Mandelbrot.

[Duer 157099]

Can someone explain this to me in very simple, yet not too simple, language?

I've watched the videos, looked at the Wiki entry, even watched a whole documentary with Arthur C. Clarke and Mandelbrot talking about it.

Yet there's something that still eludes me, and I don't even know what it is.

Somebody (in the doc) was saying how if a point goes to zero, then it ends up in the black spot (the "bug" looking thing) but otherwise it goes off into infinity (all the other colors that you can zoom in on forever.

So is the "bug" thing sort of like a black hole? (and why does it look like a Buddha?)

I wish that my brain was better at understanding math!

And that's another thing, how exactly is this thing "graphed?" Simple coordinates, like x,y,z?

Is there a 3D rendering, or is it already?

[Flaxbee]

1. I'll ask my husband, who knew Arthur well;

also, a good friend of ours spent some time with Mandelbrot.

Will let you know if I get a reply.

[elleng]

2. Voila!

http://www.cookitsimply.com/recipe-0010-0137v38.html

:rofl:

[Liberal_in_LA]

3. lots of books on the subject at the library in the math section

[HopeHoops]

4. There are two ways to understand it - the algorithm and what it really is.

The algorithm is deceptively simple. First of all, you are working in the complex plane, not the real plane. Without going into the specific math, for each point you are plotting, you iterate the equation until it either escapes an outer boundary (distance from zero,zero) or you've reached a large set number of maximum iterations. The drawing is done by coloring that dot black (usually) if it reaches maximum iterations - those points are part of the set (the bug). If it escapes, the iteration at which it escaped is the color in the color table (often a gradient) corresponding to that iteration.

Now for what it REALLY is. The Mandelbrot set is the map of all points for which the corresponding Julia set is connected. The Julia sets are plotted using the same approach as Mandelbrot but with a slightly different equation. Connected means exactly what it sounds like - all of the points are connected by some thread. That isn't always obvious in a plot, but mathematically it is true. For example, if you start at 0,0 and plot Julia sets moving along the X axis (the vertical i axis staying at 0), the Julia set will be a big black blob becoming sort of a rounded upright rectangle. At the point right in the butt crack of the huge cardioid, the Julia set moves from connected to disconnected. At that point and outward, it looks like four quadrants of old-style telephone receivers.

If you trace around the outside of the Mandelbrot set plotting Julia sets, they sort of fold over to the left over the top and get wider. When you reach the far left tip, they are like a long stream of bubbles. Continuing around the bottom, it reverses (they're symmetrical with respect to the X axis).

Another interesting fact. If you map the perimeter of the Mandelbrot set onto a circle, the intersections of major features are all rational fractions (not what you would expect. The large circle attached to the cardioid connects at 1/3 and 2/3 (top and bottom). Just some weird trivia.

I started studying these about 1986 and did my mathematics seminar on the sets. I had my Apple II, my C=64, and three Apple II GI(?) systems going at the college almost around the clock for weeks. A buddy of mine had just gotten a Mac II and we wrote an animation program that plotted zooms and played them back (not in real time - the plots took hours each). It had like 36 frames for the Mandelbrot and 56 for the Julia sets. The "escape from the butt crack" I described above is the one we used for the Julia sets.

[Duer 157099]

5. Fantastic! Thanks

I've found the resident DU expert on Mandelbrot!

I guess my first problem is that I need to brush up on the difference between the complex and real planes.

Can you suggest any books/websites that are written to the lay audience and yet somehow manage to communicate the more complex subtleties (sort of like how you did, but with pictures).

I've only taken basic college math up through calculus, but even that was lost almost as soon as acquired. Geometry was the only math that I really really enjoyed learning (although trig was also fun). But it was all too many years ago.

Also, what program do people typically use to generate those massive zoom videos that are on YouTube?

Thanks again for your help, I really do want to get some sort of handle on this! I remember back in the 80s when this was being discovered, but I didn't pay enough attention.

:hi:

[HopeHoops]

6. As for the zoom videos, the best ones are from self-written programs.

There are countless apps out there and a lot of them are java applets on web sites.

Now, as for the complex plane,...

I trust you are familiar with i, the square root of -1. Just keep in mind that i^2 = -1.

With the complex plane, you have an X axis and an i axis in place of Y. A point is expressed as aX +bi.

The fun comes in when you do mathematical operations. From algebra, (a + b)^2 = a^2 + 2ab + b^2. Take the simple case of (X + i)^2. That's going to be X^2 + 2Xi + i^2 and since i^2 = -1, it is really X^2 + 2Xi -1. The more complicated case would be (aX + bi)^2 = (aX)^2 + 2(aXbi) - b^2 {substituting the -1}.

The Mandelbrot and Julia sets involve squaring the current value and adding a constant for each iteration. With the negatives being thrown in, that's where you get the fun and everything bounces around. I wrote a program many years ago that plotted the bouncing like a spyrograph. That was fun.

If you are really interested, I suggest you start with "The Beauty of Fractals". In a strict manner of speaking, Mandelbrot and Julia sets are NOT fractals, but common culture has labeled them as such.

Another interesting bit of history. Julia discovered the magic of the Julia sets around 1917 - obviously without a computer. He was working from the mathematical beauty of it and obviously never plotted an image. The first Mandelbrot image was produced on a dot matrix printer (by Benoit Mandelbrot) in 1980.