Some of this will repeat earlier comments, but I'm including them here for completeness. I'm not an expert in this, so some of it might be wrong in the detail.
Firstly, there is a comparatively simple pattern that grows endlessly - the Gosper Glider Gun.
you'll see that he's a hacker's hacker. If you analyse the GGG you'll see that it can be broken down into smaller pieces, each of which is a small modification of something small and easy to understand. In essence, you can build a GGG from small components, and then optimise it, the entire process being like writing a computer program, perhaps in something like BrainFuck: http://en.wikipedia.org/wiki/Brainfuck
I might meet Gosper in March next year. I've made a note to ask him if that's actually what he did. From what I've read, it is.
The question about "growth in which it was not an oscillating reaction" is hard to pin down, and it's not clear what you mean. I would guess you mean something like:
A pseudo-periodic pattern is one which can be
divided into two areas, P and P'. Inside P the
pattern is periodic, and in P' the pattern is
comprised of units, each of which is periodic
under some isometry.
Well, using GGGs you can beat that definition too, because you can construct patterns that build GGGs at a distance, and then you can make a Turing Machine that computes the digits of pi, then you can build GGGs at distances that get further away by the amount that is a digit of pi, and then you have something that's still sort of predictable, and still isn't what you want, but still satisfies the above definition.
Part of the problem is making precise make precise what you mean. I suspect you're trying the equivalent of proving the existence of non-computable numbers, and then asking to show one. Counting arguments suggest that amorphous patterns that grow indefinitely exist, but they're the equivalent of uncomputable numbers. I can't show you one.
Firstly, there is a comparatively simple pattern that grows endlessly - the Gosper Glider Gun.
http://en.wikipedia.org/wiki/Gun_%28cellular_automaton%29
Someone asked about the processes used to create these sorts of things. If you read about Bill Gosper:
http://en.wikipedia.org/wiki/Bill_Gosper
you'll see that he's a hacker's hacker. If you analyse the GGG you'll see that it can be broken down into smaller pieces, each of which is a small modification of something small and easy to understand. In essence, you can build a GGG from small components, and then optimise it, the entire process being like writing a computer program, perhaps in something like BrainFuck: http://en.wikipedia.org/wiki/Brainfuck
I might meet Gosper in March next year. I've made a note to ask him if that's actually what he did. From what I've read, it is.
The question about "growth in which it was not an oscillating reaction" is hard to pin down, and it's not clear what you mean. I would guess you mean something like:
Well, using GGGs you can beat that definition too, because you can construct patterns that build GGGs at a distance, and then you can make a Turing Machine that computes the digits of pi, then you can build GGGs at distances that get further away by the amount that is a digit of pi, and then you have something that's still sort of predictable, and still isn't what you want, but still satisfies the above definition.Part of the problem is making precise make precise what you mean. I suspect you're trying the equivalent of proving the existence of non-computable numbers, and then asking to show one. Counting arguments suggest that amorphous patterns that grow indefinitely exist, but they're the equivalent of uncomputable numbers. I can't show you one.