Could you just explain a bit “will only be likely to contain arbitrary patterns of sizes up to log(10^120)” please ? Or give some pointers with other usage of such calculation ?
This is very much a heuristic, but good enough in this case.
Suppose we want to know how many times we expect to see a pattern with n cells in a random field of area A. Ignoring edge effects, there are A different offsets at which the pattern could appear. Each of these has a 1/2^n chance of being the pattern. So we expect at least one copy of the pattern if n < log_2(A).
In this case the area is (10^60)^2, so we expect patterns of size up to 398.631. In other words, we expect the ash to contain any pattern you can fit in a 20 by 20 box.
So just to connect this back to your original point: if we knew that it were possible to construct some kind of intelligent entity in a region with area of, say, 1,000,000,000 cells, then if our overall grid had 21,000,000,000 total cells and we initialized it at random, then we would expect an intelligent entity to pop up by chance at least once in the whole grid.
Yeah, although probably you’d want to include a ‘buffer’ at the edge of the region to protect the entity from gliders thrown out from the surroundings. A 1,000,000 cell thick border filled randomly with blocks at 0.1% density would do the job.
Could you just explain a bit “will only be likely to contain arbitrary patterns of sizes up to log(10^120)” please ? Or give some pointers with other usage of such calculation ?
This is very much a heuristic, but good enough in this case.
Suppose we want to know how many times we expect to see a pattern with n cells in a random field of area A. Ignoring edge effects, there are A different offsets at which the pattern could appear. Each of these has a 1/2^n chance of being the pattern. So we expect at least one copy of the pattern if n < log_2(A).
In this case the area is (10^60)^2, so we expect patterns of size up to 398.631. In other words, we expect the ash to contain any pattern you can fit in a 20 by 20 box.
So just to connect this back to your original point: if we knew that it were possible to construct some kind of intelligent entity in a region with area of, say, 1,000,000,000 cells, then if our overall grid had 21,000,000,000 total cells and we initialized it at random, then we would expect an intelligent entity to pop up by chance at least once in the whole grid.
Yeah, although probably you’d want to include a ‘buffer’ at the edge of the region to protect the entity from gliders thrown out from the surroundings. A 1,000,000 cell thick border filled randomly with blocks at 0.1% density would do the job.