attractor fractals anyone?

(Update: got some decent detail on this one. n=10, f(x)=4x(1-x) read below for more info...)

Ok, so we were learning about a new sort of fractal in class today, and I used my computer to display some data for some of the functions that we were looking at.

These are all fractals of the sort where we have a function f and we calculate a point xn by plugging in a value x0 into the recursive definition xn+1=f(xn) n number of times.

The graph that you can see is made up of a series of xi values on the vertical axis (from x0-xn, for example x0-x20 or so), where the horizontal axis is the value of the starting input x0, ranging over the unit interval (0.0, 1.0). I made low-detail renderings by limiting n to 5, and I made high-detail renderings by letting n go up to 20. That means on the high-detail graphs, we have 20 plotted values for x at each x0.

The first one that was mentioned was f(x)=3/2*x for x between 0 and 1/2 and f(x)=-3/2*(x-1) for x between 1/2 and 1. Here is a low-detail rendering (n=5).
(****Note: for a larger view of these pictures, left click on them. My blogger displays shrunken "thumbs" instead of full resolution. On some of these renderings, getting the larger view is critical. -Ed)

Here is a high-detail rendering (n=20). See the other equations below for better results.


The second function that was mentioned was f(x)=2*x for x between 0 and 1/2 and f(x)=-2*(x-1) for x between 1/2 and 1. This is a low-detail rendering of the above equation (n=5).here is the most highly detailed rendering (n=20). I think this is a really awesome illustration of the infinite detail contained in such simple math.

This is another new shot. I decreased the line width and increased the increment in order to show more detail. I still am unclear about two aspects of these fractals: I don't understand exactly what an "orbit" is for one of these attractors and how it causes the detail. Also, and this is related to the first question, why is it called an attractor in the first place? if you know the answer to either of these questions, then by all means post a comment!
In the closup here, you can really see the layers upon layers of lines that seem to reflect off the "walls" of the graph over and over again.
The third one is f(x)=4x*(1-x). Here is a low detail shot of the fractal (by low detail, once again I mean that each x0-xn series is only 5 terms long, as in n=5)


This one is a slightly more detailed rendering (n=7), If you look closely at the two renderings, you can see exactly what the value n controls. If you look at any point on the horizontal axis, there are 6 curves that are crossing that spot (x0 is not drawn). If you refer to the previous shot, you can clearly see the 4 individual curves. As you increase n, you get more curves and thus more detail!

here we can see the most highly detailed rendering (n=20). I am still trying to figure out a way to zoom properly or display it so that the infinity is clear to the eye, but I think that this shot hints at the infinite detail of the fractal.

This final image is a rendering of an attractor-like equation that I discovered myself after playing around with the math for awhile.
This one is an n=10 rendering of the function f(x)= - (x-0.5)^3 for x less than 0.5 and f(x)=x^3 for x greater than 0.5.

here is a listing of the source code that renders these images. For this code to work, you need to have your opengl window set up already.

(note: code was removed, as blogger thought that less-than signs were html tags...
once i figure out how to post code, I will add the source here. -Ed)

another note on self-similarity

here are a few shots from xaos. These are good examples of the self-similar quality of fractals. You can clearly see the original mandelbrot "hole" over and over again, each time slightly different but also the same.

The original mandelbrot

now, after zooming, we can find the same "hole" only as if viewed through a curved piece of glass.

Eventually, the "hole" gets so warped that it is barely recognizable, though still it is mathematically similar to the original.

If you go further and further into the set, you get to a point where your brain doesnt understand the similarity, although what you are seeing is still similar to the original.



but if you go far enough, at some point you hit the wall. This is based on what you set the iterations to in the beginning. This has to be a finite number, though, and it is reliant on computing power, so we can never go forever into the fractal. The fractal itself has no limit, however.
neat huh?

fractal zooming software

I have been playing with a real-time fractal zoomer that deals with a bunch of escape-time type fractals. Here is a link.

(windows download... if you want to get the source code, just search google for "xaos")
http://prdownloads.sourceforge.net/xaos/winxaos31.zip?download

I think this software provides a dramatic view of the concept of infinite detail (hold the left mouse button while clicking on the spot that you want to focus on to zoom into the fractal). Also to be clearly seen is the calculation limit. If you zoom in far enough, you hit the "wall" which is the threshold of the iterations that is set to cap computing time. Don't forget to hold the right mouse button (zoom out) to see the equally mysterious phenomenon of the infinite detail disappearing into single pixels.

Also, dont forget to set the iterations higher than the default, where it doesnt look great. On my laptop i can set it at well over 750 iterations without slowdown. Depending on your machine, you should be able to set it from 500-1000 iterations. It looks decent on 500 though. You can change that on the calculations menu->iterations. Also, dont forget that you can look at lots of different representations of the mandelbrot set, and also other fractals besides mandelbrot. There is lots to mess around with here.

see my last post for discussion related to the calculation threshold and its possible implications.

Self-Similarity and the Human Condition

The fundamental goal of the human race is to understand our place in the universe. All schools of thought, every organized religion, and indeed every branch of science is just another avenue of exploration towards some beacon that will reveal to us the truth of our existence.

A never-ending and impossible search you say? I disagree. There is only one relevant bit of evidence to examine, and that is each and every sublime instant of time, which, no matter where you are, is an amalgam of every last atom and force EVERYWHERE. This, of course, goes back to the concept of chaos theory, that (simply, *ehem*) when a butterfly flaps its wings, a hurricane comes into existence on the other side of the world (or at least has the possibility of coming into existence, according to quantum theory). Indeed, even though some forces appear weak (i.e. gravity), and they might propagate through space in an exponentially diminishing fashion, the universe as a whole still feels the effects. And so the current instant in time is a combination of all the forces in the universe (and not just the forces of this moment. Each moment also contains a record of each force that has EVER existed, as the universe can be viewed as a gigantic state machine where the current state is determined by the previous one). As such, anywhere in the universe, each moment of time provides all the evidence that is needed to discover the secret of everything.

So what do we do with this incredible resource that is each and every instant of our lives?

Well, most of us, including myself, watch television.

Wait a second! No, no, sorry. That's not what we REALLY do (*cough*). The correct answer is that we conduct experiments on Now (I will henceforth refer to the rich resource of enlightenment that is each moment of existence as Now). Not only are we currently conducting experiments on Now, but we have been conducting experiments on it since we first sprouted legs and crawled up some slimy shoreline onto the firm, grassy bosom of the Earth.

So how does this lead to the article that I am currently authoring? Here is a brief timeline:

790,000 years ago, In Eurasia, some guy picks up some flint and some wood and dramatically increases the bedtime of humans everywhere, much to the dismay of protective parents. Each and every night, he would light a fire in the center of his village and ruminate on the meaning of it all. Everyone else in the village assumed it was his force of will that kept the flames from dying, and so they agreed to support him collectively, as doing manual labor or any other kind of real work would break his concentration. The first philosopher is born!

Then, about 789,600 years later, Newton or someone proposes a radical idea called the Scientific Method. Here is a Wikipedia definition for you:

"Scientific method as envisaged by one of its early exponents, Sir Isaac Newton, is fundamental to the investigation and acquisition of new knowledge based upon physical evidence. Scientists use observations, hypotheses, and logic to propose explanations for natural phenomena in the form of theories. Predictions from these theories that can be reproducibly tested by experiment are the basis for developing new technology."

While humans had been utilizing the power of Now since the beginning, it was in the 17th century that someone finally pulled his head out of the sand, looked around, and pronounced boldly that Now was more powerful than had been previously realized.

What followed over the next 400 years was the most explosive era of new understanding in the history of mankind (electricity, general relativity, nuclear power, flight, space-flight, computers, etc.)

Scientific experimentation is, at its heart, a rigid process designed around producing knowledge from Now. Indeed the unspoken basis of the scientific philosophy (yes, thats right, I said it. I am not taking it back, either.) is that Now contains the sum knowledge of the universe. Scientists everywhere are affirming the power of Now without realizing the greater significance of that affirmation.

Now I bring up self-similarity. The concept of self-similar sets in mathematics is that you have some set S which contains elements S0-Sn, each of which can be morphed back into the whole set with some function Fi. I propose a grand self-similar set (the universe) which contains (among other things...) each and every instant of time in each and every single location in the universe. That is S=the universe, and Si=a given instant in a given location.

The reason that this ties into the discussion above is the concept of Now. If we can derive all of the information of the universe from Now, then the two are similar. Specifically, we use experimentation to test values of functions of Now (or "points") and we use different input values (like altering conditions) to plot functions of Now. These functions become theories and sometimes (eventually) "laws" of Now. These "laws" end up creating a collection of functions (F0-Fn) that map Now, and we call it Physics. This collective function (Physics) attempts to map Now to the universe. Many believe this ongoing effort is the development of a Grand Unification Theory, which would be a complete function to map an instant of time onto the universe.

If we assume that this theory is possible (and the triumphs of science and mathematics are a reasonably good indication that we are on the right track with this line of thinking), then we admit the notion that the universe is self-similar. If we admit that the universe is self-similar, then we arrive on a fascinating array of consequences.

First off, is the entire universe a fractal? Our book is titled Fractals Everywhere, by Michaul F. Barnsley (I refer to the text used in Professor Tom Taylor's fractal math class at Arizona State University... more on this later). One of the startling realizations that came with the mathematics of fractals was that they were all over the place in nature. The shape of a fern, for example: (from http://www.home.aone.net.au/~byzantium/ferns/fractal.html)The fact that fractals are everywhere in nature seems to be further indication that there is a greater self-similarity to everything around us.

So if the entire universe is a self-similar fractal, then wouldn't it follow that all of the fractals that arise within the universe be part of that self-similar set as well?


Is this the universe?


Next, consider the human condition. What is the purpose of art, if not to touch upon the universal themes that move the human soul? What causes millions of humans everywhere to connect to the same story? I argue that this happens due to the similarity of our lives. If each moment truly is the whole, then each and every day all of us experience the same struggle. This is something that we realize intuitively, but the fact that mathematics may prove that we are all literally the same has dramatic ramifications.

And why do such small bits of things move us, such as the scent of a perfume or the brilliant red of a rose? If the universe is infinitely self-similar, then Now is also self-similar and individual constituents of the moment would remind us of the whole.

The last item to mention is the human brain. A brain is a perfect example of the beautiful chaos that infinite detail suggests.

Zoom out to the view of earth from the moon. You are looking at a round ball of rock covered in a swirling atmosphere of white surrounding an ocean of blue. After a moment, (in your godliness) you realize that the earth itself can be mapped onto the entire universe using function F. You laugh to yourself as you realize the simplicity of it all and how ironic it is that the planet's inhabitants falsely believe that their planet isn't the center of the universe after thousands of years of intuitively but unwittingly knowing the startling truth.

Then you think about going down to check things out, and so you drift down towards the Earth at incredible speed (in your godliness). At some point you are struck with the sight of the new detail of the landscape which had before been hidden in the fog of distance. What had before seemed a ball is now a great plane beneath you, and you head for the shining dot that is a huge metropolis.

As you approach, you realize again that your vision had been lacking. This shining dot was not a dot at all, but a "concrete jungle" of girders, glass, and steel. Soon you start to see that what was before a gray mass was, in fact, life. People are everywhere in this city, and they cover the ground in a living carpet of splooging color (yes, splooging is a word). Being of the curious sort, you want to investigate what makes these guys tick.

You go down to microscopic size and enter the human brain. After flying around for what seems like an eternity, you have a complete understanding of the entire brain (in your godliness). Something hits you like a flash, and it manifests as a vision in your mind. You remember looking at the world and realizing... Yes another vision hits you instantly, which was where this leg of the adventure all started.

You see yourself looking at what you would come to know as this universe, and although you hadn't expected it at the time it seems blindingly obvious now. You had flown down on a wildly different scene, though perhaps you could call it a city, and after reaching "microscopic" size, you had entered a brain in an attempt to understand the life that dwelled there.

The moral of this story: If the universe is infinitely self-similar, then the human brain can be mapped back onto the universe. Thus, the universe is life, and life is the universe.

Perhaps we would find life if we could break the conventional limit on computing power and calculate known fractals down to such detail. Maybe space-exploration is an effort in the wrong direction...

(more to follow!)

Initial Post

Greetings all!

My name is Max Rebuschatis, and this is my blog! If this is your first time to my blog, let me tell you that you are in for a real treat. If this is your second time visiting my blog, then shame on you!! You should have learned your lesson the first time.

I'll teach you to read the words of a raving lunatic!

Anyway, here is some info about myself:

name: Max Rebuschatis
math background: I have been highly interested in math since high-school. I have taken calculus I, II, and III, Discrete Math Structures, Linear Algebra, CAGD, and, most recently, Fractals.
programming languages: I am a computer scientist, and as such am familiar with several languages (though not perl, which is my next goal) including: C, C++, C#, Java, x86 assembly, lisp, scheme, prolog.....
Goals for (Fractals) course: I want to gain a more clear understanding of fractals so that I can do two things. The first is to be able to use fractals in many different aspects of the video games that I create!!! The second is to increase my understanding of reality, as fractals seem to be wrapped up in the concept of life itself.
Extent to which goals have been reached: not much yet, although the concept of self-similarity has got me thinking.....
Unanswered questions: if "fractals are everywhere," do we live in one?

anyway, keep reading my blog for scintillating thoughts on fractals, god, string theory, and the universe. enjoy!