From: dbell@pdact.pd.necisa.oz.au (David I. Bell)
Newsgroups: comp.theory.cell-automata
Subject: Spaceships in Conway's Life (Part 5a)
Keywords: life
Date: 9 Oct 92 03:16:47 GMT
Organization: NEC Information Systems Australia, Canberra


		Spaceships in Conway's Life (Part 5)
			by David I. Bell
			dbell@pdact.pd.necisa.oz.au
			9 Oct 1992


This is the fifth in a series of articles concerning Conway's Game of Life.
In this article, I will survey the results for the new orthogonal period 4
c/2 spaceships.

The well-known LWSS, MWSS, and HWSS belong to the class of period 4 c/2
spaceships.  Since they will appear often in this article, I will use the
term "standard spaceship" to refer to any one of the LWSS, MWSS, or HWSS.

As might be expected, many larger spaceships require one or more standard
spaceships as essential components.  For many of these spaceships, the
standard spaceships appear at the front, and the rest of the spaceship is
then a tagalong to them.  But for some of these spaceships, the front of the
spaceship is not a standard spaceship, and yet one or more standard spaceships
are still needed alongside or in back of the rest of the spaceship to
support it.

Perhaps more interestingly, there are also many spaceships which do not
require the use of any standard spaceships at all.  So these can be thought
of as truly different and new.

More obscurely, there are some spaceships which are not easy to categorize.
This is because they have components which are not exactly like a standard
spaceship, but which are obviously based on standard spaceships.  I will
treat these fuzzy cases along with the standard spaceships.

Period 4 c/2 spaceships can either be glide-reflective or not.  As mentioned
in my first article, a period 4 glide-reflective spaceship means that the
original spaceship reappears in generation 2, except that it is reflected
across a line parallel to its motion.  (All of the standard spaceships are
glide-reflective spaceships.)

Glide reflective spaceships are easy to search for using a search program
because the effective period is actually only 2, and a mapping which
performs the required mirroring is easy to implement.  Therefore almost all
of the new c/2 period 4 spaceships that have been found are glide-reflective.
The search results shown here were done by Hartmut Holzwart and myself,
with Hartmut providing the majority of the results.

Now I will begin the survey of period 4 c/2 spaceships by starting with the
ones which require the use of standard spaceships.

One way to construct large spaceships is by reacting the standard spaceships
together in various simple ways.  These constructions are rather obvious,
and mostly old, so I will only give two examples of them here.  According
to Bill Gosper, spaceships like these are the "world's most boring".

["Boring" period 4 spaceships (speed c/2)]
 ..O..O        .OO...
 .O....        OO.OO.
 .O...O        .OOOO.
 .OOOO.        ..OO..
 ......        ......
 ......        ..OOO.
 ..OOO.        .OOOOO
 .OOOOO        OO.OOO
 OO.OOO        .OO...
 .OO...


Another way to make larger spaceships from the standard spaceships is to find
tagalongs for them.  Tagalongs are expected to exist because of the convenient
sparks that the standard spaceships have.  But many tagalongs do not have a
period of 4.  (For example, the Schick engine has a period of 12.)  The first
tagalong that was found that actually had a period of 4 is quite large, and so
required a search program to find.  I found this in April, 1992.

[First period 4 tagalong for pair of LWSSs (speed c/2)]
 .....................O...
 ..................OOOO...
 .............O..O.OO.....
 .............O...........
 OOOO........O...O.OO.....
 O...O.....OO.OO.O.O.OOOOO
 O.........OO.O.O.O..OOOOO
 .O..O..OO..O...OOO..O.OO.
 ......O..O.OO............
 ......O....OO............
 ......O..O.OO............
 .O..O..OO..O...OOO..O.OO.
 O.........OO.O.O.O..OOOOO
 O...O.....OO.OO.O.O.OOOOO
 OOOO........O...O.OO.....
 .............O...........
 .............O..O.OO.....
 ..................OOOO...
 .....................O...


Unfortunately this tagalong does not have any new side sparks for use in
perturbing puffer output.  However, the back of this tagalong does have
two sparks similar to the LWSS, and spaced just like the leading pair of
LWSSs.  Therefore this tagalong can be repeatedly attached to itself (in
the next phase) to make an arbitrarily large spaceship.  The following
shows three copies of the tagalong.

[Two LWSSs pulling three copies of a period 4 tagalong (speed c/2)]
 .....................O..................O.O.................O.O...
 ..................OOOO.................O..O................O.OO...
 .............O..O.OO..............OO..OO...................OO.....
 .............O...................OOOO........O.......OOOO..O..OOO.
 OOOO........O...O.OO...............OO.OO.............O...O.OOOOOOO
 O...O.....OO.OO.O.O.OOOOO......O.OO...O......O......OO......O.OOOO
 O.........OO.O.O.O..OOOOO.....OO.O..O.O.OOOOO......OO..O..O.OO....
 .O..O..OO..O...OOO..O.OO....O..O...O....O.......OO.O....OOOO......
 ......O..O.OO..............OOO.O....O..O.......O...O....OOO.......
 ......O....OO.............OO..OO..............O...O...............
 ......O..O.OO..............OOO.O....O..O.......O...O....OOO.......
 .O..O..OO..O...OOO..O.OO....O..O...O....O.......OO.O....OOOO......
 O.........OO.O.O.O..OOOOO.....OO.O..O.O.OOOOO......OO..O..O.OO....
 O...O.....OO.OO.O.O.OOOOO......O.OO...O......O......OO......O.OOOO
 OOOO........O...O.OO...............OO.OO.............O...O.OOOOOOO
 .............O...................OOOO........O.......OOOO..O..OOO.
 .............O..O.OO..............OO..OO...................OO.....
 ..................OOOO.................O..O................O.OO...
 .....................O..................O.O.................O.O...


One problem with searching for new period 4 c/2 spaceships is that the
standard spaceships keep appearing everywhere there is room.  Special
techniques have to be used to skip over these common results, so that
new things can be found.  One way to do this is to manually backtrack the
search slightly past the point where a standard spaceship has been found,
and then let the search progress from there.  Doing this results in finding
reactions with almost-standard spaceships.  That is, the spaceships are
slightly perturbed at the back.  I named these ships "smashalongs", since
it appears that the trailing tagalongs had run into the base ships instead
of merely attaching to them.

Following are some of the "smashalongs" that I found.  All of these start
with the same convoy of a LWSS and an almost-LWSS.   The first one shown
has an interesting spark at its edge.

[Period 4 "smashalongs" based on modified LWSSs (speed c/2)]
 ....................O..............
 ..OO.............O.................
 .OOOO..........O.OO...O............
 OO.OO.OO.....OO........O...........
 .OO.......OOOO...OO.OO.OO.O.OO.OOOO
 ......O......OO.O..OO...O.....OO...
 .......O.O...OO.O...OO.OO.O.OOO.OOO
 ..OO.OO.OOO..O..O..........O.O..O..
 .OOOO.OO....O..OO..O.OO...O.....O.O
 OO.OO........OOO....O.O...O.....O.O
 .OO...O.......OO....O..............
 ................OO.O...............
 ..................O................


The following smashalong can be extended at the back to make it larger
by one or two columns, exactly like an LWSS can be extended to make a
MWSS or HWSS.

[Period 4 smashalong (speed c/2)]
 .........................O..O.....
 ....................O....O........
 ..OO.............O......O...O.....
 .OOOO..........O.OO...O.O.OO......
 OO.OO.OO.....OO...................
 .OO.......OOOO...OO.OO.O.O.O.O.OO.
 ......O......OO.O..OO......OOOOOOO
 .......O.O...OO.O...OO.OO.....OOOO
 ..OO.OO.OOO..O..O.......OOOOOO....
 .OOOO.OO....O..OO..O.OO...........
 OO.OO........OOO....O.O.O.O.......
 .OO...O.......OO....O.............
 ................OO.O....O...O.....
 ..................O.....OOOO......


The next two spaceships require "escorts" of standard spaceships in order to
survive.  In the first ship, a pair of LWSSs are necessary, whereas in the
second spaceship, a HWSS is inside of the the spaceship to stabilize it.

[Two period 4 smashalongs with necessary escorts (speed c/2)]
 ..............................OO........
 .............................OOOO.......
 ....................O.......OO.OO.......
 ..OO.............O...........OO......O.O
 .OOOO..........O.OO...O.............OO.O
 OO.OO.OO.....OO........O...........O.O..
 .OO.......OOOO...OO.OO.OO.O.OO.OOOOOOO..
 ......O......OO.O..OO...O.....OO....O...
 .......O.O...OO.O...OO.OO.O.OOO.OOOO.O..
 ..OO.OO.OOO..O..O..........O.O..O...OO..
 .OOOO.OO....O..OO..O.OO...O.....O..O....
 OO.OO........OOO....O.O...O.....O..O....
 .OO...O.......OO....O.........OOOO...OO.
 ................OO.O.........OOOO....OO.
 ..................O.........OO.OO.......
 .............................OO.........


 ................................O......
 ...............................OOOOO...
 ....................O........OO...OO...
 ..OO.............O...........O....O....
 .OOOO..........O.OO...O..OOOOO.........
 OO.OO.OO.....OO..........OO............
 .OO.......OOOO...OO.OO.O..O.......OOOO.
 ......O......OO.O..OO....OOOOO...OOOOOO
 .......O.O...OO.O...OO.OO.....O.OO.OOOO
 ..OO.OO.OOO..O..O.......O........OO....
 .OOOO.OO....O..OO..O.OO..O.O...........
 OO.OO........OOO....O.OOOOO............
 .OO...O.......OO....O.....O...O....O.O.
 ................OO.O.......OO...O......
 ..................O...........O..OOOO..
 ..............................O........
 ...............................O.OO....


A MWSS spaceship can replace the HWSS as the escort in the last spaceship.

Here are two more of these smashalongs with the same beginning.  After
finding these, I stopped searching for more, so there are probably more.

[Two more large period 4 smashalongs (speed c/2)]
 .........................................O........O.......
 .........................................O.....OOOO.O.....
 ....................O............O..O.OOO.OO.OO......O...O
 ..OO.............O..............O.OO.OO....OOO.OOO........
 .OOOO..........O.OO...O...OOO..OO..OO.O..........O.......O
 OO.OO.OO.....OO........O.O.OO.O.O.OO........O.....OOOOOOO.
 .OO.......OOOO...OO.OO.O.....O..O..OO..OOO.....OOO........
 ......O......OO.O..OO....OO......OO.OO........OOOOOO......
 .......O.O...OO.O...OOOOOO..O.O.O...OO...OOO.OO.OOO.O.....
 ..OO.OO.OOO..O..O.........O...O....O...O.O....OO.......O..
 .OOOO.OO....O..OO..O.OOOO.O.OOO....O..O..O.........O.....O
 OO.OO........OOO....O.O.OOOO...O.O....O.OO.........O......
 .OO...O.......OO....O.......OOOO.O.....OO..OO..O..O......O
 ................OO.O........OO..OO.O..OO.O....O..OO.OOOOO.
 ..................O..............OOOO..OO...O.O.....O.....
 ...................................OO..OO.O....OOO........
 .................................................O..O.....


 ....................................OO......
 ....................O..O.........OO.........
 ..OO.............O....OOOOOO..OO.OO...OO....
 .OOOO..........O.OO.......OOO.....O.O.O.....
 OO.OO.OO.....OO........O.O....O.O...........
 .OO.......OOOO...OO.OOOO.O.OO.OOOOOO.O.....O
 ......O......OO.O..OO.......................
 .......O.O...OO.O...OOOOO.OOOOOOOOOOO......O
 ..OO.OO.OOO..O..O.........O..........OOOOOO.
 .OOOO.OO....O..OO..O.OO.O.OO..OO.OO.........
 OO.OO........OOO....O...OO...O...O...OO.....
 .OO...O.......OO....O.O...O..O.O.O..O...O...
 ................OO.O..O...OOOO...O..........
 ..................O....O..OO.....OOO.OO.O...
 ..................................OO........


After finding the above series of spaceships, I went on to other things.
Then Hartmut Holzwart began looking for more c/2 period 4 spaceships.

Here are several small spaceships found by Hartmut.  The first one shown
is interesting because the top part is an almost-spaceship which requires
the HWSS for support.  The HWSS can support one of these on each side.  Two
copies of the almost-spaceship alongside each other can not support each
other because they require growth on the row between them.

[Period 4 small spaceships (speed c/2)]
 .O......
 O.....O.
 O.....O.
 OOOOO.O.
 ........
 ....OO..
 ..O....O
 .O......
 .O.....O
 .OOOOOO.


 ...OO...
 .O....O.
 O.......
 O.....O.
 OOOOOO..
 ........
 .....O.O
 ....O..O
 .....O..
 ...OO..O
 .O.....O
 O.......
 O.....O.
 OOOOOO..


 OOOO.........
 O...O.O......
 O.....OO.....
 .O....OO.....
 ......O......
 ...OOOOOOOOO.
 ...O........O
 ...O.........
 ....O...O...O
 .............
 OOOO.OO......
 O...OO.O.....
 O....O.O.....
 .O..O........


The last spaceship above is a good example of my comments about almost-
standard spaceships being used in a spaceship.  The back end can be extended
similarly to extending standard spaceships, using other standard spaceships
as escorts.  Following are two examples of this.

[Two related period 4 spaceships (speed c/2)]
                         ...........OO......
 OOOO........OO...       OOOO......OO.OO....
 O...O.O....OO.OOO       O...O.O....OOOO....
 O.....OO....OOOOO       O.....OO....OO.....
 .O....OO.....OOO.       .O....OO...........
 ......O..........       ......O............
 ...OOOOOOOOOO....       ...OOOOOOOOOOOOOOO.
 ...O.........O...       ...O..............O
 ...O.............       ...O...............
 ....O...O........       ....O...O.........O
 .................       ...................
 OOOO.OO.....OO...       OOOO.OO............
 O...OO.O...OO.OOO       O...OO.O...OO......
 O....O.O....OOOOO       O....O.O..OO.OO....
 .O..O........OOO.       .O..O......OOOO....
                         ............OO.....


The following results are based on a convoy of two LWSSs and a MWSS which
produce two copies of the spark produced by the second "boring" spaceship
shown previously.  The central MWSS in the convoy can also be replaced by
an OWSS which is two cells longer than the MWSS.  Hartmut found most of
these results in the longer forms first, but I am only showing the shorter
MWSS forms here.

The small tagalong below uses the sparks from the boring spaceship reaction,
plus the end of the middle MWSS.

[Period 4 small tagalong to standard spaceship convoy (speed c/2)]
 OOOO.........
 O...O........
 O............
 .O..O.....O..
 ....O...OO.O.
 .O...O..OO...
 O......O....O
 O....O.OOO..O
 OOOOO...O.O..
 .......OO.O..
 .........O...
 ..........O..
 OOOO.........
 O...O........
 O............
 .O..O........


A larger tagalong for the convoy which only requires the sparks is
the following.

[Period 4 tagalong for standard spaceship convoy (speed c/2)]
 .OO......OO....
 OO.OO...OOO....
 .OOOO..OOO...OO
 ..OO....O....OO
 .............O.
 ..OOO.....OO...
 .OOOOO...O..O..
 OO.OOO...O...O.
 .OO......O....O
 ...........O...
 .....OO..OOO...
 ........OOO.OO.
 .OO....OO.O...O
 OO.OO...OOO....
 .OOOO..........
 ..OO...........


The following larger tagalong starts similar to the one just shown, but
then diverges.  It has a HWSS-like back end.  In the same way that the
standard spaceships come in different sizes, this back end can also be
modified to have various sizes.  By adding extra spaceships as escorts the
back end can be made quite long.

[Period 4 tagalong with HWSS-like back end (speed c/2)]
 .OO......OO..................
 OO.OO...OOO..................
 .OOOO..OOO....O..............
 ..OO....O.....OO.............
 .................O.......OO..
 ..OOO.....OOOO....O...O.....O
 .OOOOO...O..OOO....OO........
 OO.OOO...O...O....OOO.......O
 .OO......O..O..OO.O..OOOOOOO.
 ...........OO..OO...OO.......
 .....OO..OO.OO...............
 ........OOOOOO...............
 .OO....OO.O....OO............
 OO.OO...OOO..................
 .OOOO........................
 ..OO.........................


By adding two accompanying LWSS to that back end, sparks can be produced
which are just like those produced by the front convoy.  This allows an
arbitrarily long spaceship to be constructed by repeatedly attaching the
tagalong to itself.  The following shows three copies of the tagalong.

[Period 4 spaceship with three copies of repeatable tagalong (speed c/2)]
 .................................................OO.....................O..O.
 .OO......OO.............OOOO....O.O.............OOOO...................O.....
 OO.OO...OOO.............O...O..O...............OO.OO...OOO.............O...O.
 .OOOO..OOO....O.........O......O..O...OO........OO....OO.O....OO.......OOOO..
 ..OO....O.....OO.........O..O..OOO....OO...............OOOOOO................
 .................O..........O......OOOO.............OO..OO.OO................
 ..OOO.....OOOO....O...O......O....OO..O...OO..............OO..OO...OO........
 .OOOOO...O..OOO....OO............O....O.....OO..........O..O..OO.O..OOOOOOOO.
 OO.OOO...O...O....OOO........O..OOO....O.OO...OOOOOOO...O...O....OOO........O
 .OO......O..O..OO.O..OOOOOOOO.....OOOOOOO.O...OOOOOOO...O..OOO....OO.........
 ...........OO..OO...OO...........O....OOOO..OO.OOOOO.....OOOO....O...O......O
 .....OO..OO.OO..................O.....O.........................O..........O.
 ........OOOOOO................O......OO..........OO....O.....OO.........O..O.
 .OO....OO.O....OO.......OOOO...O....O...........OOOO..OOO....O.........O.....
 OO.OO...OOO.............O...O..OO.O............OO.OO...OOO.............O...O.
 .OOOO...................O........O..............OO......OO.............OOOO..
 ..OO.....................O..O................................................


The following tagalongs are larger and closer to "space-dust".

[Several large period 4 tagalongs to standard spaceship convoy (speed c/2)]
 .OO......OO........OOOO........
 OO.OO...OOO........O...OO......
 .OOOO..OOO....O....O...........
 ..OO....O.....OO....O..OO...O.O
 .................O.......O..O.O
 ..OOO.....OOOO....O...O....O...
 .OOOOO...O..OOO....OO......OO..
 OO.OOO...O...O....OOO...O.OO...
 .OO......O..O..OO.O..OOO..O..O.
 ...........OO..OO...OO.....O..O
 .....OO..OO.OO..........OOOOOO.
 ........OOOOOO..........OO.....
 .OO....OO.O....OO..OOOOOO......
 OO.OO...OOO........O....O......
 .OOOO..............O....O......
 ..OO................O..........


 ..........................O..O......
 .OO......OO..............O....OO..OO
 OO.OO...OOO..............O..O.OO.O..
 .OOOO..OOO....O..........OO.OO.O....
 ..OO....O.....OO.....OO...O.O.......
 ...................OO...O..O.O...O.O
 ..OOO.....OOOO...O.OOOO..OO......O.O
 .OOOOO...O..OOO.........OO.O..O..O..
 OO.OOO...O...O....O.O.O...OOOOOO.OOO
 .OO......O..O..OO.O...O.OO.....OO...
 ...........OO..OO.O..O......OOO.OOOO
 .....OO..OO.OO.....O..O....O.OO.....
 ........OOOOOO......OO....OO.OO.O...
 .OO....OO.O....OO........O...OO.O...
 OO.OO...OOO..............O...O......
 .OOOO....................OOOO...O.O.
 ..OO............................O...


 ...............O.O..O.O......
 .OO......O....O..O..O.O......
 OO.OO...OOOOO.O...O...OO.....
 .OOOO..OO.OO.......OO..OOO...
 ..OO....O..O...O.OO.OO.OOOO.O
 ..........OO.O.OO.OO........O
 ..OOO.....O....O.OOO..OOOO...
 .OOOOO........OOO..OO.O......
 OO.OOO.......OOO..OOO.OOOOOO.
 .OO..........O....O..........
 ..........O...........OOO....
 .....OO..OO..O.OOO...O...OOOO
 ........OOOOO..OO....OOO..O..
 .OO....OO.....OO.OOOOO..O..OO
 OO.OO...O....OO.O....O.......
 .OOOO....O....OOOO.OO.O......
 ..OO...........O.O.O.........


The following spaceship has another back end which comes in various sizes.
The pair of LWSSs at the back are required in order to stabilize the tagalong.
The back pair of MWSSs are required for this particular sized back end, but
for smaller back ends, they are not required.  I selected this particular
sized spaceship to demonstrate an uncommon use of the MWSS to perturb some
objects.

[Period 4 tagalong to standard spaceship convoy with escorts (speed c/2)]
 ........................O..O.....O..
 .OO......OO............O.......O...O
 OO.OO...OOO............O...O..O.....
 .OOOO..OOO...OOO..O....OOOO...O....O
 ..OO....O....OOO.O..O.........OOOOO.
 .............O.O.OO.................
 ..OOO......O...OO.....O.............
 .OOOOO...OO.O...OO..OO.O............
 OO.OOO...O..O.O..O.O.OO.OOOOOOOO....
 .OO......O......OOOOO...OOOOOOOO....
 ..........OO...O.O....OOOOOOOOOOOO..
 .....OO..OOO...O.OO.O..........O...O
 ........OOO.OO.O..O.....O..O..O.....
 .OO....OO.O...OOOO.....O......O....O
 OO.OO...OOO....OO......O...O..OOOOO.
 .OOOO..................OOOO.........
 ..OO................................


Hartmut also found the following tagalong which requires two LWSS escorts.
This produces a nice spark at the back.

[Tagalong for standard spaceship convoy (speed c/2)]
 .OO......OO.............OOOO.......
 OO.OO...OOO.............O...O......
 .OOOO..OOO....O.........O..........
 ..OO....O.....OO.........O..O..O...
 .................O...........OOOO..
 ..OOO.....OOOO....O...O.........O..
 .OOOOO...O..OOO....OO...........O..
 OO.OOO...O...O....OOO..............
 .OO......O..O..OO.O..OOOOOOOOOO....
 ...........OO..OO...OO........OO.OO
 .....OO..OO.OO................O....
 ........OOOOOO................O.OO.
 .OO....OO.O....OO.......OOOO.......
 OO.OO...OOO.............O...O......
 .OOOO...................O..........
 ..OO.....................O..O......


When I saw this spaceship, I realized that the back part of the tagalong
could be extracted and slightly modified to run on its own by changing
one column.  This is shown below.

[Period 4 spaceship (speed c/2)]
 ..OOOO.......
 ..O...O......
 ..O..........
 ...O..O..O...
 .......OOOO..
 .O........O..
 O.........O..
 O............
 OOOOOOOOO....
 ........OO.OO
 ........O....
 ........O.OO.
 ..OOOO.......
 ..O...O......
 ..O..........
 ...O..O......


The same basic spaceship was later found by Hartmut's search program in a
smaller form, as shown below.

[Smaller period 4 spaceship (speed c/2)]
 OOOO.......
 O...O......
 O..........
 .O..O..O...
 .....OOOO..
 .O......O..
 O.......O..
 O..........
 OOOOOOO....
 ......OO.OO
 ......O....
 ......O.OO.
 OOOO.......
 O...O......
 O..........
 .O..O......
 ...........


Hartmut found the following tagalong for the above spaceship.  This tagalong
is unusual in that most of it is actually of period 2.  Most tagalong periods
are either the same or are multiples of the base spaceship period.  Having a
tagalong which is a sub-multiple is very rare.  By using this tagalong, many
of the period 2 results from a previous article can be attached to this
spaceship.

[Period 4 spaceship with period 2 compatible tagalong (speed c/2)]
 OOOO...............
 O...O...........O.O
 O..............O..O
 .O..O..O......OO...
 .....OOOO.....O.O..
 .O......O....O.O..O
 O.......O...OOO.OO.
 O...........O......
 OOOOOOO.....OOO.OO.
 ......OO.OO..O.O..O
 ......O.......O.O..
 ......O.OO....OO...
 OOOO...........O..O
 O...O...........O.O
 O..................
 .O..O..............


By modifying the end of the base spaceship, the following much larger
spaceship was found.

[Larger period 4 spaceship (speed c/2)]
 ...............O...OO.
 OOOO........OOO..OO...
 O...O......OOOO......O
 O.........O.....OO.O..
 .O..O..OOOO.OO.OO.O...
 .....OOO.O..O.OO......
 .O......O...O.OO......
 O.......O..OOO........
 O.......O..OO.........
 OOOOOOO....O...O......
 ......OO...O..........
 ......O..O.O.OO.......
 ......O.O..O.OO....O..
 OOOO......OO.OOO.O.O..
 O...O......OO..O......
 O............OO..OOO.O
 .O..O..........O...O.O

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Newsgroups: comp.theory.cell-automata