From: torbenm@diku.dk (Torben AEgidius Mogensen)
Newsgroups: comp.theory.cell-automata
Subject: Symmetric Life rules
Date: 17 Nov 92 12:24:45 GMT
Organization: Department of Computer Science, U of Copenhagen

I have been doing some investigations of what I call "symmetric" Life
rules. The idea is to have a set of rule for a variant of Conways Game
of Life, which are symmetric in the sense that a pattern of empty
cells in a field of set cells has exactly the same behaviour as the
corresponding pattern of set cells on an empty field. As an example,
the pattern

OOOOOOOOOOOO
OOOOOOOOOOOO
OOOOOOOOOOOO
OOOOOO.OOOOO
OOOOOO..OOOO
OOOOOO..OOOO
OOOOOOOOOOOO
OOOOOOOOOOOO
OOOOOOOOOOOO

will have the same behaviour as the pattern

............
............
............
......O.....
......OO....
......OO....
............
............
............

(which with the rules given below is a period 4 repetition).

But the rules should also be nontrivial in the sense that small
patterns will not normally grow arbitrarily big. Also, it should be
possible to find patterns with large periods and even spaceships.

I have found such a set of rules and carried out some experiments with
these. The rules are also interesting in that the central cell in the
3x3 neighbourhood counts the same as the other cells. In short, given
that N is the sum sum of all the cells in the 3x3 neighbourhood, a
cell is generated for the next generation based on the following table

	N Cell
	0  .
	1  .
	2  .
	3  O
	4  .
	5  O
	6  .
	7  O
	8  O
	9  O

It is easy to see that this does indeed yield a symmetric behaviour.

I have found some interesting patterns using these rules. The are many
period 2 and 4 patterns (one is shown above). Below is shown a small
period 30 pattern

............
............
.....OO.....
....OOOO....
.....OOO....
......O.....
............
............

I have made some programs to search for spaceships, and I have so far
found two. They are both period two, speed c/2 and have similar edges.
I have found no small spaceships with short periods, even though I
have made a thorough search. The two spaceships (both 38x5, moving
left) are shown below.

..............
..............
.....O........
....OOO.......
....OOOOO.....
.....O.OO.....
..............
......O.O.....
.....OOO......
.....OO.......
......O.O.....
.......OO.....
.......O......
.....OO.O.....
....OOO.......
......O.......
.....O........
......OO......
..............
.......O......
.......OOO....
.......OOO....
.......O......
..............
......OO......
.....O........
......O.......
....OOO.......
.....OO.O.....
.......O......
.......OO.....
......O.O.....
.....OO.......
.....OOO......
......O.O.....
..............
.....O.OO.....
....OOOOO.....
....OOO.......
.....O........
..............
..............


..............
..............
.....O........
....OOO.......
....OOOOO.....
.....O.OO.....
..............
......O.O.....
.....OOO......
.....OO.......
....O.O.......
.....OOO......
.....O..O.....
......O.......
....OO........
....OOO.......
....O.........
..............
.....OO.......
.....OO.......
....O.........
....O.........
.....OO.......
.....OO.......
..............
....O.........
....OOO.......
....OO........
......O.......
.....O..O.....
.....OOO......
....O.O.......
.....OO.......
.....OOO......
......O.O.....
..............
.....O.OO.....
....OOOOO.....
....OOO.......
.....O........
..............
..............

I have some negative results for spaceships.

For period 2 spaceships:

  No spaceships less than 5 deep.
  No spaceships of depth 5 less than 30 wide.
  No spaceships less than 7 wide.

For period 3 spaceships:

  No speed (c/3,0) spaceships less than 4 deep.
  No speed (c/3,0) spaceships less than 6 wide.
  No speed (c/3,c/3) spaceships less than 6 wide/deep.

For period 4 spaceships:

  No speed (c/4,0) spaceships less than 4 deep.
  No speed (c/2,0) spaceships less than 5 deep.
  No speed (c/4,c/4) spaceships less than 5 wide/deep.

nonexhaustive searches of higher periods and sizes have yielded no
further conclusions.

If you start with a random distribution of set and clear cells, the
space will after a while be divided into areas with empty and full
interiors with activity mainly along the borders. The shape of these
areas change relatively slowly.

There are no small patterns I know of that grow very large (unlike in
Conways life, where e.g. the r-pentanomino grows fairly big). I have
not tried to add tagalongs to the spaceships above, so it is possible
that this might yield puffer trains that provide unbounded growth.

Most Game of Life programs are fairly easy to modify to use the
symmetric rules, so if anyone are interested in making experiments of
their own, they should have little problems doing so. Note that the
optimizations that look for large empty areas will have less effect
than usual, unless they are modified to handle large filled areas too.

	Torben Mogensen (torbenm@diku.dk)
Newsgroups: comp.theory.cell-automata