patterns and designs in the sand,,, even if it produces something of order,,, the act of wind is one of randomness and chaos? even though there were causes acting under laws which prompted that gust of wind to exist in precisely the moment it did to create those patterns.

REPLACE INFINITY WITH A CIRCLE