Mandelbrot's Fractal: Why the Simple Set Creates Mesmerizing Shapes

[Sorento7]

Do you know why a simple set used for construction of the Mandelbrot's fractal results in a jaw dropping shape?

 

[Leacy]

Well a Mandelbrot's set (Fractal) is created by graphing the basic equation
z_n+1 = z_n² + c
This is graphed on a complex plane also known as an Argen Diagram. The Reason why the 2D graphic is so good is because the depth of the equation. The Equation is bounded so to give an infinite depth. Much like looking into space and seeing the stars, if you increase your magnification you would see more stars. The same can be said about the mandelbrot's set. No matter how deep you get the patterns say the same.

 

[DaveC426913]

It is the nature of chaotic systems that they are sensitive to initial conditions.

In the case of the Mandelbrot Set and Julia sets, the initial conditions are x and y coordinates. Very, very tiny changes in input (x,y) result in large changes in output (the result of the calculation). So, when you plot the result at the coord 0.00000001 x 0.00000001 it can be quite different than the result at 0.00000002 x 0.00000002. This is what forms the infinitely regressive patterns. You can zoom into 0.00000000000001 x 0.0000000000001 and its still a different result from 0.00000000000002 x 0.0000000000002.

 

[laurent711]

http://www.infoocean.info/avatar1.jpg It is the nature of chaotic systems that they are sensitive to initial conditions.

 

[SteveL27]

Do you know why a simple set used for construction of the Mandelbrot's fractal results in a jaw dropping shape?

It's pretty wild. Also if you've never seen Ulam's prime spiral, it's also very strange. There are definite patterns there but nobody knows exactly how to explain them.

http://en.wikipedia.org/wiki/Ulam_spiral

 

[Sorento7]

These explanations are quite correct, but I don't think that chaos can explain the repetetive patterns in Mandelbrot which are very "similar" and very "different" at the same time. Transitions of these patterns everywhere in plane happens smoothly, and the whole shape is replicated in bizarre places when it's zoomed. (Actually that is the key criterion of a fractal.) But surprisingly, for this wild behavior, I havn't seen any mathematical analysis yet. I think Steve's quote "nobody knows exactly how to explain them." feels good. Any better idea?

 

[Bill Simpson]

Contrary position.

In the places where there are modest numbers of floating point calculations, and thus modest numbers of tiny floating point roundoff errors introduced before stopping, the picture is very simple.

In the places where there are vast vast numbers of floating point calculations, and thus vast vast numbers of tiny floating point roundoff errors introduced before stopping, the picture is very complicated.

Decades and decades ago an author in Byte magazine was desperately trying to get his Mandelbrot images done in time for the publishing deadline. He was using the official blessed Intel software floating point library. Because of the processor speed it was taking him hours and days to produce each image. Then someone loaned him a wildly expensive 80287 floating point chip so that he could get done in time. This was supposed to produce EXACTLY the same results. He accidentally happened to compare the results from the software and the hardware and found that in the simple places the pictures were the same, but when he looked at the more complicated parts of the pictures he saw small differences and the more complicated the more different. At that point he hit the deadline, ran out of time, mentioned all this in a few sentences in his article and that was the end of it.

Since reading that I've wondered what it would look like if someone would "subtract" the Mandelbrot done with floating point approximations from the same done with arbitrary precision exact calculations. I suspect the difference would look almost exactly like the Mandelbrot pictures that everyone displays. If so that would go a long way towards saying that the pictures we all recognize are to a substantial extent just floating point errors in color.

Years ago someone sent me a link to a journal article supposedly on exactly this subject. Unfortunately I lost the link before I was able to follow up on that.

 

[Sorento7]

Contrary position.

In the places where there are modest numbers of floating point calculations, and thus modest numbers of tiny floating point roundoff errors introduced before stopping, the picture is very simple.

In the places where there are vast vast numbers of floating point calculations, and thus vast vast numbers of tiny floating point roundoff errors introduced before stopping, the picture is very complicated.

Decades and decades ago an author in Byte magazine was desperately trying to get his Mandelbrot images done in time for the publishing deadline. He was using the official blessed Intel software floating point library. Because of the processor speed it was taking him hours and days to produce each image. Then someone loaned him a wildly expensive 80287 floating point chip so that he could get done in time. This was supposed to produce EXACTLY the same results. He accidentally happened to compare the results from the software and the hardware and found that in the simple places the pictures were the same, but when he looked at the more complicated parts of the pictures he saw small differences and the more complicated the more different. At that point he hit the deadline, ran out of time, mentioned all this in a few sentences in his article and that was the end of it.

Since reading that I've wondered what it would look like if someone would "subtract" the Mandelbrot done with floating point approximations from the same done with arbitrary precision exact calculations. I suspect the difference would look almost exactly like the Mandelbrot pictures that everyone displays. If so that would go a long way towards saying that the pictures we all recognize are to a substantial extent just floating point errors in color.

Years ago someone sent me a link to a journal article supposedly on exactly this subject. Unfortunately I lost the link before I was able to follow up on that.

Good point on the impact of rounding errors on final shape. But this only changes the question without answering it, why should this simple set introduce such weird rounding errors producing a fractal?

 

[Bill Simpson]

Good point on the impact of rounding errors on final shape. But this only changes the question without answering it, why should this simple set introduce such weird rounding errors producing a fractal?

Complicated things can often produce very complicated results and floating point errors can be very complicated things.

But in centuries of thinking a handful of people have found what look like very simple things that produce very complicated results.

Look at Wolfram's "A New Kind of Science" where he produces wildly complicated things with what look like very simple rules for cellular automata.

Look at "the busy beaver problem" where vast long calculations are the result of what seems to be a very simple problem statement.

Look at some of the "chaotic systems" problems where seemingly simple starting conditions gradually evolve into what looks like wildly complicated behavior. But, to be contrary again, in the real world we never seem to see something as simple as the variation in the behavior of water drops become more and more and more complicated and finally rip the universe in half. Many or perhaps even all these complications seem to only exist inside our manufactured ability to do vast numbers of calculations, each with their own tiny contribution of error.

 

[Sorento7]

You can also think of creation of universe and evolution, simplest subatomic structures changing to an intelligent human being with key contribution of genetic mutations. I think that makes sense.
(However, philosophically speaking, I think a concept such as Mandelbrot's fractal did exist before we construct it with our rules.)