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?