I believe that such a 3x3 square does not necessarily exist.

A counterexample would take the form of an infinite [still life][1] pattern in the [life-like cellular automaton][2] rule B123678/S34 (these rules are chosen so that the only patterns that remain stable are the ones in which the number of live cells in each 3x3 box is 4 or 5). Additionally, both the live and dead cells of the pattern should be connected.

But as the following partial double spiral shows (copy and paste it into [Golly][3] to view and test) it's possible to form partial double-spiral patterns that, at least in the center of the pattern, have the desired properties. I don't see any good reason why it shouldn't be possible to continue the spiral infinitely.

<pre>x = 31, y = 31, rule = B123678/S34
14b4o$12b3o2b3o$10b3o6b3o$8b3o3b4o3b3o$6b3o3b3o2b3o3b3o$5b2o3b3o6b3o3b
2o$5bo2b3o3b4o3b3o2bo$4b2ob2o3b3o2b3o3b2ob2o$4bo2bo2b3o6b3o2bo2bo$3b2o
b2ob2o3b4o3b2ob2ob2o$3bo2bo2bo2b3o2b3o2bo2bo2bo$2b2ob2ob2ob2o6b2ob2ob
2ob2o$2bo2bo2b2obo2b4o2bo2bo2bo2bo$2bo2bo2bo2bob2o2b2ob2ob2ob2ob2o$b2o
b2ob2ob2o2bo2bo2bo2bo2bo2bo$b2ob2ob2ob2ob2ob2ob2ob2ob2ob2o$bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo$2ob2ob2ob2ob2ob2o2bo2bo2bo2bo$o2bo2bo2bo2b2o2bob2ob
2ob2ob2o$2ob2ob2ob2o4b2obo2bo2bo2bo$bo2bo2bo2b6o2bo2bo2bo2bo$b2ob2ob2o
3b2o3b2ob2ob2ob2o$2bo2bo2b3o4b3o2bo2bo2bo$2b2ob2o3b6o3b2ob2ob2o$3bo2b
3o3b2o3b3o2bo2bo$3b2o3b3o4b3o3b2ob2o$4b3o3b6o3b3o2bo$6b3o3b2o3b3o3b2o$
8b3o4b3o3b3o$10b6o3b3o$12b2o3b3o!</pre>

Here's a screenshot:

![alt text][4]


  [1]: http://en.wikipedia.org/wiki/Still_life
  [2]: http://en.wikipedia.org/wiki/Life-like_cellular_automaton
  [3]: http://golly.sourceforge.net/
  [4]: http://www.ics.uci.edu/~eppstein/0xDE/b123678s34-spiral.png