I strongly believe that - given the rules of Conway's [Game of Life](http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life) and an initial configuration - it is not decidable by a Turing Machine whether a given pattern will emerge, let alone as a stable pattern, be it static, moving, and/or rotating.



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> How can this be proven?
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I guess, this kind of uncomputability would go far beyond the "simple" unpredictability of non-linear systems.



Undecidability in Conway's Game of Life
Conway's game of Life [can simulate a universal Turing machine](http://rendell-attic.org/gol/tm.htm) which means that it is indeed undecidable by reduction from the halting problem. 


You can program this Turing machine in the game of Life so that it builds some pattern when it halts that doesn't occur while it's still running. Then the pattern will be built if and only if the Turing machine halts.


