Re: adjusting an image

David Epstein <dbae@xxxxxxxxxxxxxxxxxxx> wrote:

Let A be a matrix of zeros and ones. We imagine A to be extended in all
directions by zeros, so we don't need to worry about edge effects. For
each entry x of A, we look at a 3x3 window centred at x, see whether
zeros or ones predominate, and replace x by whichever entry
predominates.

I'm assuming that the replacement is done simultaneously for all points;
otherwise the result depends on the order in which the changes are made.

Since the situation is finite, the procedure must terminate in a finite
cycle. My question is: does it always terminate in a cycle of length
one (that is, no change)?

No. It can result in a cycle of length 2:

.oo...... .oo......
oooo..... oooo.....
oooo..... oooo.....
.ooo..... .oooo....
...ooo... <--> ....o....
.....ooo. ....oooo.
.....oooo .....oooo
.....oooo .....oooo
......oo. ......oo.

(Here 'o'=1 and '.'=0.)

I don't know if other periods are possible.

Meta questions: is this sort of question more appropriate to some
specialized mathematical newsgroup?

Try comp.theory.cell-automata. You're asking about periodic patterns in
the cellular automaton with the Conway neighborhood and the totalistic
rule B5678/S45678.

Dean Hickerson
dean@xxxxxxxxxxxxxxxx

.

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