Article 36601 of rec.puzzles:
From: hoey@pooh.tec.army.mil (Dan Hoey)
Newsgroups: rec.puzzles
Subject: Re: Dense plane-coverings in Conway's "life".
Date: 7 Dec 1995 23:40:40 GMT
Organization: Naval Research Laboratory, Washington, DC

Bill Taylor (wft@math.canterbury.ac.nz) wrote:
: I'm interested in finding regular patterns in Conway's "life",
: that cover the plain as densely as possible.

: What are the greatest densities achievable -

: (i) with a time-periodic pattern, preferably stationary;

The cute name for a period-1 pattern is a "still life".  It seems 1/2
is the greatest possible density.  Noam Elkies outlined a proof
approach for the stronger result that no pattern of maximum adjacency
degree 3 can have density over 1/2.  I'm not sure he's polished it
into a complete proof yet.  Here are some density-1/2 still-lifes,
mostly from Noam and earlier sources.

                                 Chicken wire
 . x x x x . . . . x x x x    . . x x . . x x . .
 . x . . x . x x . x . . x    x x . . x x . . x x
 . x . . x . x x . x . . x    . . x x . . x x . .
 . x x x x . . . . x x x x    x x . . x x . . x x
 x . . . . x x x x . . . .    . . x x . . x x . .
 x . x x . x . . x . x x .    x x . . x x . . x x
 x . x x . x . . x . x x .    . . x x . . x x . .
 x . . . . x x x x . . . .    x x . . x x . . x x

 x x . . x x x x . . x x          x x x x . x x x x . x x x x
 . x . x . . . x . x . .          x . . x . x . . x . x . . x
 x . . x x x x . . x x x          x . . x . x . . x . x . . x
 . x x . . . . x x . . .          x x x x . x x x x . x x x x
 x . . x x x x . . x x x          . . . . x . . . . x . . . .
 . x . x . . . x . x . .          . x x . x . x x . x . x x .
 x x . . x x x x . . x x          . x x . x . x x . x . x x .
 . . x x . . . . x x . .          . . . . x . . . . x . . . .
                                  x x x x . x x x x . x x x x
 . . . . x . . . . x . . . . x    x . . x . x . . x . x . . x
 x x x x . x x x x . x x x x .    x . . x . x . . x . x . . x
 x . . x . x . . x . x . . x .    x x x x . x x x x . x x x x
 x . x . . x . x . . x . x . .    . . . . x . . . . x . . . .
 x . x x x x . x x x x . x x x    . x x . x . x x . x . x x .
 . x . . . . x . . . . x . . .    . x x . x . x x . x . x x .
 . x . x x . x . x x . x . x x    . . . . x . . . . x . . . .
 . x . x x . x . x x . x . x x
 . x . . . . x . . . . x . . .
 x . x x x x . x x x x . x x x        x x x x . x x x x .
 x . x . . x . x . . x . x . .        . . . . x . . . . x
 . . x . x . . x . x . . x . x        . x x . x . x x . x
 x x x . x x x x . x x x x . x        . x . x x . x . x x
                                      . x . . . . x . . .
                                      x . x x x x . x x x
 x x x x x x x x . . . . . . . .      x . x . . x . x . .
 x . . . . . . x . x x x x x x .      x . x . . x . x . .
 x . x x x x . x . x . . . . x .      x . x x x x . x x x
 x . x . . x . x . x . x x . x .      . x . . . . x . . .
 x . x . . x . x . x . x x . x .      . x . x x . x . x x
 x . x x x x . x . x . . . . x .      x x . x . x x . x .
 x . . . . . . x . x x x x x x .      . . . x . . . . x .
 x x x x x x x x . . . . . . . .      x x x . x x x x . x
 . . . . . . . . x x x x x x x x
 . x x x x x x . x . . . . . . x
 . x . . . . x . x . x x x x . x      x x x x . . x x x x . .
 . x . x x . x . x . x . . x . x      x . . . x . x . . . x .
 . x . x x . x . x . x . . x . x      . x x x x . . x x x x .
 . x . . . . x . x . x x x x . x      . x . . . x . x . . . x
 . x x x x x x . x . . . . . . x      . . x x x x . . x x x x
 . . . . . . . . x x x x x x x x      x . x . . . x . x . . .

: (ii) that preferably contains no groups of live cells infinite in extent.

Well, the above qualify, but my favorite still life (because it's my
own invention) is the following:

 . . . x x x x x x x x x x
       . . . . . . . . . x
       x x x x x x x x . x
       x . . . . . . x . x
       x . x x x x . x . x
       x . x . . x . x . x
       x . x . . x . x . x
       x . x . x x . x . x
       x . x . x . . x . x
       x . x . x . . x . x
       x . x . x x x x . x
       x . x . . . . . . x
       x . x x x x x x x x
       x . . . . . . . . .
       x x x x x x x x x x . . .

Not only does it have an infinite connected group, it _is_ an infinite
connected group.  Rich Schroeppel has shown that there is no finite
connected still-life other than the 2x2 block, but this can be
modified by adding spirals to make uncountably many infinite connected
still-lifes.  I don't know if there's any i.c.s-l. that is not made up
of such spirals, though.  Could there be one that has lacunae?

Dan                        Posted and e-mailed
Hoey@AIC.NRL.Navy.Mil