Mandelbrot 3D

Incredible site. Thanks for posting. Literally hundreds of totally unbelievable images.

yeah cliff, that's awesome. i forwarded it to friends.

b.w.

Fuck off Cliff....keep things on topic.

"Mandelbrot 3D"

Trimming the tree, thanks!

but, this is REALLY REALLY interesting, and you can quite a lot from fractals - much more than you can learn from politics. A few decades ago I wrote a mandelbrot set drawing program (see the "software I wrote" link on my web page,

if you want to DL it and play with it - it has several interesting options beyond the basic 2-D mandelbrot set.

Now, for you CNC guys - figure out how to machine something like this - wonderful stuff

>

How long is the boundary?

Oh! Now there's an idea for a fun project. Take a tiny center drill or ball endmill, and some soft aluminum or machinable wax, and drip feed points from a PC program, maybe VB, that does the same thing pixel generating programs do. Calculate points that are in the set, and put a spot in the work at every valid pair of XY coordinates. Maybe, instead of using colors to represent how quickly a calculated point resolves itself, you could use Z depth, and get a 2 1/2 D relief of the

2D graphic.

Anybody got some spare time on a small 3-axis machine?

I wonder what would happen on a 4-axis machine. Calculate XY pixel locations, as above, but machine them as XC or YC on a rotary, with some scale factor so degrees would make sense in place of linear coordinates. You'd get (I think) the normal shape wrapped around a cylindrical surface.

What about 5 axes? Plot AC instead of XY? I can't even guess what that would look like.

KG

if you want to try it, you are more than welcome to DL the program from my web site - the actual source code is just a few lines out of something over

10,000 lines of turbo pascal - all the rest was to make a GUI, read the mouse, and so on - if the source code isn't there, drop me a note and I'll see if I can find a disk with it on it still - otherwise, it's pretty simple code - just square a complex number for up to a fixed amount of times - if the absolute value ever exceeds 4, stop. select color based on number of iterations - at the time (mid 80s) I used 10,000 iterations - as you reduce iterations your figure gets smoother.

I doubt you used 10,000 iterations (though you may have considered than many starting points) and always wonder about truncation & rounding errors. With each iteration the number of significant digits needed for an accurate calculation process (for THIS problem) just about doubles. So you need extreme-precision math ... over about 4 iterations.

Why machine one, when you can grow one?

Cheers! Rich

in this case, I'm sorry to say, I wrote the code. It just squares (in double precision) a number up to the required number of times. If you want to play with it, download it from my site - but honestly, you can't tell me how the code that I wrote and you have never seen works.

But I can tell you about the computations related to the Mandelbrot Set. Unless you can compute with all the significant digits you don't know what is the "set" (to the level of your computation) & what an artifact of your truncation & rounding errors.

Arbitrary-precision arithmetic:

Applications: [ Arbitrary precision arithmetic is also used to compute fundamental mathematical constants such as p to millions or more digits and to analyze the properties of the digit strings (e.g. [1]), or more generally to investigate the precise behaviour of functions such as the Riemann Zeta function where certain questions are difficult to explore via analytical methods. Another example is in rendering Fractal images with an extremely high magnification, such as those found in the Mandelbrot set. ]

Naturally, for coordinates for which the value went out of bounds before that computation limit was reached you need not worry.

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