Thread: Fractals and other mathematical forms

First off if you havent seen it, you must see this video on fractals. It's a PBS NOVA. Shows how our body uses fratals to see and pump our blood and many other aspects of fractals in nature. If you have the time watch it, it's about 54min in length.

http://www.pbs.org/video/video/10509...gram/979359664

I want to ask if anyone knows if something like M-theory uses fractals? Or Chaos theory? Do all mathematics interlink eventually. I think they must. I'm of the opinion that if it exists in mathematics then it exists in the world. If there is going to be a TOE then all mathematics must be addressed.

Originally Posted by Wildstar

First off if you havent seen it, you must see this video on fractals. It's a PBS NOVA. Shows how our body uses fratals to see and pump our blood and many other aspects of fractals in nature. If you have the time watch it, it's about 54min in length.

http://www.pbs.org/video/video/10509...gram/979359664

I want to ask if anyone knows if something like M-theory uses fractals? Or Chaos theory? Do all mathematics interlink eventually. I think they must. I'm of the opinion that if it exists in mathematics then it exists in the world. If there is going to be a TOE then all mathematics must be addressed.

http://74.125.155.132/search?q=cache...&ct=clnk&gl=us

At first it seemed like a flop transition. But the mathematics there have a paradox. what about mathematics with no paradoxes. I assume you gave me that link to illustrate that not all mathematics are physically possible.

Originally Posted by Wildstar

At first it seemed like a flop transition. But the mathematics there have a paradox. what about mathematics with no paradoxes. I assume you gave me that link to illustrate that not all mathematics are physically possible.

There is NO paradox there.

What there is is a theorem that is a) true and is proved using perfectly rigorous mathematics and that is b) wildly counter-intuitive.

I offer it as an example of something that is a valid mathematicala theorem that is physically impossible, basically inapplicable to physics.

BTW the reason that this theorem works is that the sets into which the spheres are decomposed that are subsequently re-assembles are not measurable in the sense of the theory of measure and integration.

I see. My question really was if fractals and chaos and any of these mathematics all weave into one another?

Originally Posted by Wildstar

I see. My question really was if fractals and chaos and any of these mathematics all weave into one another?

Yes and no.

Chaos is a an often abused notion. I have seen it used properly by people who study topological dynamics -- Bob Devaney for instance. In other situations I have seen a lot of hand waving, complete lack of understanding of what is really going on, and zero mathematics. That is unfortunately the case with a lot of what goes out under the heading of "applied mathematics".

Fractals and the notion of fractional topological dimension have also been studied rigorously. There is also a lot of hand waving in some circles.

If you are looking for real understanding and real mathematics I suggest that you look at work being done by people with expertise in cellular automata and topological dynamics. But I suggest that you be very leery of many things written for the popular press on these topics.

As a general principle, virtually all of mathematics is linked in some manner. It is difficult to think of anything that is isolated, and the most beautiful mathematics comes from discovering and exploiting subtle links.

Mathematics is largely saved from much of the "hype" that shows up in popularizations of physics. Unfortunately "chaos" and " fractals' are exceptions to that immunity. There is some good work, but there is a lot of charlatanism.

The good work takes some background to understand. If you are really interested I suggest that you take a look at one of Bob Devaney's books on Topological Dynamics -- An Introduction to Chaotic Dynamical Systems for instance.

I have not looked into fractals in any great detail, but Heinz-Otto Peitgen has written a couple of books on them and the relationship to dynamical systems. When I met him years ago he was a pretty good differential equations guy, so I would expect his books to be pretty good. But they are also something of a popularization, so I would still be careful regarding interpretations and possible over-simplifications. I have not read the books so I have no first-hand knowledge.

As for the books written by popularizers, non-mathematicians, I personally would not bother with them. With regard to popularizations of science in general, I find them beneficial only when written by first-rank researchers

Okay I'll look for these. Are these text books full of equations or just a dialog?

Originally Posted by Wildstar

Okay I'll look for these. Are these text books full of equations or just a dialog?

As I said, I have not read Peitgen's book, so I don't know about that one.

Devaney's book is a real mathematics book. That means that it is full of mathematics. However, despite the impression that is given in elementary classes, equations are a very small part of mathematics. So, in Devaney's book you will find a few equations, but you will find a lot of theorems and rigorous proofs. It is NOT a popularization. It is the real McCoy. It does have a few pictures of fracatal patterns that are germane to topological dynamics.

Okay Ill look for it. I have zero math training, but the subject really intrigues me. I wish I had the mathematical skill to look at the direct law of the maths involved, but I can't so I rely on the English language rather the the mathematical language.

Originally Posted by Wildstar

Okay Ill look for it. I have zero math training, but the subject really intrigues me. I wish I had the mathematical skill to look at the direct law of the maths involved, but I can't so I rely on the English language rather the the mathematical language.

If that is the case you may find Devaney's book tough sledding. It is written at a level that is accessible to someone with a pretty firm understanding of calculus and a general level of "mathematical maturity" at the advanced undergraduate or beginning graduate level.

No thats not me. I'm just a layperson. But I like to know whats going on as much as I can.