Some Images from the Mandelbrot Set Fractal

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The above are some images from the  "Mandelbrot Set" fractal. It may be hard to believe, but each one of these pictures is just a colored graph of results from one equation, an equation as simple as 2*2+1=5.

Curious about the math? Click here.

Curious about the beauty? Click here.

Images by JWMealy, 1996/1997. Generated with thanks using FRACTINT.EXE, which is an easy-to-use public domain software (freeware) lovingly contributed by a large team of recreational programmers and available in mathematics software libraries everywhere for the PC-compatible. Get out there on the web, download it, and enjoy! Comments, questions, responses? Use this link.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

The images are more or less computerized graphs that have been colored in by me. Each image displays the results of a very simple feedback loop process applied to a series of pairs of numbers. For example, take this simple equation:

2*2+1=5

The 2 is one number of a pair and the 1 is the other. To get the Mandelbrot Set, you multiply the first number by itself, then you add the second number. The result, 5, rather than being the end of the story, replaces the first number,
and you try the process over again:

2*2+1=5  --  5*5+1=26   --   26*26+1=677   -- etc.

This is called a feedback loop process because
the results feed back into the equation, and the process repeats over and over. The second number, 1 in this particular example,  is called
a "constant", because it stays constant--that is, the same--rather than changing with each new  repetition of the process. So you'd get:

1st repetition   2nd repetition    3rd repetition
2*2+1=5  --    5*5+1=26   --   26*26+1=677   -- etc.

So, if you haven't gotten totally bored by now, all the wonderful images you have just seen simply come from testing a whole series of pairs of numbers by plugging them into this process. If you can deal with algebra, it kinda looks like this:

newfirstnumber = oldfirstnumber*oldfirstnumber+constant.

The question is, when you plug a specific pair of numbers into the feedback "machine" we have just looked at, will the process create results that

1.   keep getting bigger until they go out to
      infinity (like 2 and 1), or will they
2.    keep getting smaller and smaller, or
3.    stay about the same?

Lots of tiny numbers just get smaller, and numbers bigger than 1 or 2 always get bigger and bigger. But there are an infinite number of combinations in between that just can't be predicted at all until you try it. That's what computers are good at. You just do the feedback process  for a whole array of numbers and plot the results on a graph, showing black for combinations that go down to zero, and various other colors for numbers that get bigger, depending on how fast they get bigger.

That's how every one of the beautiful,  richly complex and unique pictures you just saw were created.

But here's one last odd question:

Why should the results be so wild and wonderful, when the equation is as simple as 2*2+1=5?