21 replies to this topic

[zanzibarjones]

Do you believe in the notion that there is a part of the universe that is not controlled, by.... something? Chaos, utter and complete chaos?

I believe Chaos does not exist. I can throw you some examples or why I believe that.

1) If chaos truly existed, would planets have formed into spheres? I mean a sphere has got to be the least favorite shape of chaos, because it is the easiest one to make. It's the path of least resistance.

2) Could chaos also create symmetry? I don't think so. Look at yourself in the mirror. Do you really think chaos made you, by mistake or by accident? Nope.

3) Could chaos have invented math? Probably the most ordered thing in our universe...

4) How about laws of nature that exist just not here but on other planets or in space....

What do you think?

[rvalkass]

Maths can be chaotic too. Take a look at the logistic difference equation:

x_n+1 = R * x_n * ( 1 - x_n )

Looks like a perfectly logical, simple equation. Watch what happens when the value of R is changed by small amounts:

 logistic-equation.png   166.17K   15 downloads

Is this true chaos? Given the equation and the starting parameters, there are cases where you have little chance predicting what the outcome will be. It's like stepping out your front door, walking a few steps, then suddenly being on Mars, then in the supermarket, then on the sun, then on your sofa.

Chaotic shapes also exist. Take a look at fractals - shapes with finite volume but infinite surface area. As simple to create as a sphere, but infinitely more complex and intricate.

[zanzibarjones]

rvalkass, on 25 August 2010 - 11:10 AM, said:

Chaotic shapes also exist. Take a look at fractals - shapes with finite volume but infinite surface area. As simple to create as a sphere, but infinitely more complex and intricate.

I don't know about that. I mean you can predict where the nest surface area is going to be right? I mean I have seen programs zoom in on fractals, and they all have symmetry to them. Is that really chaos?

[rvalkass]

You know where the surface area (or volume for a 3D shape) is. But any two points on the edge of the shape are separated by an infinite distance along the shape's edge. From a distance the points appear right next to each other. As you look closer, as you correctly say, you see more and more detail. That detail continues forever, adding more and more complexity to the perimeter of the shape. So, if the edge is getting more complex, and therefore longer, with no limit then it follows that the distance between any two points on the perimeter of the fractal must be separated by an infinite distance along the perimeter.

[Bikerman]

Thats exactly the same explanation I normally start with - the good old difference/logistics equation.
Just to add - the equation is of a type used by biologists for simple models of populations. The R corresponds to the 'fecundity' or 'randiness' of the animals and 1 is full population, 0 is extinction. The 1-X represents environmental constraints - food, competition etc.

In fact they were using the thing for years and just disregarding the results when it seemed to go bad. Not unlike the way that mathematicians are first taught to solve polynomials and ignore the ones that have no real solutions - just junk - when in fact they are probably the most interesting :-)

---

Here is something that should astonish readers (it blew my mind when one of my lecturers asked me to code it on an old BBC microcomputer back in 1982)...

Draw a triangle - any size and shape (as long as it IS a triangle of course).

1. Choose a point completely at random somewhere in the triangle and put your pencil down to make a point.
2. Repeat the following 2 steps
3. Choose one of the vertices (corners) of the triangle at random
4. Move your pencil halfway from where it is to that vertex (corner), put it down and draw a point.
5. as many times as you can

Rvalkass might know this (in fact I bet he does), but I bet nobody else can tell me what you get after repeating this a few thousand times. I was gobsmacked when I came back to the computer after an hour and saw what was on the screen - in fact I thought it must be an error in the graphics chip...

If you want to try it, or try to work it out, then don't scroll down. If you can't be bothered then scroll down to see what results - and it works everytime..

Edited by truefusion, 25 August 2010 - 07:15 PM.

[rvalkass]

Bikerman, on 25 August 2010 - 06:15 PM, said:

Rvalkass might know this (in fact I bet he does)

Sierpinski's triangle - already had one in my gallery of fractals, flames and other weird images:

http://rvalkass.co.u...is-050717-2.jpg

[Bikerman]

rvalkass, on 25 August 2010 - 06:57 PM, said:

Sierpinski's triangle - already had one in my gallery of fractals, flames and other weird images:

http://rvalkass.co.u...is-050717-2.jpg

Thought you might :-) I like this way of generating it. You can use the normal method of just connecting the sides but this way brings it to life by showing how order emerges from apparent random choices. The lecturer who gave me it knew it would be right up my street, and it got me interested enough to do some more reading and learn a bit about the whole interesting story. Since then, of course, everyone now has seen the Mandlebrot set - seems to crop up regularly. My other favourite is of course the Koch - which bangs home the bounded infinity (and is also pretty) :-)
http://math.rice.edu.../frac/koch.html

Edited by Bikerman, 26 August 2010 - 12:57 PM.

[zanzibarjones]

Hmmm... not that I am a genius or anything, but I still ask this, if chaos supposedly reigns, how do you explain the simple sphere? It's nature's, or the universe's simplest shape and it's, for lack of a better word, perfect.

[Bikerman]

zanzibarjones, on 30 August 2010 - 01:49 PM, said:

Hmmm... not that I am a genius or anything, but I still ask this, if chaos supposedly reigns, how do you explain the simple sphere? It's nature's, or the universe's simplest shape and it's, for lack of a better word, perfect.

Chaos doesn't 'reign' but it is relatively easy to create a sphere, even in a chaotic system. Water, for example, has surface tension - ie the molecules attract each other. The optimum way to arrange water (or for it to arrange itself) is in a way where the minimum number of molecules are not surrounded by others (they all pull on each other).
That arrangement is what we call a sphere. Put some water in space and wait- you will soon have a sphere.
http://stevespeeves....re-experiments/

Edited by Bikerman, 30 August 2010 - 06:24 PM.

[Rael IAK]

I don't consider the logistic equation chaotic at all. Like any other deterministic model of anything, you plug in some values (like for N(0) and R) and it spits out a result. You always get the same result when you plug in the same starting values. Stochastic models allow for some unpredictability in the output when you have a given input.

Whether chaos exists depends a lot on what you mean by "chaos", I think. I'm not sure what the point of using the existence of math as an example of chaos is all about. Math is entirely a construct of the human mind. It was not created "out there somewhere". I suspect I'm on a different page somehow.