h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

i was just looking at this:

http://video.google.com/videoplay?docid=2013533182496749803&q=fractals&total=6832&start=0&num=10&so=0&type=search&plindex=2

because i had just drawn an analogy referring to the inner parts of a fractal being building blocks for the outer edges... but watching this... i have become confused.. it looks like it is being built from within?

so just what is a fractal or mandlebrot and how does it work and what are we looking at when we look at a fractal zoom? and how can we 'create' them ourselves?

thanks

A fractal is a mathematical object with infinite complexity. Its structure is such that it looks more or less the same no matter what scale you look at it with.

The classic example is a coastline. If you look at the west coastline of Great Britain you'll see a wiggly line that has big bumps and hollows (bays and peninsulas), smaller ones and even smaller ones. If you zoom in on any point on the coast, you'll see a similar pattern at just about any scale. Even at the scale of one to one, wehre the picture on your screen represents the same size of rocky coastline, you'll still see bits that stick out and hollows (although the word "bay" isn't appropriate at this scale.

The Mandelbrot set is a particular mathematical object discovered by a guy called Mandelbrot. Each point on the screen is put through a repetitive sequence of steps based on its coordinates. Some points will be transformed to the value (0,0) by this process; these points are coloured black. Other points won't ever reach zero. These are coloured white. The Mandelbrot set is the "most fractal" object ever discovered because you can zoom in on any part of the edge of the black bit and find an infinitely complex border between white and black. The colours are added to make it look more interesting - points which tend towards black quickly are coloured dark colours, ones that take a long long time are coloured light colours.

very well explained!

got a little lost here: "Each point on the screen is put through a repetitive sequence of steps based on its coordinates. Some points will be transformed to the value (0,0) by this process; these points are coloured black. Other points won't ever reach zero. These are coloured white." ....probly cuz i dont know programming code?

is there a mathematical formula for these mandlebrot sets?
i mean do you have an example?
are they constructed within a program? like fractal generating program?

is the math huge... like programming code for a graphics file?

This reminds me of favourite song that can be downloaded free by anyone who cares for it: http://www.jonathancoulton.com/songdetails/Mandelbrot%20Set

To add to what Gnomon said: Think of a number - call it x if you like . Multiply it by itself. Now multiply this number by itself. Do it again. Keep doing it. What happens to your number? If it was between -1 and 1 (not including -1 or 1), then each time you multiply it by itself, it gets smaller - closer to 0. If the number was bigger than 1, or smaller than -1, each time you multiply it by itself it will get bigger and bigger, that is it isn't getting closer to 0. This is like a simple version of the Mandelbrot set - you'd "colour" the points between -1 and 1 black, and the points outside of that set white. The rule for the Mandelbrot set is not actually that much more complicated than this rule, but it's 2 dimensional instead of 1 dimensional because you start with a coordinate (x,y) instead of just a number x. The code is usually pretty short, that's what makes it amazing that there's this infinite amount of detail in there...

ohhhhh the Video By PIsut Wisessing on that site is amaaaaaaaaaaazing!! what a great way to do animation too! i am going to experiment! thanks for the turn-on, Br Azathoth Kai!

ohhhh thank you too, Dogster, that makes alot of sense... you and Gnomon have really made this clear!

sooo.. any examples of code? and where do you run it once you have it?

and, ohhhh wow!! i love the song!!

my favorite has always been 'She Blinded Me With Science' by Dolby...
http://www.youtube.com/watch?v=9yWahrEKL9o
...but this one comes in a strong second for sure!!

anymore science based songs out there?

Well, the... erm... (whispers) wiki page has some pseudocode (not actual computer code, but tells you how the algorithm works):

http://en.wikipedia.org/wiki/Mandelbrot_set#For_Programmers

You can see a fairly good example in action, here (using Java):

http://spaz.ca/aaron/SCS/fractals/

You have to increase the number of iterations as you zoom in a lot or you lose the detail.

I can post the code I wrote for it a while ago if you want but it was done in PowerBASIC so you might struggle to find a complier.

I wrote a Mandelbrot set generator in BASIC for my ZX Spectrum in 1985. It took about 15 hours to do its first run...

Ten years later on a computer at work (a Sparcstation, I think) I saw a program that did the same thing, in colour, in less than a hundredth of a second.

SoRB

If you want to tinker with fractals, you could do worse than get Fractint...

http://spanky.triumf.ca/www/fractint/fractint.html